solution of stochastic process and modelling

7/23/2019 Solution of Stochastic Process and Modelling

http://slidepdf.com/reader/full/solution-of-stochastic-process-and-modelling 1/31

❙ ♦ ❝ ❤ ❛ ✐ ❝ ♦ ❝ ❡ ❡ ❛ ♥ ❞ ▼ ♦ ❞ ❡ ❧ ✐ ♥ ❣ ✭ ▼ ❆ ❚ ❍ ✻ ✻ ✻ ✵ ✮

❆ ✐ ❣ ♥ ♠ ❡ ♥ ✸

● ♦ ✉ ❛ ✈ ❙ ❛ ❤ ❛ ✭ ❘ ■ ◆ ✿ ✻ ✻ ✶ ✺ ✸ ✾ ✽ ✻ ✻ ✮

✶ ❆ ♣ ♣ ❧ ✐ ❝ ❛ ✐ ♦ ♥ ❛ ♥ ❞ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♦ ♥ ♦ ❜ ❧ ❡ ♠

✶ ✳ ✶ ❘ ❛ ✐ ♥ ❛ ▼ ❛ ❦ ♦ ✈ ▼ ❛ ③ ❡

✶ ✷ ✸ ✹

✺ ✻ ✼ ✽

✾ ✶ ✵ ✶ ✶ ✶ ✷

✶ ✸ ✶ ✹ ✶ ✺ ✶ ✻

❚ ❛ ❜ ❧ ❡ ✶ ✿ ❋ ✐ ❣ ✉ ❡ ❤ ♦ ✇ ✐ ♥ ❣ ❤ ❡ ♠ ❛ ③ ❡ ❛ ❧ ♦ ♥ ❣ ✇ ✐ ❤ ✐ ♥ ❞ ✐ ❝ ❡ ❡ ♣ ❡ ❡ ♥ ✐ ♥ ❣ ❤ ❡ ❛ ❡ ✳ ❘ ❛ ❤ ❜ ❡ ❛ ❢ ♦ ♠ ❙ ❛ ❡ ✲ ✶ ✻ ✱ ❤ ❡ ❤ ♦ ❝ ❦ ✐

❧ ♦ ❝ ❛ ❡ ❞ ✐ ♥ ❙ ❛ ❡ ✲ ✽ ❛ ♥ ❞ ❙ ❛ ❡ ✲ ✾ ✱ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✐ ❧ ♦ ❝ ❛ ❡ ❞ ✐ ♥ ❙ ❛ ❡ ✲ ✸ ❛ ♥ ❞ ❙ ❛ ♠ ❛ ❢ ♦ ♠ ❙ ❛ ❡ ✲ ✶ ✳

✭ ❛ ✮ ❚ ❤ ❡ ✇ ❛ ② ❘ ❛ ❤ ❜ ❡ ♠ ♦ ✈ ❡ ✐ ♥ ❤ ❡ ♠ ❛ ③ ❡ ✐ ❣ ✉ ✐ ❞ ❡ ❞ ❜ ② ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❤ ❡ ❡ ✉ ❧ ❡ ✿

• ❘ ✉ ❧ ❡ ✲ ✶ ✿ ■ ♥ ❡ ✈ ❡ ② ❡ ♣ ♦ ❝ ❤ ❤ ❡ ❡ ✐ ❛ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② prest = 0.1 ❤ ❛ ❘ ❛ ❤ ❜ ❡ ✇ ♦ ♥ ✬ ♠ ♦ ✈ ❡ ❜ ❡ ❝ ❛ ✉ ❡ ❤ ❡ ✴ ❤ ❡ ✐ ✐ ❡ ❞ ❛ ♥ ❞

✇ ❛ ♥ ♦ ❡ ✳

• ❘ ✉ ❧ ❡ ✲ ✷ ✿ ■ ♥ ❡ ✈ ❡ ② ❡ ♣ ♦ ❝ ❤ ✱ ❘ ❛ ❤ ❜ ❡ ❝ ❛ ♥ ♠ ♦ ✈ ❡ ♦ ♥ ❧ ② ♦ ♥ ❡ ❡ ♣ ❡ ✐ ❤ ❡ ✉ ♣ ♦ ❞ ♦ ✇ ♥ ♦ ❧ ❡ ❢ ♦ ✐ ❣ ❤ ♣ ♦ ✈ ✐ ❞ ❡ ❞ ❤ ❛ ❤ ❡ ❡ ✐

♥ ♦ ♦ ❜ ❛ ❝ ❧ ❡ ✐ ♥ ❤ ❛ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✳ ❉ ✐ ❛ ❣ ♦ ♥ ❛ ❧ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ✐ ◆ ❖ ❚ ❛ ❧ ❧ ♦ ✇ ❡ ❞ ✳

• ❘ ✉ ❧ ❡ ✲ ✸ ✿ ❙ ❛ ② ❤ ❛ ✐ ♥ ❛ ❣ ✐ ✈ ❡ ♥ ❛ ❡ ❘ ❛ ❤ ❜ ❡ ❝ ❛ ♥ ♠ ♦ ✈ ❡ ✐ ♥ m ∈ {1, 2, 3, 4} ♣ ♦ ✐ ❜ ❧ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✳ ❚ ❤ ❡ ♥ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ②

❤ ❛ ✐ ✇ ✐ ❧ ❧ ♠ ♦ ✈ ❡ ✐ ♥ ♦ ♥ ❡ ♦ ❢ ❤ ❡ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✐

1− prestm

= 0.9m

✱ ✐ ✳ ❡ ✳ ❘ ❛ ❤ ❜ ❡ ❤ ❛ ❡ ✉ ❛ ❧ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ♠ ♦ ✈ ❡ ✐ ♥ ❛ ♥ ②

❞ ✐ ❡ ❝ ✐ ♦ ♥ ✳

■ ▼ ❖ ❘ ❚ ❆ ◆ ❚ ✿ ❲ ✐ ❤ ♦ ✉ ❘ ✉ ❧ ❡ ✲ ✶ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ♣ ❡ ✐ ♦ ❞ ✐ ❝ ✇ ✐ ❤ ♣ ❡ ✐ ♦ ❞ 2✳ ❚ ❤ ✐ ♠ ❛ ② ❧ ❡ ❛ ❞ ♦ ❝ ♦ ♠ ♣ ❧ ✐ ❝ ❛ ✐ ♦ ♥ ✐ ♥

❝ ♦ ♠ ♣ ✉ ❛ ✐ ♦ ♥ ✳ ❍ ❡ ♥ ❝ ❡ ✇ ❡ ✐ ♠ ♣ ♦ ❡ ❞ ❘ ✉ ❧ ❡ ✲ ✶ ♦ ♠ ❛ ❦ ❡ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❛ ♣ ❡ ✐ ♦ ❞ ✐ ❝ ✳

❚ ❤ ❡ ❛ ❜ ♦ ✈ ❡ ✉ ❧ ❡ ❧ ❡ ❛ ❞ ♦ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① P ✳

✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶ ✵ ✶ ✶ ✶ ✷ ✶ ✸ ✶ ✹ ✶ ✺ ✶ ✻

✶ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✾ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✷ ✵ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✸ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✹ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✺ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✻ ✵ ✵ ✳ ✷ ✷ ✺ ✵ ✵ ✵ ✳ ✷ ✷ ✺ ✵ ✳ ✶ ✵ ✳ ✷ ✷ ✺ ✵ ✵ ✵ ✳ ✷ ✷ ✺ ✵ ✵ ✵ ✵ ✵ ✵

✼ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✽ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✾ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✾ ✵ ✵ ✵

✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✵

✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵

✶ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✵ ✵ ✵ ✳ ✾

✶ ✸ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵ ✵

✶ ✹ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶ ✵ ✳ ✹ ✺ ✵

✶ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✸ ✵ ✵ ✳ ✸ ✵ ✳ ✶ ✵ ✳ ✸

✶ ✻ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✵ ✵ ✳ ✹ ✺ ✵ ✳ ✶

❚ ❛ ❜ ❧ ❡ ✷ ✿ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① P ❣ ♦ ✈ ❡ ♥ ✐ ♥ ❣ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❘ ❛ ❤ ❜ ❡ ✐ ♥ ❤ ❡ ♠ ❛ ③ ❡ ✳

✶



▲ ❊ ❆ ❙ ❊ ◆ ❖ ❚ ❊ ✿

✶ ✮ ❚ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ❣ ♦ ✈ ❡ ♥ ❡ ❞ ❜ ② ❤ ❡ ❛ ❜ ♦ ✈ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① ✐ ✐ ❡ ❞ ✉ ❝ ✐ ❜ ❧ ❡ ❛ ♥ ❞ ❢ ♦ ♠ ❛ ❝ ❧ ♦ ❡ ❞ ❝ ♦ ♠ ♠ ✉ ♥ ✐ ❝ ❛ ✲

✐ ♦ ♥ ❝ ❧ ❛ ✳

✷ ✮ ❙ ✐ ♥ ❝ ❡ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ❢ ♦ ♠ ❛ ❝ ❧ ♦ ❡ ❞ ❝ ♦ ♠ ♠ ✉ ♥ ✐ ❝ ❛ ✐ ♦ ♥ ❝ ❧ ❛ ✱ ✇ ❡ ❝ ❛ ♥ ✉ ❡ ❤ ❡ ❝ ♦ ♥ ❝ ❡ ♣ ❢ ♦ ♠ ▲ ❡ ❝ ✉ ❡ ✲ ✶ ✷ ✭ 19th ❖ ❝ ♦ ❜ ❡ ✱

✷ ✵ ✶ ✺ ✮ ♦ ❛ ♥ ✇ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ✳ ❚ ♦ ❛ ♥ ✇ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ✇ ❡ ♠ ✐ ❣ ❤ ♥ ♦ ❣ ✐ ✈ ❡ ❛ ❞ ❡ ❛ ✐ ❧ ❡ ❞ ❡ ① ♣ ❧ ❛ ♥ ❛ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛

✉ ❡ ❞ ✳ ❘ ❛ ❤ ❡ ✱ ✇ ❡ ✇ ✐ ❧ ❧ ❥ ✉ ♠ ❛ ♣ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ♦ ✇ ❤ ❛ ✇ ❡ ❧ ❡ ❛ ♥ ❡ ❞ ✐ ♥ ▲ ❡ ❝ ✉ ❡ ✲ ✶ ✷ ❛ ♥ ❞ ❞ ✐ ❡ ❝ ❧ ② ✉ ❡ ❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛ ❢ ♦ ♠ ▲ ❡ ❝ ✉ ❡ ✲ ✶ ✷ ✳

✸ ✮ ❲ ❡ ♠ ✐ ❣ ❤ ❤ ❛ ✈ ❡ ❞ ♦ ♥ ❡ ♦ ♠ ❡ ♥ ♦ ❛ ✐ ♦ ♥ ♦ ✈ ❡ ❧ ♦ ❛ ❞ ✐ ♥ ❣

✶

♦ ❛ ♥ ✇ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ✳

✹ ✮ ❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❛ ❧ ❧ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❆ ✳

✭ ❜ ✮ ❚ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ❝ ❛ ♥ ❜ ❡ ❛ ❜ ❛ ❝ ❡ ❞ ❛ ✿ ❙ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ i✱ ✇ ❤ ❛ ✐ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❡ ♣ ♦ ❝ ❤ ✉ ♥ ✐ ❧ ❛ ❡ j ✐

❡ ❛ ❝ ❤ ❡ ❞ ❄ ■ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ i = 16 ❛ ♥ ❞ j = 3✳ ❚ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❛ ❞ ♦ ♣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ♣ ✿

❙ ❡ ♣ ✲ ✶ ✿ ❲ ❡ ♠ ❛ ❦ ❡ ❛ ❡ j = 3 ❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳ ❚ ♦ ❞ ♦ ❤ ✐ ✇ ❡ ♠ ❛ ❦ ❡ ❛ ♠ ♦ ❞ ✐ ❢ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①

P ❛ ❢ ♦ ❧ ❧ ♦ ✇ ✿ ✶ ✮

❙ ❡

P = P ✳ ✷ ✮ ❙ ❡ ❛ ❧ ❧ ❤ ❡ ❡ ❧ ❡ ♠ ❡ ♥ ♦ ❢ ❤ ❡ 3rd ♦ ✇ ♦ ❢

P ❛ 0 ❡ ① ❝ ❡ ♣ p33 ✇ ❤ ✐ ❝ ❤ ✐ ❡ ♦ 1 ✳

❙ ❡ ♣ ✲ ✷ ✿ ❲ ❡ ❝ ♦ ♥ ✉ ❝ ❛ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① Q ✭ ❛ ✐ ❝ ❧ ② ✉ ❜ ✲ ♦ ❝ ❤ ❛ ✐ ❝ ♠ ❛ ✐ ① ✮ ✇ ❤ ✐ ❝ ❤ ❝ ♦ ♥ ❛ ✐ ♥ ❤ ❡ ♦ ♥ ❡ ✲ ❡ ♣

❛ ♥ ✐ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❢ ♦ ♠ ♦ ♥ ❡ ❛ ♥ ✐ ❡ ♥ ❛ ❡ ♦ ❛ ♥ ♦ ❤ ❡ ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✳ ■ ♥ ❤ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①

P ✱ ❛ ❧ ❧ ❛ ❡ ❛ ❡ ❛ ♥ ✐ ❡ ♥ ❡ ① ❝ ❡ ♣ ❙ ❛ ❡ ✲ ✸ ✳ ❍ ❡ ♥ ❝ ❡ ✇ ❡ ❞ ❡ ❧ ❡ ❡ ❤ ❡

3rd

♦ ✇ ❛ ♥ ❞

3rd

❝ ♦ ❧ ✉ ♠ ♥ ❢ ♦ ♠ P ♦ ❣ ❡

Q✳

❙ ❡ ♣ ✲ ✸ ✿ ▲ ❡ ❛ ② ❤ ❛ ❤ ❡ ✐ ♠ ❡ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❡ ❛ ❝ ❤ ❡ ♣ ♦ ❝ ❤ ✐ 1 ✳ ❲ ❡ ❤ ❡ ❡ ❢ ♦ ❡ ❡ ❤ ❡ ❛ ❞ ❞ ✐ ✐ ✈ ❡ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ❧ f ≡ 1 ✳ ▼ ♦ ❡

♣ ❡ ❝ ✐ ❡ ❧ ② ✇ ❡ ❞ ❡ ✜ ♥ ❡ ❛ ❝ ♦ ❧ ✉ ♠ ♥ ✈ ❡ ❝ ♦ f ∈ R15❛ ♥ ❞ ❡ ❛ ❧ ❧ ✐ ❡ ❧ ❡ ♠ ❡ ♥ ❛ 1 ✳

❙ ❡ ♣ ✲ ✹ ✿ ❲ ❡ ❝ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ✈ ❡ ❝ ♦

µτ = (I − Q)−1

f

✇ ❤ ✐ ❝ ❤ ❤ ♦ ✇ ❤ ❡ ❛ ✈ ❡ ❛ ❣ ❡ ✐ ♠ ❡ ♦ ❡ ❛ ❝ ❤ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ♥ ② ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✳

❙ ❡ ♣ ✲ ✺ ✿ ❲ ❡ ❛ ❡ ✐ ♥ ❡ ❡ ❡ ❞ ✐ ♥ ❤ ❡ ❛ ✈ ❡ ❛ ❣ ❡ ✐ ♠ ❡ ♦ ❡ ❛ ❝ ❤ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❣ ✐ ✈ ❡ ♥ ❤ ❛ ❘ ❛ ❤ ❜ ❡ ❛ ❢ ♦ ♠ ❙ ❛ ❡ ✲ ✶ ✻ ✳ ❚ ❤ ✐ ✐ ❣ ✐ ✈ ❡ ♥

❜ ② µτ (15)✱ ❤ ❡ 15th ♦ ✇ ♦ ❢ µτ ✳

❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ✐ ❢ ♦ ✉ ♥ ❞ ♦ ❜ ❡ 85.1852 ✳

✭ ❝ ✮ ❚ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ❝ ❛ ♥ ❜ ❡ ❛ ❜ ❛ ❝ ❡ ❞ ❛ ✿ ❙ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ i✱ ✇ ❤ ❛ ✐ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❡ ♣ ♦ ❝ ❤ ♣ ❡ ♥ ✐ ♥ ❛ ❡ k ✭ k

❝ ❛ ♥ ❜ ❡ ❛ ✈ ❡ ❝ ♦ ❝ ♦ ♥ ✐ ✐ ♥ ❣ ♦ ❢ ♠ ❛ ♥ ② ❛ ❡ ✮ ✱ ❜ ❡ ❢ ♦ ❡ ❛ ❡ j ✐ ✈ ✐ ✐ ❡ ❞ ❄ ■ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ i = 16 ✱ k =

8 9

❛ ♥ ❞ j = 3 ✳ ❚ ♦

❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❛ ❞ ♦ ♣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ♣ ✿

❙ ❡ ♣ ✲ ✶ ✿ ❲ ❡ ♠ ❛ ❦ ❡ ❛ ❡ j = 3 ❛ ❜ ♦ ❜ ✐ ♥ ❣ ❧ ✐ ❦ ❡ ✇ ❡ ❞ ✐ ❞ ✐ ♥ ❛ ✭ ❜ ✮ ✳

❙ ❡ ♣ ✲ ✷ ✿ ❲ ❡ ❝ ♦ ♥ ✉ ❝ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① Q ❧ ✐ ❦ ❡ ✇ ❡ ❞ ✐ ❞ ✐ ♥ ❛ ✭ ❜ ✮ ✳

❙ ❡ ♣ ✲ ✸ ✿ ❉ ❡ ✜ ♥ ❡ ❤ ❡ ❛ ❞ ❞ ✐ ✐ ✈ ❡ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ❧ ❛ f ≡ δ km ✇ ❤ ❡ ❡ ❤ ❡ ❞ ✐ ❛ ❝ ❞ ❡ ❧ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ δ km ❢ ♦ ❛ ✈ ❡ ❝ ♦ k =

k1 k2 · · · kn

✐ ❞ ❡ ✜ ♥ ❡ ❞ ❛

δ km =1 ; m = k1 ♦ m = k2 · · · ♦ m = kn

0 ; ♦ ✳ ✇ ✳

▼ ♦ ❡ ♣ ❡ ❝ ✐ ❡ ❧ ② f ∈ R15✐ ❛ ❝ ♦ ❧ ✉ ♠ ♥ ✈ ❡ ❝ ♦ ✇ ❤ ♦ ❡ 7th ❛ ♥ ❞ 8th ♦ ✇ ✭ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❤ ❡ 8th ❛ ♥ ❞ ❤ ❡ 9th ❛ ❡ ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✮

✐ ❡ ♦ 1 ❛ ♥ ❞ ❛ ❧ ❧ ♦ ❤ ❡ ❡ ❧ ❡ ♠ ❡ ♥ ✐ ❡ ♦ 0 ✳

❙ ❡ ♣ ✲ ✹ ✿ ❲ ❡ ❝ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ✈ ❡ ❝ ♦

µshock = (I − Q)−1

f

✶

■ ✐ ❥ ✉ ❛ ❡ ♠ ✇ ❡ ❝ ♦ ✐ ♥ ❡ ❞ ✇ ❤ ✐ ❝ ❤ ✐ ♠ ♣ ❧ ✐ ❡ ❤ ❡ ✉ ❡ ♦ ❢ ❛ ♠ ❡ ♠ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ❧ ♥ ♦ ❛ ✐ ♦ ♥ ✐ ♥ ♠ ✉ ❧ ✐ ♣ ❧ ❡ ❝ ♦ ♥ ❡ ① ❤ ❛ ✈ ✐ ♥ ❣ ❞ ✐ ✛ ❡ ❡ ♥ ♠ ❡ ❛ ♥ ✐ ♥ ❣ ✦ ✦ ✦

✷



✇ ❤ ✐ ❝ ❤ ❤ ♦ ✇ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❘ ❛ ❜ ❡ ❡ ❝ ❡ ✐ ✈ ❡ ❜ ❡ ❢ ♦ ❡ ❡ ❛ ❝ ❤ ✐ ♥ ❣ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✱ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ♥ ② ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✳

❙ ❡ ♣ ✲ ✺ ✿ ❲ ❡ ❛ ❡ ✐ ♥ ❡ ❡ ❡ ❞ ✐ ♥ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❣ ✐ ✈ ❡ ♥ ❤ ❛ ❘ ❛ ❤ ❜ ❡ ❛ ❢ ♦ ♠ ❙ ❛ ❡ ✲ ✶ ✻ ✳ ❚ ❤ ✐ ✐ ❣ ✐ ✈ ❡ ♥ ❜ ②

µshock (15)✱ ❤ ❡ 15th ♦ ✇ ♦ ❢ µshock ✳

❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❘ ❛ ❜ ❡ ❡ ❝ ❡ ✐ ✈ ❡ ❜ ❡ ❢ ♦ ❡ ❡ ❛ ❝ ❤ ✐ ♥ ❣ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✐ 6.2963✳

✭ ❞ ✮ ❚ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ❝ ❛ ♥ ❜ ❡ ❛ ❜ ❛ ❝ ❡ ❞ ❛ ✿ ❙ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ i ✱ ✇ ❤ ❛ ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❛ ❡ j ✐ ✈ ✐ ✐ ❡ ❞ ❜ ❡ ❢ ♦ ❡ ❛ ❡ ❦

✭ ❝ ❛ ♥ ❜ ❡ ❛ ✈ ❡ ❝ ♦ ✮ ❄ ■ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ i = 16 ✱ j = 3 ❛ ♥ ❞ k = 8 9✳ ❚ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❛ ❞ ♦ ♣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ♣ ✿

❙ ❡ ♣ ✲ ✶ ✿ ❲ ❡ ♠ ❛ ❦ ❡ ❛ ❡ j = 3 ❛ ♥ ❞ k =

8 9

❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳ ❚ ♦ ❞ ♦ ❤ ✐ ✇ ❡ ♠ ❛ ❦ ❡ ❛ ♠ ♦ ❞ ✐ ❢ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①

P

❛ ❢ ♦ ❧ ❧ ♦ ✇ ✿ ✶ ✮ ❙ ❡

P = P ✳ ✷ ✮ ❙ ❡ ❛ ❧ ❧ ❡ ❧ ❡ ♠ ❡ ♥ ♦ ❢ ❤ ❡ 3rd ✱ 8th ❛ ♥ ❞ 9th ♦ ✇ ♦ ❢

P ❛ 0 ✳ ✸ ✮ ❙ ❡ p33 = p88 = p99 = 1 ✳

❙ ❡ ♣ ✲ ✷ ✿ ❲ ❡ ❝ ♦ ♥ ✉ ❝ ❛ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① Q ✇ ❤ ✐ ❝ ❤ ❝ ♦ ♥ ❛ ✐ ♥ ❤ ❡ ♦ ♥ ❡ ✲ ❡ ♣ ❛ ♥ ✐ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❢ ♦ ♠ ♦ ♥ ❡ ❛ ♥ ✲

✐ ❡ ♥ ❛ ❡ ♦ ❛ ♥ ♦ ❤ ❡ ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✳ Q ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠

P ❜ ② ❞ ❡ ❧ ❡ ✐ ♥ ❣ ✐ 3rd ✱ 8th ❛ ♥ ❞ 9th ♦ ✇ ❛ ♥ ❞ 3rd ✱ 8th ❛ ♥ ❞ 9th ❝ ♦ ❧ ✉ ♠ ♥ ✳

❙ ❡ ♣ ✲ ✸ ✿ ❲ ❡ ❝ ♦ ♥ ✉ ❝ ❛ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① R ✇ ❤ ✐ ❝ ❤ ❝ ♦ ♥ ❛ ✐ ♥ ❤ ❡ ♦ ♥ ❡ ✲ ❡ ♣ ❛ ♥ ✐ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❢ ♦ ♠ ❛ ❛ ♥ ✲

✐ ❡ ♥ ❛ ❡ ♦ ❛ ❡ ❝ ✉ ❡ ♥ ❛ ❡ ✳ Q ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠

P ❜ ② ❞ ❡ ❧ ❡ ✐ ♥ ❣ ✐ 3rd ✱ 8th ❛ ♥ ❞ 9th ♦ ✇ ❛ ♥ ❞ ❞ ❡ ❧ ❡ ✐ ♥ ❣ ❛ ❧ ❧ ❝ ♦ ❧ ✉ ♠ ♥ ❡ ① ❝ ❡ ♣

❤ ❡ 3rd ✱ 8th ❛ ♥ ❞ 9th ✳

❙ ❡ ♣ ✲ ✹ ✿ ❲ ❡ ❝ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ♠ ❛ ✐ ①

U = (I − Q)−1

R

✇ ❤ ♦ ❡ U ij ❡ ❧ ❡ ♠ ❡ ♥ ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❛ ♥ ✐ ❡ ♥ ❛ ❡ i✱ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✇ ✐ ❧ ❧ ❣ ❡ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❡ ❝ ✉ ❡ ♥

❛ ❡ j ✳

❙ ❡ ♣ ✲ ✺ ✿ ❚ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❘ ❛ ❤ ❜ ❡ ❡ ❛ ❝ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✇ ✐ ❤ ♦ ✉ ❣ ❡ ✐ ♥ ❣ ❤ ♦ ❝ ❦ ❡ ❞ ✐ ❣ ✐ ✈ ❡ ♥ ❜ ② ❤ ❡ (13, 1)th

❡ ❧ ❡ ♠ ❡ ♥ ♦ ❢ U ✳ ■ ♥

❤ ❡ U ♠ ❛ ✐ ① ♦ ✇ 13 ❝ ♦ ❡ ♣ ♦ ♥ ❞ ♦ ❙ ❛ ❡ ✲ ✶ ✻ ❛ ♥ ❞ ❝ ♦ ❧ ✉ ♠ ♥ 1 ❝ ♦ ❡ ♣ ♦ ♥ ❞ ♦ ❙ ❛ ❡ ✲ ✸ ✳

❚ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❘ ❛ ❜ ❡ ✜ ♥ ❞ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❜ ❡ ❢ ♦ ❡ ❣ ❡ ✐ ♥ ❣ ❤ ♦ ❝ ❦ ❡ ❞ ✐ ❢ ♦ ✉ ♥ ❞ ♦ ❜ ❡ 0.2143 ✳

✭ ❡ ✮ ❲ ❡ ✇ ✐ ❧ ❧ ❞ ✐ ✈ ✐ ❞ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✐ ♥ ✇ ♦ ♣ ❛ ✿ ✐ ✮ ❋ ♦ ♠ ✉ ❧ ❛ ✐ ♥ ❣ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ▼ ♦ ❞ ❡ ❧ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✳ ✐ ✐ ✮

❈ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♥ ❣ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ✇ ❛ ✐ ✉ ♥ ✐ ❧ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ✜ ♥ ❞ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✳

❋ ♦ ♠ ✉ ❧ ❛ ✐ ♥ ❣ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ▼ ♦ ❞ ❡ ❧ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✿

❚ ❤ ✐ ❡ ♣ ❝ ❛ ♥ ❜ ❡ ❢ ✉ ❤ ❡ ✉ ❜ ✲ ❞ ✐ ✈ ✐ ❞ ❡ ❞ ✐ ♥ ♦ ✇ ♦ ❡ ♣ ✿ ■ ✮ ▼ ♦ ❞ ❡ ❧ ❧ ✐ ♥ ❣ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠ ❛ ❧ ♦ ♥ ❡ ✉ ✐ ♥ ❣ ❛ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥

♠ ♦ ❞ ❡ ❧ ✳ ■ ■ ✮ ▼ ♦ ❞ ❡ ❧ ❧ ✐ ♥ ❣ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✉ ✐ ♥ ❣ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ♠ ♦ ❞ ❡ ❧ ✳

❲ ❡ ✇ ✐ ❧ ❧ ✜ ♠ ♦ ❞ ❡ ❧ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠ ❛ ❧ ♦ ♥ ❡ ✳ ▼ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠ ✐ ♥ ❤ ❡ ♠ ❛ ③ ❡ ✐ ❣ ✉ ✐ ❞ ❡ ❞ ❜ ② ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❤ ❡ ❡ ✉ ❧ ❡ ✿

• ❘ ✉ ❧ ❡ ✲ ✶ ✿ ■ ♥ ❡ ✈ ❡ ② ❡ ♣ ♦ ❝ ❤ ❤ ❡ ❡ ✐ ❛ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ 0.2 ❤ ❛ ❙ ❛ ♠ ✇ ♦ ♥ ✬ ♠ ♦ ✈ ❡ ❜ ❡ ❝ ❛ ✉ ❡ ❤ ❡ ✴ ❤ ❡ ✐ ✐ ❡ ❞ ❛ ♥ ❞ ✇ ❛ ♥ ♦ ❡ ✳

• ❘ ✉ ❧ ❡ ✲ ✷ ✿ ■ ♥ ❡ ✈ ❡ ② ❡ ♣ ♦ ❝ ❤ ✱ ❙ ❛ ♠ ❝ ❛ ♥ ♠ ♦ ✈ ❡ ♦ ♥ ❧ ② ♦ ♥ ❡ ❡ ♣ ❡ ✐ ❤ ❡ ✉ ♣ ♦ ❞ ♦ ✇ ♥ ♦ ❧ ❡ ❢ ♦ ✐ ❣ ❤ ✳ ❉ ✐ ❛ ❣ ♦ ♥ ❛ ❧ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ✐ ◆ ❖ ❚

❛ ❧ ❧ ♦ ✇ ❡ ❞ ✳

• ❘ ✉ ❧ ❡ ✲ ✸ ✿ ❙ ✐ ♥ ❝ ❡ ❙ ❛ ♠ ✐ ❜ ❧ ✐ ♥ ❞ ✱ ✐ ❝ ❛ ♥ ② ♦ ♠ ♦ ✈ ❡ ✐ ♥ ❛ ♥ ② ♦ ♥ ❡ ♦ ❢ ❤ ❡ ❢ ♦ ✉ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✇ ✐ ❤ ❡ ✉ ❛ ❧ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ✳ ■ ❢ ❛ ❛ ❣ ✐ ✈ ❡ ♥

❡ ♣ ♦ ❝ ❤ ❙ ❛ ♠ ✐ ❡ ♦ ♠ ♦ ✈ ❡ ✐ ♥ ❤ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✇ ❤ ✐ ❝ ❤ ❤ ❛ ❛ ♥ ♦ ❜ ❛ ❝ ❧ ❡ ✱ ❤ ❡ ♥ ✐ ✇ ✐ ❧ ❧ ❣ ❡ ❜ ♦ ✉ ♥ ❝ ❡ ❞ ❜ ❛ ❝ ❦ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ ✇ ✐ ❧ ❧ ❡ ♠ ❛ ✐ ♥

✐ ♥ ❤ ❡ ❛ ♠ ❡ ❛ ❡ ✳

❙ ❛ ② ❤ ❛ ✐ ♥ ❛ ❣ ✐ ✈ ❡ ♥ ❛ ❡ ✱ ❙ ❛ ♠ ❝ ❛ ♥ ♠ ♦ ✈ ❡ ✐ ♥ m ∈ {1, 2, 3, 4} ♣ ♦ ✐ ❜ ❧ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✳ ❚ ❤ ❡ ♥ ✉ ♥ ❞ ❡ ❤ ❡ ✉ ❧ ❡ ❞ ❡ ✜ ♥ ❡ ❞ ❛ ❜ ♦ ✈ ❡ ✱ ❤ ❡

♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❙ ❛ ♠ ❡ ♠ ❛ ✐ ♥ ✐ ♥ ❤ ❡ ❛ ♠ ❡ ❛ ❡ ✐ 0.2 + (1−0.2)4 (4 − m) = 1 − 0.2m ❛ ♥ ❞ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❙ ❛ ♠ ♠ ♦ ✈ ❡

✐ ♥ ♦ ♥ ❡ ♦ ❢ ❤ ❡ m ♣ ♦ ✐ ❜ ❧ ❡ ❞ ✐ ❡ ❝ ✐ ♦ ♥ ✐

(1−0.2)4 = 0.2 ✳ ❚ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❛ ♥ ✐ ✐ ♦ ♥ ♠ ❛ ✐ ① W ❣ ♦ ✈ ❡ ♥ ✐ ♥ ❣ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠

✐ ❤ ♦ ✇ ♥ ❜ ❡ ❧ ♦ ✇ ✳

✸



✶ ✷ ✸ ✹ ✺ ✻ ✼ ✽ ✾ ✶ ✵ ✶ ✶ ✶ ✷ ✶ ✸ ✶ ✹ ✶ ✺ ✶ ✻

✶ ✵ ✳ ✽ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✷ ✵ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✸ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✹ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✺ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✻ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✷ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵

✼ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✽ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵

✾ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✽ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵

✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✵

✶ ✶ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✵ ✵ ✵ ✳ ✷ ✵

✶ ✷ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✽ ✵ ✵ ✵ ✵ ✳ ✷

✶ ✸ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✵ ✳ ✻ ✵ ✳ ✷ ✵ ✵

✶ ✹ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻ ✵ ✳ ✷ ✵

✶ ✺ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✳ ✷ ✵ ✳ ✹ ✵ ✳ ✷

✶ ✻ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✵ ✳ ✷ ✵ ✵ ✵ ✳ ✷ ✵ ✳ ✻

❚ ❛ ❜ ❧ ❡ ✸ ✿ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① W ❣ ♦ ✈ ❡ ♥ ✐ ♥ ❣ ❤ ❡ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❙ ❛ ♠ ✐ ♥ ❤ ❡ ♠ ❛ ③ ❡ ✳

◆ ♦ ✇ ❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ♦ ❢ ❘ ❛ ❜ ❡ ❛ ♥ ❞ ❙ ❛ ♠ ✳ ❋ ♦ ❡ ❛ ❝ ❤ ♦ ❢ ❤ ❡ 16 ❛ ❡ ♦ ❢ ❘ ❛ ❜ ❡ ✱ ❙ ❛ ♠ ❝ ❛ ♥ ❜ ❡ ✐ ♥ 16 ♣ ♦ ✐ ❜ ❧ ❡

❛ ❡ ✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ✉ ✐ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❡ ✐ ♥ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ♦ ❝ ❛ ♣ ✉ ❡ ❤ ✐ ♠ ♦ ❞ ❡ ❧ ✐

16 × 16 = 256✳ ❲ ❡ ❤ ❛ ✈ ❡ ♦ ✜

❣ ✐ ✈ ❡ ♣ ❤ ② ✐ ❝ ❛ ❧ ♠ ❡ ❛ ♥ ✐ ♥ ❣ ♦ ❤ ❡ ❡ ❛ ❡ ✳ ■ ❢ ❘ ❛ ❜ ❡ ✐ ✐ ♥ ❛ ❡ sr ∈ {1, 2, . . . , 16} ❛ ♥ ❞ ❙ ❛ ♠ ✐ ✐ ♥ ❛ ❡ ss ∈ {1, 2, . . . , 16}✱ ❤ ❡ ♥

✐ ♥ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✐

i = 16 (sr − 1) + (ss − 1) ✭ ✶ ✮

◆ ♦ ❡ ❤ ❛ i ∈ {0, 1, 2, . . . , 255}✳ ❲ ❡ ✇ ✐ ❧ ❧ ♥ ♦ ✇ ❝ ♦ ♥ ❝ ❡ ♥ ❛ ❡ ♦ ♥ ❞ ❡ ✐ ❣ ♥ ✐ ♥ ❣ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① M ♦ ❢ ❤ ❡

❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✳ ❆ ✉ ♠ ✐ ♥ ❣ ❤ ❛ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ✬ ♠ ♦ ✈ ❡ ♠ ❡ ♥ ✐ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✱ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ②

❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① M ✐ ❣ ✐ ✈ ❡ ♥ ❜ ②

M ij = P sirs

jr

W siss

js

✭ ✷ ✮

✇ ❤ ❡ ❡ sir ❛ ♥ ❞ sjr ✭ sis ❛ ♥ ❞ sjs ✮ ✐ ❤ ❡ ❛ ❡ ♦ ❢ ❘ ❛ ❜ ❡ ✭ ❙ ❛ ♠ ✮ ✇ ❤ ❡ ♥ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✐ i ❛ ♥ ❞ j ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✳

▼ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ❧ ❧ ② ✱

sir = ✢ ♦ ♦

i

16

+ 1 ✭ ✸ ✮

sis = i − 16

sir − 1

+ 1 ✭ ✹ ✮

sjr = ✢ ♦ ♦

j

16

+ 1 ✭ ✺ ✮

sjs = j − 16

sjr − 1

+ 1 ✭ ✻ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✷ ❛ ❧ ♦ ♥ ❣ ✇ ✐ ❤ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✱ ✹ ✱ ✺ ❛ ♥ ❞ ✻ ❝ ♦ ♠ ♣ ❧ ❡ ❡ ❧ ② ❝ ❛ ♣ ✉ ❡ ❤ ❡ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ♠ ♦ ❞ ❡ ❧ ♦ ❢ ❤ ❡ ❘ ❛ ❜ ❡ ✲ ❙ ❛ ♠ ② ❡ ♠ ✳

❈ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♥ ❣ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ✇ ❛ ✐ ✉ ♥ ✐ ❧ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ✜ ♥ ❞ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✿

❋ ✐ ♦ ❢ ❛ ❧ ❧ ❧ ❡ ✜ ♥ ❞ ❤ ❡ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❡ ♦ ❢ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✳ ❲ ❡ ❦ ♥ ♦ ✇ ❤ ❡ ❢ ♦ n = 0 ✱ sr = 16 ❛ ♥ ❞ ss = 1 ✳ ❲ ❤ ❡ ♥

✉ ❜ ✐ ✉ ❡ ❞ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✇ ❡ ✜ ♥ ❞ ❤ ❛ ❤ ❡ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❡ ♦ ❢ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ✐ i = 240 ✳ ❖ ✉ ♦ ❢ ❤ ❡ 256 ❛ ❡

❤ ❡ ❡ ❛ ❡ ♦ ♥ ❧ ② 16 ❛ ❡ ✇ ❤ ✐ ❝ ❤ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ♦ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ♠ ❡ ❡ ✐ ♥ ❣ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✳ ❚ ❤ ❡ ❡ ❛ ❡ ❛ ❡ ❝ ❤ ❛ ❛ ❝ ❡ ✐ ❡ ❞ ❜ ②

sr = ss ✳ ❯ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✱ ✇ ❡ ❝ ❛ ♥ ❛ ② ❤ ❛ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ♠ ❡ ❡ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✐ ❢ ❛ ❡ j ♦ ❢ ❤ ❡ ❝ ♦ ♠ ❜ ✐ ♥ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥

❜ ❡ ❧ ♦ ♥ ❣ ♦ ❤ ❡ ❡ C = {0, 17, 34, . . . , 255}✳

■ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❛ ❡ ✐ ♥ ❡ ❡ ❡ ❞ ✐ ♥ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ❡ ❛ ❝ ❤ ❛ ♥ ② ♦ ♥ ❡ ♦ ❢ ❤ ❡ ❛ ❡ ✐ ♥ C ✳ ❚ ♦ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ✐ ✇ ❡ ❝ ❛ ♥ ♠ ❛ ❦ ❡

❡ ❛ ❝ ❤ ❛ ❡ ✐ ♥ C ❛ ❛ ❝ ❧ ♦ ❡ ❞ ✭ ❡ ❝ ✉ ❡ ♥ ✮ ❝ ♦ ♠ ♠ ✉ ♥ ✐ ❝ ❛ ✐ ♦ ♥ ❝ ❧ ❛ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♣ ❡ ♥ ✐ ♥ ❤ ❡ ❛ ♥ ✐ ❡ ♥

❛ ❡ ✳ ❲ ❡ ❝ ❛ ♥ ✉ ♠ ♠ ❛ ✐ ③ ❡ ❤ ❡ ❡ ✉ ✐ ❡ ❞ ❡ ♣ ❛ ❢ ♦ ❧ ❧ ♦ ✇ ✿

❙ ❡ ♣ ✲ ✶ ✿ ▼ ❛ ❦ ❡ ❛ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①

M ✉ ❝ ❤ ❤ ❛ ❛ ❧ ❧ ❤ ❡ ❛ ❡ ✐ ♥ ❡ C ❛ ❡ ❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳

✹



❙ ❡ ♣ ✲ ✷ ✿ ❡ ❢ ♦ ♠ ❤ ❡ ❝ ❛ ♥ ♦ ♥ ✐ ❝ ❛ ❧ ❞ ❡ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢

M ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❤ ❡ Q ♠ ❛ ✐ ① ✳ ● ✐ ✈ ❡ ♥ ❤ ❛ ❤ ❡ ❡ ❛ ❡ 16 ❛ ❜ ♦ ❜ ✐ ♥ ❣ ❛ ❡ ✱

❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ♥ ✐ ❡ ♥ ❛ ❡ ✐ 256 − 16 = 240 ✳ ❍ ❡ ♥ ❝ ❡ Q ∈ R240×240

✳

❙ ❡ ♣ ✲ ✸ ✿ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❡ µτ = (I − Q)−1

f ✇ ❤ ❡ ❡ f ∈ R240×1✐ ❛ ❝ ♦ ❧ ✉ ♠ ♥ ✈ ❡ ❝ ♦ ✇ ✐ ❤ ❛ ❧ ❧ ❡ ♥ ✐ ❡ ❛ 1 ✳

❙ ❡ ♣ ✲ ✹ ✿ ■ ♥ ❤ ❡ ❝ ♦ ❧ ✉ ♠ ♥ ✈ ❡ ❝ ♦ µτ ✱ ❤ ❡ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❡ ♦ ❢ ❤ ❡ ② ❡ ♠ i = 240 ✱ ✐ ❛ ♦ ❝ ✐ ❛ ❡ ❞ ✇ ✐ ❤ ♦ ✇ ♥ ✉ ♠ ❜ ❡ 226✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡

❡ ✉ ✐ ❡ ❞ ❛ ♥ ✇ ❡ ✐ µτ (226)✳

❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ✇ ❛ ✐ ✉ ♥ ✐ ❧ ❙ ❛ ♠ ❛ ♥ ❞ ❘ ❛ ❜ ❡ ✜ ♥ ❞ ❡ ❛ ❝ ❤ ♦ ❤ ❡ ✐ ❡ ✉ ❛ ❧ ♦ 54.4488 ✳

✭ ❢ ✮ ■ ♥ ❛ ✭ ❜ ✮ ✱ ✇ ❡ ❢ ♦ ✉ ♥ ❞ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ❢ ♦ ❘ ❛ ❜ ❡ ♦ ✜ ♥ ❞ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❣ ✐ ✈ ❡ ♥ ❤ ❛ ✐ ❛ ❛ ❙ ❛ ❡ ✶ ✻ ✳ ❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞

✐ ♠ ❡ ✐ 85.1852✳ ❆ ❢ ❡ ❡ ❛ ✐ ♥ ❣ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❢ ♦ ❤ ❡ 1st ✐ ♠ ❡ ✱ ❘ ❛ ❜ ❡ ❤ ❛ ♦ ❡ ❛ ❝ ❤ ❡ ❡ ❡ C − 1 ✐ ♠ ❡ ❜ ❡ ❢ ♦ ❡ ✐ ❣ ♦ ❡ ♦ ❡ ✳ ❚ ❤ ❡

❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ♦ ✜ ♥ ❞ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ❢ ♦ ❤ ❡ ❧ ❛ C − 1 ✐ ♠ ❡ ❛ ❡ ❡ ✉ ❛ ❧ ✳ ▲ ❡ ❤ ✐ ✐ ♠ ❡ ❜ ❡ τ ✳ ❖ ♥ ❡ ✐ ❝ ❦ ♦ ✜ ♥ ❞ τ ✐ ♦ ♦ ❜ ❡ ✈ ❡

❤ ❛ ✐ ✐ ❤ ❡ ♠ ❡ ❛ ♥ ✜ ❡ ✉ ♥ ✐ ♠ ❡ ♦ ❢ ❛ ❡ ✲ ✸ ✳ ● ✐ ✈ ❡ ♥ ❤ ❛ ❤ ✐ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ✐ ♥ ✐ ❡ ❞ ✉ ❝ ✐ ❜ ❧ ❡ ✱ ✐ ❤ ❛ ❛ ✜ ♥ ✐ ❡ ♠ ❡ ❛ ♥ ❡ ✉ ♥

✐ ♠ ❡ ✳ τ ✐ ✐ ♥ ❞ ❡ ❡ ❞ ❣ ✐ ✈ ❡ ♥ ❜ ② ❛ ✈ ❡ ② ✐ ♠ ♣ ❧ ❡ ❢ ♦ ♠ ✉ ❧ ❛

τ = 1

π3

✇ ❤ ❡ ❡ π3 ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❛ ❡ ✸ ♦ ❢ ❤ ❡ ❛ ♦ ❝ ✐ ❛ ❡ ❞ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ π ✳ ❲ ❡ ❝ ❛ ♥ ✜ ♥ ❞ π ✉ ✐ ♥ ❣ ❤ ❡

❢ ♦ ♠ ✉ ❧ ❛

π = 1T (I − P + ONES )

−1

❯ ✐ ♥ ❣ ▼ ❆ ❚ ▲ ❆ ❇ ✇ ❡ ❢ ♦ ✉ ♥ ❞ ❤ ❛ τ = 16 ✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ✐ ♠ ❡ ❜ ❡ ❢ ♦ ❡ ❘ ❛ ❜ ❡ ❝ ❛ ♥ ❛ ❦ ❡ ❡ ✐ 85.1852 + 16 (C − 1) ✳

✭ ❣ ✮ ❚ ❤ ❡ ❡ ♣ ♦ ♦ ❧ ✈ ❡ ❤ ✐ ♣ ♦ ❜ ❧ ❡ ♠ ❛ ❡ ❛ ❢ ♦ ❧ ❧ ♦ ✇ ✿

❙ ❡ ♣ ✲ ✶ ✿ ▼ ❛ ❦ ❡ ❛ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ①

P ✐ ♥ ✇ ❤ ✐ ❝ ❤ ❛ ❡ ✸ ✭ ❝ ❤ ❡ ❡ ❡ ❛ ❡ ✮ ✱ ✽ ❛ ♥ ❞ ✾ ✭ ❤ ♦ ❝ ❦ ❛ ❡ ✮ ❛ ❡

❛ ❜ ♦ ❜ ✐ ♥ ❣ ❛ ❡ ✳

❙ ❡ ♣ ✲ ✷ ✿ ❈ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❣ ❡ ✐ ♥ ❣ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✸ ✱ ✽ ♦ ✾ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ ✶ ✻ ✳ ❚ ❤ ✐ ❝ ❛ ♥ ❜ ❡ ❞ ♦ ♥ ❡ ❜ ② ✉ ✐ ♥ ❣

❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛ U = (I − Q)−1

R ✇ ❤ ❡ ❡ Q ❛ ♥ ❞ R ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ② ❝ ❛ ♥ ♦ ♥ ✐ ❝ ❛ ❧ ❞ ❡ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢

P ✳ ▲ ❡ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❣ ❡ ✐ ♥ ❣

❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✸ ✱ ✽ ❛ ♥ ❞ ✾ ❜ ❡ p3 ✱ p8 ❛ ♥ ❞ p9 ✳ ❖ ❜ ❡ ✈ ❡ ❤ ❛ p3 + p8 + p9 = 1 ✳

❙ ❡ ♣ ✲ ✸ ✿ ■ ❢ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❣ ❡ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✸ ❤ ❡ ♥ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ✐ 0 ✳ ❚ ❤ ✐ ❤ ❛ ♣ ♣ ❡ ♥

✇ ✐ ❤ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② p3 ✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❤ ✐ ❝ ❛ ❡ ✐ p3 · 0✳

❙ ❡ ♣ ✲ ✹ ✿ ❙ ❛ ② ❤ ❛ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❣ ❡ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✽ ✳ ❚ ❤ ✐ ❤ ❛ ♣ ♣ ❡ ♥ ✇ ✐ ❤ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② p8 ✳ ❚ ♦

❤ ✐ ❡ ♥ ❞ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❡ ❝ ❡ ✐ ✈ ❡ ❞ ✐ 1 ❛ ♥ ❞ ❤ ❡ ❤ ♦ ❝ ❦ ❣ ❡ ♥ ❡ ❛ ♦ ✐ ♥ ❛ ❡ ✽ ✇ ✐ ❧ ❧ ❜ ❡ ✉ ♥ ❡ ❞ ♦ ✛ ✳ ◆ ♦ ✇ ✇ ❡ ❛ ❦ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣

✉ ❜ ❡ ♣ ✿

• ❲ ❡ ♠ ❛ ❦ ❡ ❛ ❡ ✽ ❛ ♥ ❞ 9 ✉ ♥ ❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳ ❚ ❤ ✐ ✐ ❞ ♦ ♥ ❡ ❜ ② ❡ ✐ ♥ ❣ ❤ ❡ 8th ❛ ♥ ❞ ❤ ❡ 9th ♦ ✇ ♦ ❢

P ❡ ✉ ❛ ❧ ♦ ❤ ❡ 8th ❛ ♥ ❞ ❤ ❡

9th ♦ ✇ ♦ ❢ P ✳ ❲ ❡ ✐ ❧ ❧ ❦ ❡ ❡ ♣ ❛ ❡ ✸ ❛ ❜ ♦ ❜ ✐ ♥ ❣ ✳

• ◆ ♦ ✇ ✇ ❡ ❤ ❛ ✈ ❡ ♦ ✜ ♥ ❞ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❡ ❝ ❡ ✐ ✈ ❡ ❞ ✐ ♥ ❛ ❡ ✾ ❜ ❡ ❢ ♦ ❡ ❡ ❛ ❝ ❤ ✐ ♥ ❣ ❤ ❡ ❝ ❤ ❡ ❡ ❡ ✳ ❚ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ❝ ❛ ♥ ❜ ❡

❛ ❜ ❛ ❝ ❡ ❞ ❛ ✿ ❙ ❛ ✐ ♥ ❣ ❢ ♦ ♠ ❛ ❡ ✽ ✱ ✇ ❤ ❛ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❡ ♣ ♦ ❝ ❤ ♣ ❡ ♥ ✐ ♥ ❛ ❡ ✾ ✱ ❜ ❡ ❢ ♦ ❡ ❛ ❡ ✸ ✐ ✈ ✐ ✐ ❡ ❞ ❄ ❚ ♦

❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ✇ ✐ ❧ ❧ ❛ ❦ ❡ ❡ ♣ ✐ ♠ ✐ ❧ ❛ ♦ ❛ ✭ ❝ ✮ ✳

• ▲ ❡ ❤ ❡ ❛ ♥ ✇ ❡ ♦ ❤ ❡ ❛ ❜ ♦ ✈ ❡ ✉ ❡ ✐ ♦ ♥ ❜ ❡ N 8 ✳ ❆ ❧ ♦ ✐ ❡ ❝ ❡ ✐ ✈ ❡ ❞ 1 ❤ ♦ ❝ ❦ ✐ ♥ ❤ ❡ ❜ ❡ ❣ ✐ ♥ ♥ ✐ ♥ ❣ ✳ ❍ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡

♦ ❢ ❤ ♦ ❝ ❦ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❤ ✐ ❝ ❛ ❡ ✐

p8 (N 8 + 1)✳

❙ ❡ ♣ ✲ ✺ ✿ ❙ ❛ ② ❤ ❛ ❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ♠ ❛ ❦ ♦ ✈ ❝ ❤ ❛ ✐ ♥ ❣ ❡ ❛ ❜ ♦ ❜ ❡ ❞ ✐ ♥ ❛ ❡ ✾ ✳ ❚ ❤ ✐ ❤ ❛ ♣ ♣ ❡ ♥ ✇ ✐ ❤ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② p9 ✳ ◆ ♦ ✇

✇ ❡ ❡ ♣ ❡ ❛ ❤ ❡ ❛ ♠ ❡ ❡ ♣ ❛ ❙ ❡ ♣ ✹ ✳ ❚ ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ❤ ✐ ❝ ❛ ❡ ✐ p9 (N 9 + 1) ✇ ❤ ❡ ❡ N 9 ✐

❤ ❡ ✏ ❊ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❡ ♣ ♦ ❝ ❤ ♣ ❡ ♥ ✐ ♥ ❛ ❡ ✽ ✱ ❜ ❡ ❢ ♦ ❡ ❛ ❡ ✸ ✐ ✈ ✐ ✐ ❡ ❞ ✱ ❣ ✐ ✈ ❡ ♥ ❤ ❛ ✇ ❡ ❛ ❢ ♦ ♠ ❛ ❡ ✾ ✑ ✳

❋ ✐ ♥ ❛ ❧ ❧ ② ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ✐ p3 · 0 + p8 (N 8 + 1) + p9 (N 9 + 1) = p8 (N 8 + 1) + p9 (N 9 + 1) ✳ ❲ ❡ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❞ p8 ✱

p9 ✱ N 8 ❛ ♥ ❞ N 9 ✉ ✐ ♥ ❣ ▼ ❆ ❚ ▲ ❆ ❇ ❛ ♥ ❞ ✜ ♥ ❛ ❧ ❧ ② ❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ✇ ❛ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❞ ♦ ❜ ❡ 1.791✳ ❆ ♦ ♥ ❡ ♠ ❛ ② ❡ ① ♣ ❡ ❝ ✱

❤ ❡ ❡ ① ♣ ❡ ❝ ❡ ❞ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❤ ♦ ❝ ❦ ✐ ♥ ❤ ✐ ❝ ❛ ❡ ❤ ♦ ✉ ❧ ❞ ❜ ❡ ❧ ❡ ❤ ❛ ♥ ❤ ❛ ✐ ♥ ❛ ✭ ❝ ✮ ❜ ❡ ❝ ❛ ✉ ❡ ✐ ♥ ❤ ✐ ❝ ❛ ❡ ♦ ♥ ❡ ❤ ♦ ❝ ❦ ❣ ❡ ♥ ❡ ❛ ♦

❣ ❡ ✇ ✐ ❝ ❤ ❡ ❞ ♦ ✛ ✳ ❚ ❤ ✐ ✐ ♥ ✉ ✐ ✐ ♦ ♥ ✐ ❢ ♦ ✉ ♥ ❞ ♦ ❜ ❡ ❝ ♦ ❡ ❝ ❛ 1.791 < 6.2963✳

❆ ❣ ❛ ✐ ♥ ❛ ❧ ❧ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ❆ ✳

✺



✶ ✳ ✷ ■ ❲ ❛ ❛ ❙ ♥ ♦ ✇ ❜ ❛ ❧ ❧ ✐ ♥ ❍ ❡ ❧ ❧

❋ ✐ ❣ ✉ ❡ ✶ ✿ ❋ ✐ ❣ ✉ ❡ ❤ ♦ ✇ ✐ ♥ ❣ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ❛ ♥ ❞ ✐ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳

▲ ❊ ❆ ❙ ❊ ◆ ❖ ❚ ❊ ✿ ❛ ✭ ❛ ✮ ✱ ✭ ❜ ✮ ✱ ✭ ❝ ✮ ❛ ♥ ❞ ✭ ❞ ✮ ❡ ✉ ✐ ❡ ✉ ♦ ✐ ♠ ♣ ❧ ❡ ♠ ❡ ♥ ❤ ❡ ❝ ♦ ❞ ❡ ❛ ♥ ❞ ❡ ❤ ❡ ❝ ♦ ❞ ❡ ✉ ✐ ♥ ❣ ❛ ♥ ❡ ① ❛ ♠ ♣ ❧ ❡ ✳ ❚ ❤ ❡

▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ❢ ♦ ❤ ❡ ❡ ♣ ❛ ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ❇ ✳ ❆ ❢ ♦ ❤ ❡ ❡ ① ❛ ♠ ♣ ❧ ❡ ✱ ✇ ❡ ❤ ❛ ✈ ❡ ♠ ❛ ❞ ❡ ❛ ❡ ♣ ❡ ❛ ❡ ❡ ❝ ✐ ♦ ♥ ✐ ♥ ❤ ❡ ❡ ♥ ❞

♦ ❢ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❤ ❡ ❡ ✇ ❡ ❝ ✉ ♠ ✉ ❧ ❛ ✐ ✈ ❡ ❧ ② ❤ ❛ ♥ ❞ ❧ ❡ ❛ ❧ ❧ ❤ ❡ ❡ ① ❛ ♠ ♣ ❧ ❡ ✳

✭ ❛ ✮ ❋ ✐ ❣ ✉ ❡ ✶ ❤ ♦ ✇ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ❛ ♥ ❞ ✐ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳ ❖ ✉ ❣ ♦ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✐ ❤ ❡ ❡ ♣ ♦ ❝ ❤ ✐ ♥ ✇ ❤ ✐ ❝ ❤ ■ ❡ ♥ ❞ X n ♠ ❛ ✐ ❧ ✳ ■ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤

✐ ❤ ❡ ❡ ♣ ♦ ❝ ❤ ✐ ♥ ✇ ❤ ✐ ❝ ❤ ■ ❡ ❝ ❡ ✐ ✈ ❡ Z n ♠ ❛ ✐ ❧ ✳ ◆ ♦ ✇ ✇ ❡ ✇ ✐ ❧ ❧ ❞ ❡ ✈ ❡ ❧ ♦ ♣ ❤ ❡ ♦ ❝ ❤ ❛ ✐ ❝ ✉ ♣ ❞ ❛ ❡ ✉ ❧ ❡ ♦ ❢ ❤ ✐ ② ❡ ♠ ✳

Z n+1 =

Xni=1

M I n,i ✭ ✼ ✮

X n+1 =

Z n+1j=1

M On+1 , j ✭ ✽ ✮

■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✼ ✱ M I n,i ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ✐ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ith ♠ ❛ ✐ ❧ ♦ ❢ ❤ ❡ nth ♦ ✉ ❣ ♦ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳ M I n,i ∼ pI ❛ ♥ ❞

P I (s) ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ pI ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✼ ✱ M On+1 , j ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣

♦ jth ♠ ❛ ✐ ❧ ♦ ❢ ❤ ❡ (n + 1)th

✐ ♥ ❝ ♦ ♠ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳ M On+1 , j ∼ pO ❛ ♥ ❞ P O (s) ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ pO ✳ ▲ ❡

P Xn (s) ❛ ♥ ❞ P Z n (s) ❜ ❡ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ X n ❛ ♥ ❞ Z n ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✳ ❋ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣

▲ ❡ ❝ ✉ ❡ ✶ ✺ ✭ 5th ◆ ♦ ✈ ❡ ♠ ❜ ❡ ✮ ✇ ❡ ❝ ❛ ♥ ❞ ✐ ❡ ❝ ❧ ② ✇ ✐ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ❧ ❛ ✐ ♦ ♥ ❤ ✐ ♣

P Z n+1 (s) = P Xn (P I (s)) ✭ ✾ ✮

P Xn+1 (s) = P Z n+1 (P O (s)) ✭ ✶ ✵ ✮

❯ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✾ ❛ ♥ ❞ ✶ ✵ ✇ ❡ ❣ ❡

P Xn+1 (s) = P Xn (P I (P O (s))) ✭ ✶ ✶ ✮

▲ ❡ P Y (s) = P I (P O (s)) ✭ ♦ P Y = P I ◦ P O ✮ ✳ ❆ ❧ ♦ ❧ ❡ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ P Y (s) ❜ ❡ pY ✳ ❚ ❤ ❡ ♥

❤ ❡ ❡ ❧ ❛ ✐ ♦ ♥ ❜ ❡ ✇ ❡ ❡ ♥ X n ❛ ♥ ❞ X n+1 ❝ ❛ ♥ ❜ ❡ ✐ ♠ ♣ ❧ ② ❝ ❛ ♣ ✉ ❡ ❞ ❜ ② ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♦ ❝ ❤ ❛ ✐ ❝ ✉ ♣ ❞ ❛ ❡ ✉ ❧ ❡

X n+1 =

Xnk=1

Y n,k ✭ ✶ ✷ ✮

✻



✇ ❤ ❡ ❡ Y n,k ∼ pY ✳ ❚ ❤ ✐ ✐ ❛ ❛ ♥ ❞ ❛ ❞ ● ❛ ❧ ♦ ♥ ✲ ❲ ❛ ♦ ♥ ❇ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ♦ ❝ ❡ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ ✇ ❡ ❝ ❛ ♥ ❛ ♣ ♣ ❧ ② ❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛ ✇ ❤ ✐ ❝ ❤ ✇ ❡

❧ ❡ ❛ ♥ ❡ ❞ ✐ ♥ ▲ ❡ ❝ ✉ ❡ ✶ ✺ ✱ ✶ ✻ ❛ ♥ ❞ ✶ ✼ ✳ ◆ ♦ ✇ ✇ ❡ ♣ ♦ ❝ ❡ ❡ ❞ ✇ ✐ ❤ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♥ ❣ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ X n ✳

E [X n+1] =

s

d

ds

P Xn+1 (s)

s=1

=

s

d

ds

P Xn

(P I (P O (s)))

s=1

= sP ′

O (s) P ′

I (P O (s)) P ′

Xn (P I (P O (s)))s=1

= P ′

O (1) P ′

I (P O (1)) P ′

Xn (P I (P O (1)))

= P ′

O (1) P ′

I (1) P ′

Xn (P I (1))

= P ′

O (1) P ′

O (1) P ′

Xn (1)

= E [O] E [I ] E [X n]

= µOµI E [X n] ✭ ✶ ✸ ✮

E

(X n+1)

2

=

s

d

ds2

P Xn+1 (s)

s=1

=

s d

ds

sP

′

O (s) P ′

I (P O (s)) P ′

Xn (P I (P O (s)))

s=1

= sP O′

(s) P I ′

P O (s)

P ′

Xn

P I

P O (s)

+s2P ′′

O (s) P ′

I (P O (s)) P ′

Xn (P I (P O (s)))

+s2P ′

O (s) P ′

O (s) P ′′

I (P O (s)) P ′

Xn (P I (P O (s)))

+s2P ′

O (s) P ′

I (P O (s)) P ′

O (s) P ′

I (P O (s)) P ′′

Xn (P I (P O (s)))

s=1

= P ′

O (1) P ′

O (1) P ′

Xn (1) + P

′′

O (1) P ′

I (1) P ′

Xn (1)

+

P ′

O (1)

2

P ′′

I (1) P ′

Xn (1) +

P ′

I (1) P ′

O (1)

2

P ′′

Xn (1)

= P ′O (1) + P ′′O (1)P ′I (1) P ′Xn (1) + P ′O (1)2 P ′′I (1) P ′Xn

(1)

+

P ′

I (1) P ′

O (1)2

P ′′

Xn (1)

= E

O2

E [I ] E [X n] + E [O]2

E

I 2

− E [I ]

E [X n] + (E [I ] E [O])2

E

(X n)2

− E [X n]

=

E

O2

− E [O]2

E [I ] +

E

I 2

− E [I ]2

E [O]2

E [X n] + (E [I ] E [O])2

E

(X n)2

=

❱ ❛ [O] E [I ] + ❱ ❛ [I ] E [O]2

E [X n] + (E [I ] E [O])2

E

(X n)2

=

σ2OµI + σ2

I µ2O

E [X n] + (µOµI )

2E

(X n)2

✭ ✶ ✹ ✮

■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✸ ❛ ♥ ❞ ✶ ✹ ✱ µI ❛ ♥ ❞ σI ✭ µO ❛ ♥ ❞ σO ✮ ✐ ❤ ❡ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ❛ ♥ ❞ ❛ ❞ ❞ ❡ ✈ ✐ ❛ ✐ ♦ ♥ ♦ ❢ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ pI ✭ pO ✮

❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✳ ❲ ❡ ❝ ❛ ♥ ✇ ✐ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✸ ❛ ♥ ❞ ✶ ✹ ✐ ♥ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♠ ❛ ✐ ① ❢ ♦ ♠ E [X n+1]

E

(X n+1)2

=

µOµI 0

σ2OµI + σ2

I µ2O (µOµI )

2

E [X n]

E

(X n)2

✭ ✶ ✺ ✮

❯ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✺ ✐ ♥ ❛ ❡ ❝ ✉ ✐ ✈ ❡ ♠ ❛ ♥ ♥ ❡ ✇ ❡ ❣ ❡ E [X n]

E

(X n)2

=

µOµI 0

σ2OµI + σ2

I µ2O (µOµI )

2

n E [X 0]

E

(X 0)2

=

µOµI 0

σ2OµI + σ2

I µ2O (µOµI )

2

n m

m2

✭ ✶ ✻ ✮

❙ ♦ ❧ ✈ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✻ ✇ ❡ ❣ ❡ ✱

✼



E [X n] = m (µOµI )n

❱ ❛ [X n] = E

(X n)2

− (E [X n])2

= m

σ2OµI + σ2

I µ2O

(µOµI )

n−1 (µOµI )n

− 1

µOµI − 1

▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ♦ ❝ ♦ ♠ ♣ ✉ ❡ E [X n] ❛ ♥ ❞ ❱ ❛ [X n] ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳

✭ ❜ ✮ ❲ ❡ ✇ ✐ ❧ ❧ ❞ ✐ ❡ ❝ ❧ ② ✉ ❡ ❝ ♦ ♥ ❝ ❡ ♣ ❢ ♦ ♠ ▲ ❡ ❝ ✉ ❡ ✶ ✻ ❛ ♥ ❞ ✶ ✼ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ● ✐ ✈ ❡ ♥ ❤ ❛ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣

♠ ❛ ✐ ❧ ✐ ♥ ❤ ❡ 0th ❡ ♣ ♦ ❝ ❤ ✐ m ✱ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② pne ❤ ❛ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✇ ✐ ❧ ❧ ❜ ❡ ❡ ① ✐ ♥ ❝ ❛ ❤ ❡ nth ❡ ① ❝ ❤ ❛ ♥ ❣ ❡ ✐ ❣ ✐ ✈ ❡ ♥ ❜ ②

pne = am ✇ ❤ ❡ ❡

a = P Y ◦ P Y · · · ◦ P Y (0)

n ❢ ♦ ❧ ❞ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥

❆ ❧ ♦ P Y = P I ◦ P O ✳ ■ ♥ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ❛ ♦ ❢ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ✜ ❧ ❧ ✉ ❡ ❤ ❡ ❤ ♦ ❤ ❛ ♥ ❞ ♥ ♦ ❛ ✐ ♦ ♥ P nY ♦ ❡ ♣ ❡ ❡ ♥ ❤ ❡ n ❢ ♦ ❧ ❞

❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ P Y ◦ P Y · · · ◦ P Y ✳

▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ♦ ❝ ♦ ♠ ♣ ✉ ❡ pne ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳

✭ ❝ ✮

♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ■ ❛ ♠ ❢ ♦ ❡ ✈ ❡ ❞ ❡ ❛ ❧ ✐ ♥ ❣ ✇ ✐ ❤ ❡ ♠ ❛ ✐ ❧ ❂ ✶ ✲ ❊ ① ✐ ♥ ❝ ✐ ♦ ♥ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❤ ❡ ❡ ♠ ❛ ✐ ❧

▲ ❡ ❞ ❡ ♥ ♦ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❜ ② pe ✳ ❯ ✐ ♥ ❣ ❝ ♦ ♥ ❝ ❡ ♣ ❢ ♦ ♠ ▲ ❡ ❝ ✉ ❡ ✶ ✻ ❛ ♥ ❞ ✶ ✼ ✇ ❡ ❝ ❛ ♥ ✇ ✐ ❡

pe = αm✇ ❤ ❡ ❡

α = P Y (α)

❲ ❡ ❝ ❛ ♥ ❛ ❧ ♦ ♦ ❜ ❛ ✐ ♥ α ✉ ✐ ♥ ❣ ❤ ❡ ❢ ♦ ♠ ✉ ❧ ❛

α = limn→∞

P nY (0)

❋ ✐ ♥ ❛ ❧ ❧ ② ✱ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ■ ❛ ♠ ❢ ♦ ❡ ✈ ❡ ❞ ❡ ❛ ❧ ✐ ♥ ❣ ✇ ✐ ❤ ❡ ♠ ❛ ✐ ❧ ✐ 1 − pe ✳ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳

✭ ❞ ✮ ■ ♥ ❤ ✐ ❝ ❛ ❡ ❤ ❡ ♦ ❝ ❤ ❛ ✐ ❝ ✉ ♣ ❞ ❛ ❡ ✉ ❧ ❡ ❣ ✐ ✈ ❡ ♥ ❜ ② ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✷ ✇ ✐ ❧ ❧ ❣ ❡ ♠ ♦ ❞ ✐ ✜ ❡ ❞ ❛

X n+1 =

Xnk=1

Y n,k + N n

✇ ❤ ❡ ❡ N n ∼ pN ✳ N n ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ♥ ❤ ❡ nth ♦ ✉ ❣ ♦ ✐ ♥ ❣ ❡ ♣ ♦ ❝ ❤ ✳ N n ∼ pN ❛ ♥ ❞ P N (s) ✐ ❤ ❡

♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ pN ✳ ◆ ♦ ✇

P Xn+1 (s) = E sXn+1= E

Xnk=1

sY n,k · sN n

= E

Xnk=1

sY n,k

E

sN n

✭ ✶ ✼ ✮

= P Xn (P Y (s)) P N (s) ✭ ✶ ✽ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✶ ✼ ✐ ♣ ♦ ✐ ❜ ❧ ❡ ❞ ✉ ❡ ♦ ❤ ❡ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧ ♠ ❛ ✐ ❧ N n ❛ ♥ ❞ ❤ ❡ ❡ ❣ ✉ ❧ ❛ ♠ ❛ ✐ ❧ Y n,k ✳

❍ ♦ ✇ ✇ ✐ ❧ ❧ ❛ ♥ ✇ ❡ ♦ ❛ ✭ ❛ ✮ ❝ ❤ ❛ ♥ ❣ ❡ ✿

✽



❋ ✐ ❧ ❡ ♣ ♦ ❝ ❡ ❡ ❞ ✇ ✐ ❤ ❤ ❡ ♠ ❡ ❛ ♥ ♦ ❢ X n ✳

E [X n+1] =

s

d

ds

P Xn+1 (s)

s=1

=

s

d

ds

P Xn

(P Y (s)) P N (s)

s=1

= sP N (s) d

ds

(P Xn (P Y (s))) + sP Xn

(P Y (s)) d

ds

(P N (s))s=1

✭ ✶ ✾ ✮

= sP N (s) P ′

Y (s) P ′

Xn (P Y (s)) + sP Xn

(P Y (s)) P ′

N (s)s=1

= sP N (s) P ′

Y (s) P ′

Xn (P Y (s))

s=1

+ sP Xn (P Y (s)) P

′

N (s)s=1

= P N (1) sP ′

Y (s) P ′

Xn (P Y (s))

s=1

+ 1 · P Xn (P Y (1)) P

′

N (1)

= 1 · sP ′

Y (s) P ′

Xn (P Y (s))

s=1

+ 1 · P Xn (1) · P

′

N (1)

= µOµI E [X n] + 1 · 1 · P ′

N (1) ✭ ✷ ✵ ✮

= µOµI E [X n] + µN ✭ ✷ ✶ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✶ ✾ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ✉ ✐ ♥ ❣ ❝ ❤ ❛ ✐ ♥ ✉ ❧ ❡ ❛ ♥ ❞ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✵ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✸ ✳ ◆ ❡ ① ✇ ❡ ❝ ♦ ♠ ♣ ✉ ❡ ❡ ❝ ♦ ♥ ❞ ♠ ♦ ♠ ❡ ♥

♦ ❢ X n ✇ ❤ ✐ ❝ ❤ ✇ ✐ ❧ ❧ ❜ ❡ ✉ ❡ ❢ ✉ ❧ ♦ ✜ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ X n ✳

E

(X n+1)2

=

s

d

ds

2

P Xn+1 (s)

s=1

=

s

d

ds

sP N (s) P

′

Y (s) P ′

Xn (P Y (s)) + sP Xn

(P Y (s)) P ′

N (s)

s=1

=

s

d

ds

P N (s) sP

′

Y (s) P ′

Xn (P Y (s))

s=1

+

s

d

ds

sP Xn

(P Y (s)) P ′

N (s)

s=1

= s

P N (s)

d

ds

sP

′

Y (s) P ′

Xn (P Y (s))

+ P

′

N (s) sP ′

Y (s) P ′

Xn (P Y (s))

s=1

+ s P Xn (P Y (s)) P ′

N (s) + sP ′

Y (s) P ′

Xn (P Y (s)) P ′

N (s) + sP Xn (P Y (s)) P ′′

N (s)s=1

✭ ✷ ✷ ✮

= P N (1)

s

d

ds

sP

′

Y (s) P ′

Xn (P Y (s))

s=1

+ 1 · P ′

N (1) · 1 · P ′

Y (1) P ′

Xn (P Y (1))

1 · P Xn (P Y (1)) P

′

N (1) + 12P ′

Y (1) P ′

Xn (P Y (1)) P

′

N (1) + 12P Xn (P Y (1)) P

′′

N (1)

=

s

d

ds

sP

′

Y (s) P ′

Xn (P Y (s))

s=1

+ E [N ] E [Y ] E [X n]

+E [N ] + E [N ] E [Y ] E [X n] +

E

N 2

− E [N ]

=

s

d

ds

sP

′

Y (s) P ′

Xn (P Y (s))

s=1

+ 2µN µOµI E [X n] +

σ2N + (µN )

2

✭ ✷ ✸ ✮

= σ2OµI + σ2

I µ2OE [X n] + (µOµI )

2E (X n)

2

✭ ✷ ✹ ✮

+2µN µOµI E [X n] +

σ2N + (µN )

2=

σ2OµI + σ2

I µ2O + 2µN µOµI

E [X n] + (µOµI )

2E

(X n)2

+

σ2N + (µN )

2

✭ ✷ ✺ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✷ ✷ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ✉ ✐ ♥ ❣ ❝ ❤ ❛ ✐ ♥ ✉ ❧ ❡ ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✸ ✇ ❡ ✉ ❡ ❞ ❤ ❡ ❢ ❛ ❝ ❤ ❛ E [Y ] = E [O] E [I ] = µOµI ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✷ ✹

✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✹ ✳ ❊ ① ♣ ❡ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✶ ❛ ♥ ❞ ✷ ✺ ✐ ♥ ♠ ❛ ✐ ① ❢ ♦ ♠

E [X n+1]

E

(X n+1)2

=

µOµI 0

σ2OµI + σ2

I µ2O + 2µN µOµI (µOµI )

2

E [X n]

E

(X n)2

+

µN

σ2N + (µN )

2

✭ ✷ ✻ ✮

✾



❊ ✉ ❛ ✐ ♦ ♥ ✷ ✻ ✐ ❜ ❛ ✐ ❝ ❛ ❧ ❧ ② ❛ ❞ ✐ ❝ ❡ ❡ ✐ ♠ ❡ ▲ ❚ ■ ② ❡ ♠ ✳ ❚ ♦ ❣ ❡ E [X n] ❛ ♥ ❞ E

(X n)2

✇ ❡ ❝ ❛ ♥ ❡ ❝ ✉ ✐ ✈ ❡ ❧ ② ♦ ❧ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✻

✐ ♥ ▼ ❆ ❚ ▲ ❆ ❇ ❛ ✐ ♥ ❣ ❢ ♦ ♠ E [X 0] = m ❛ ♥ ❞ E

(X 0)2

= m2✳ ◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ♣ ❡ ❛ ❦ ✐ ♥ ❣ ✱ E [X n] ❢ ♦ ❤ ❡ ❝ ❛ ❡ ✇ ✐ ❤ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧

♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ❞ ❡ ✜ ♥ ✐ ❡ ❧ ② ❧ ❛ ❣ ❡ ❤ ❛ ♥ E [X n] ✇ ❤ ❡ ♥ ❤ ❡ ❡ ✇ ❤ ❡ ❡ ♥ ♦ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✳

❍ ♦ ✇ ✇ ✐ ❧ ❧ ❛ ♥ ✇ ❡ ♦ ❛ ✭ ❜ ✮ ❝ ❤ ❛ ♥ ❣ ❡ ✿

❚ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❞ ❡ ✐ ✈ ❛ ✐ ♦ ♥ ✐ ✐ ♠ ♣ ♦ ❛ ♥ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ❋ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✽ ✇ ❡ ❤ ❛ ✈ ❡

P Xn (s) = P Xn−1 (P Y (s)) P N (s)

P Xn−1 (s) = P Xn−2 (P Y (s)) P N (s)

P Xn−1 (P Y (s)) = P Xn−2 (P Y (P Y (s))) P N (P Y (s))

P Xn (s) = P Xn−2 (P Y (P Y (s))) P N (P Y (s)) P N (s)

P Xn−2 (s) = P Xn−3 (P Y (s)) P N (s)

P Xn−2 (P Y (P Y (s))) = P Xn−3 (P Y (P Y (P Y (s)))) P N (P Y (P Y (s)))

P Xn (s) = P Xn−3 (P Y (P Y (P Y (s)))) P N (P Y (P Y (s))) P N (P Y (s)) P N (s)

✳

✳

✳

✳

✳

✳

✳

✳

✳

P Xn (s

) = P X0 (P

n

Y (s

))

n−1

k=0 P N P

k

Y (s

)P Xn

(s) = (P nY (s))m

n−1k=0

P N

P kY (s)

✭ ✷ ✼ ✮

❧ ❡ ❛ ❡ ◆ ♦ ❡ ✿ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✼ ✱ P kY (s) ❡ ♣ ❡ ❡ ♥ ❤ ❡ k ❢ ♦ ❧ ❞ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ P Y (s)✳

◆ ♦ ✇ ✇ ❡ ✇ ✐ ❧ ❧ ♣ ♦ ❝ ❡ ❡ ❞ ✇ ✐ ❤ ❤ ❡ ♠ ❛ ✐ ♥ ✉ ❡ ✐ ♦ ♥ ✳ ❚ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② pne ❤ ❛ ❤ ❡ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✇ ✐ ❧ ❧ ❜ ❡ ❡ ① ✐ ♥ ❝ ❛ ❤ ❡ nth

❡ ① ❝ ❤ ❛ ♥ ❣ ❡ ✐ ❡ ✉ ❛ ❧ ♦ P Xn (0) ✳ ❲ ❡ ❝ ❛ ♥ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ✐ ❜ ② ✉ ❜ ✐ ✉ ✐ ♥ ❣ s = 0 ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✼ ✳ ◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ✱ pne ✇ ✐ ❤ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧

♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✇ ✐ ❧ ❧ ❞ ❡ ❢ ♥ ✐ ❡ ❧ ② ❜ ❡ ❧ ❡ ❤ ❛ ♥ ✭ ♦ ❡ ✉ ❛ ❧ ♦ ✮ ❤ ❡ ❝ ❛ ❡ ✇ ❤ ❡ ♥ ❤ ❡ ❡ ❛ ❡ ♥ ♦ ✇ ❛ ❞ ❞ ✐ ✐ ♦ ♥ ❛ ❧ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✳

❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✐ ❣ ✐ ✈ ❡ ♥ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳

❍ ♦ ✇ ✇ ✐ ❧ ❧ ❛ ♥ ✇ ❡ ♦ ❛ ✭ ❝ ✮ ❝ ❤ ❛ ♥ ❣ ❡ ✿

♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ■ ❛ ♠ ❢ ♦ ❡ ✈ ❡ ❞ ❡ ❛ ❧ ✐ ♥ ❣ ✇ ✐ ❤ ❡ ♠ ❛ ✐ ❧ ❂ ✶ ✲ ❊ ① ✐ ♥ ❝ ✐ ♦ ♥ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❤ ❡ ❡ ♠ ❛ ✐ ❧

❚ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② pe ✐ ❣ ✐ ✈ ❡ ♥ ❜ ② limn→∞

P Xn (0)

pe = limn→∞

P Xn (0)

= limn→∞

(P nY (0))m

n−1k=0

P N

P kY (0)

❚ ♦ ✜ ♥ ❞ pe ✇ ❡ ❥ ✉ ❛ ❦ ❡ ❛ ❧ ❛ ❣ ❡ n ❛ ♥ ❞ ❝ ♦ ♠ ♣ ✉ ❡ P Xn (0) ✉ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✼ ✳ ❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✐ ❣ ✐ ✈ ❡ ♥ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❇ ✳

✭ ❡ ✮ ❚ ♦ ❛ ♥ ✇ ❡ ❤ ✐ ♣ ❛ ✇ ❡ ❣ ♦ ❜ ❛ ❝ ❦ ♦ ❡ ✉ ❛ ✐ ♦ ♥ ✶ ✽

P Xn+1 (s) = P Xn (P Y (s)) P N (s) ✭ ✷ ✽ ✮

P Xn (s) ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ ■ ❢ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥

❡ ① ✐ ❤ ❡ ♥ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ X n ✇ ✐ ❧ ❧ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❤ ❡ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ❢ ♦ ❧ ❛ ❣ ❡ n✳ ❙ ✐ ♥ ❝ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ②

♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❤ ❡ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ✱ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ✇ ✐ ❧ ❧ ❛ ❧ ♦ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ P π (s)✭ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ❛ ✐ ♦ ♥ ❛ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ✮ ✳ ❚ ❤ ✐ ✐ ❜ ❡ ❝ ❛ ✉ ❡ ❛ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❣ ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❛ ♥ ❞

❛ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♠ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❤ ❛ ❛ ♦ ♥ ❡ ✲ ♦ ✲ ♦ ♥ ❡ ❡ ❧ ❛ ✐ ♦ ♥ ❤ ✐ ♣ ✳ ▼ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ❧ ❧ ② ✱ P Xn+1 (s) → P π (s) ❛ ♥ ❞ P Xn (s) → P π (s) ❢ ♦

❧ ❛ ❣ ❡ n ✳ ❙ ✉ ❜ ✐ ✉ ✐ ♥ ❣ ❤ ✐ ✐ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✽ ✇ ❡ ❣ ❡

✶ ✵



P π (s) = P π (P Y (s)) P N (s)

P π (s) = P π (P I (P O (s))) P N (s) ✭ ✷ ✾ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✷ ✾ ✐ ❤ ❡ ❡ ✉ ✐ ❡ ❞ ❡ ❧ ❛ ✐ ♦ ♥ ❤ ✐ ♣ ✳

✭ ❢ ✮ ■ ♠ ❛ ② ♥ ♦ ❜ ❡ ♣ ♦ ✐ ❜ ❧ ❡ ♦ ❞ ✐ ❡ ❝ ❧ ② ♦ ❧ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✾ ♦ ❣ ❡ P π (s)✳ ❲ ❡ ✐ ♥ ❡ ❛ ❞ ✉ ❡ ❤ ❡ ✐ ❞ ❡ ❛ ❤ ❛ ✐ ❢ P π (s) ❡ ① ✐ ✱ ❤ ❡ ♥ ✇ ❡

❝ ❛ ♥ ❡ ① ♣ ❡ ✐ ❛ ❛ ❧ ✐ ♠ ✐ ♦ ❢ P Xn (s) ✳ ❯ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✷ ✼ ✇ ❡ ❝ ❛ ♥ ❛ ❧ ❡ ❛ ❞ ② ❡ ① ♣ ❡ P Xn

(s) ✐ ♥ ❡ ♠ ♦ ❢ ❡ ♣ ❡ ❛ ❡ ❞ ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥

♦ ❢ P N (s) ✱ P O (s) ❛ ♥ ❞ P I (s)✳ ❍ ❡ ♥ ❝ ❡

P π (s) = limn→∞

P Xn (s)

= limn→∞

(P nY (s))m

n−1k=0

P N

P kY (s)

✭ ✸ ✵ ✮

✇ ❤ ❡ ❡ P Y (s) = P I (P O (s))✳

❊ ❳ ❆ ▼ ▲ ❊

Epoch0 5 10 15 20

M e a n

0

1

2

3

4

5 Without Extra Mails

With Extra Mails

Epoch

0 5 10 15 20

V a r i a n c e

0

2

4

6

8

10

12

14 Without Extra Mails

With Extra Mails

Epoch0 5 10 15 20

E x t i n c t i o n P r o b a b i l i t y

0

0.2

0.4

0.6

0.8

1

Without Extra Mails

With Extra Mails

Epoch

0 5 10 15 20

M e a n

×104

0

1

2

3

4

5

6

7

8Without Extra Mails

With Extra Mails

Epoch0 5 10 15 20

V a r i a n c e

×109

0

0.5

1

1.5

2

2.5Without Extra Mails

With Extra Mails

Epoch0 5 10 15 20


0

0.01

0.02

0.03

0.04

0.05Without Extra Mails

With Extra Mails

µI µO=0.77µN = 0.67

µN = 0.67

µI µO=1.6

µN = 0.67 µN = 0.67

µI µO=1.6

µN = 0.67

µI µO=0.77µN = 0.67

µI µO=0.77

µI µO=1.6

❋ ✐ ❣ ✉ ❡ ✷ ✿ ❋ ✐ ❣ ✉ ❡ ♦ ✉ ❞ ② ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢ µOµI ❛ ♥ ❞ ❡ ① ❛ ♠ ❛ ✐ ❧ ♦ ♥ ❤ ❡ ♠ ❡ ❛ ♥ ✱ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ❛ ♥ ❞ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❤ ❡

♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✳

❲ ❡ ❛ ♥ ❞ ♦ ♠ ❧ ② ❣ ❡ ♥ ❡ ❛ ❡ ❞ ✇ ♦ ❡ ♦ ❢ pO ❛ ♥ ❞ pI ✱ ♦ ♥ ❡ ✇ ✐ ❤ µI µO ≈ 0.8 < 1 ❛ ♥ ❞ ❤ ❡ ♦ ❤ ❡ ✇ ✐ ❤ µI µO ≈ 1.5 > 1 ✳ ❲ ❡ ❛ ❧ ♦

❣ ❡ ♥ ❡ ❛ ❡ ❞ ❛ ❝ ♦ ♠ ♠ ♦ ♥ pN ❢ ♦ ❜ ♦ ❤ ❤ ❡ ❡ ❡ ✇ ✐ ❤ µN ≈ 0.7 ✳ ❲ ❡ ♥ ♦ ✇ ❞ ♦ ✇ ♦ ❦ ✐ ♥ ❞ ♦ ❢ ❝ ♦ ♠ ♣ ❛ ❛ ✐ ✈ ❡ ✉ ❞ ② ✿ ■ ✮ ❚ ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢

µOµI ✳ ■ ■ ✮ ❚ ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢ ❡ ① ❛ ♠ ❛ ✐ ❧ ✳

❚ ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢ µOµI ✿ ❲ ❤ ❡ ♥ µI µO < 1 ✱ ❤ ❡ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❛ ✜ ♥ ✐ ❡ ✈ ❛ ❧ ✉ ❡

❛ ✐ ♠ ❡ ✐ ♥ ❝ ❡ ❛ ❡ ✳ ■ ❢ µI µO > 1 ✱ ❜ ♦ ❤ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ♥ ❝ ❡ ❛ ❡ ❡ ① ♣ ♦ ♥ ❡ ♥ ✐ ❛ ❧ ❧ ② ✇ ✐ ❤

✐ ♠ ❡ ✳ ❚ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ✐ ♥ ❝ ❡ ❛ ❡ ✇ ✐ ❤ ❞ ❡ ❝ ❡ ❛ ❡ ✐ ♥ µOµI ✱ ■ ♥ ❞ ❡ ❡ ❞ ✱ ✐ ❢ ❤ ❡ ❡ ❛ ❡ ♥ ♦ ❡ ① ❛ ♠ ❛ ✐ ❧ ❛ ♥ ❞ µOµI < 1 ❤ ❡ ♥

❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ 1 ✇ ❤ ✐ ❧ ❡ ✐ ❢ µOµI > 1 ❤ ❡ ♥ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❛ ✜ ♥ ✐ ❡ ✈ ❛ ❧ ✉ ❡ ✳

❚ ❤ ❡ ❡ ✛ ❡ ❝ ♦ ❢ ❡ ① ❛ ♠ ❛ ✐ ❧ ✿ ❲ ✐ ❤ ❡ ① ❛ ♠ ❛ ✐ ❧ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✉ ❣ ♦ ✐ ♥ ❣ ♠ ❛ ✐ ❧ ✐ ♥ ❝ ❡ ❛ ❡ ✳ ❲ ✐ ❤ ♥ ♦

❡ ① ❛ ♠ ❛ ✐ ❧ ✱ ❤ ❡ ♠ ❡ ❛ ♥ ❛ ♥ ❞ ❤ ❡ ✈ ❛ ✐ ❛ ♥ ❝ ❡ ✇ ❡ ❡ ❝ ♦ ♥ ✈ ❡ ❣ ✐ ♥ ❣ ♦ 0 ✐ ❢ µI µO < 1 ✳ ❍ ♦ ✇ ❡ ✈ ❡ ✇ ✐ ❤ ❡ ① ❛ ♠ ❛ ✐ ❧ ✐ ✐ ❝ ♦ ♥ ✈ ❡ ❣ ✐ ♥ ❣ ♦

❛ ✜ ♥ ✐ ❡ ✈ ❛ ❧ ✉ ❡ ✳ ■ ❢ µOµI < 1 ❤ ❡ ♥ ✇ ✐ ❤ ♥ ♦ ❡ ① ❛ ♠ ❛ ✐ ❧ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ 1 ❜ ✉ ✇ ✐ ❤ ❡ ① ❛ ♠ ❛ ✐ ❧ ✐

❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ ❛ ✜ ♥ ✐ ❡ ✈ ❛ ❧ ✉ ❡ ✳ ■ ❢ µOµI > 1 ❛ ♥ ❞ ❤ ❡ ❡ ❛ ❡ ❡ ① ❛ ♠ ❛ ✐ ❧ ✱ ❤ ❡ ♥ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♦ 0 ✳

✶ ✶



✷ ◆ ✉ ♠ ❡ ✐ ❝ ❛ ❧ ❈ ♦ ♠ ♣ ✉ ❛ ✐ ♦ ♥

✷ ✳ ✶ ❇ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ❖ ✉ ✐ ♥ ❚ ✇ ♦ ❲ ❛ ②

✭ ❛ ✮ ❚ ❤ ❡ ❙ ♦ ❝ ❤ ❛ ✐ ❝ ❯ ♣ ❞ ❛ ❡ ❘ ✉ ❧ ❡ ❢ ♦ ❤ ❡ ✇ ♦ ✇ ❛ ② ❜ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ♣ ♦ ❝ ❡ ✐

X An+1 =

XAn

kA=1

Y (AA)n,kA

+

XBn

kB=1

Y (BA)n,kB

; X Bn+1 =

XAn

kA=1

Y (AB)n,kA

+

XBn

kB=1

Y (BB)n,kB

✭ ✸ ✶ ✮

✇ ❤ ❡ ❡ X An ❛ ♥ ❞ X Bn ❛ ❡ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ X An ❛ ♥ ❞ X Bn ❛ ❡ ❛ ❧ ♦

❤ ❡ ❛ ❡ ♦ ❢ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ❈ ❤ ❛ ✐ ♥ ✳ Y (AA)n,kA

✭ Y (AB)n,kA

✮ ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✭ ❚ ② ♣ ❡ ✲ ❇ ✮ ♣ ♦ ❞ ✉ ❝ ❡ ❞ ❛ ❣ ❡ ♥ kA ♦ ❢ ❚ ② ♣ ❡ ✲ ❆

✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ Y (AA)n,kA

❛ ♥ ❞ Y (AB)n,kA

❛ ❡ ❣ ♦ ✈ ❡ ♥ ❡ ❞ ❜ ② ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ p(A) ✱ ✐ ✳ ❡ ✳

Y

(AA)n,kA

, Y (AB)n,kA

∼ p(A)

✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ②

Y (BA)n,kB

✭ Y (BB)n,kB

✮ ✐ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✭ ❚ ② ♣ ❡ ✲ ❇ ✮ ♣ ♦ ❞ ✉ ❝ ❡ ❞ ❛ ❣ ❡ ♥ kB ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ Y (BA)n,kB

❛ ♥ ❞

Y (BB)n,kB

❛ ❡ ❣ ♦ ✈ ❡ ♥ ❡ ❞ ❜ ② ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ p(B)✱ ✐ ✳ ❡ ✳

Y

(BA)n,kB

, Y (BB)n,kB

∼ p(B)

✳

❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ❢ ♦ ❤ ❡ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ♦ ❢ ❚ ✇ ♦ ❲ ❛ ② ❇ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ♦ ❝ ❡ ✐ ❣ ✐ ✈ ❡ ♥ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❈ ✳

▲ ❊ ❆ ❙ ❊ ◆ ❖ ❚ ❊ ✿ ❘ ❛ ❤ ❡ ❤ ❛ ♥ ❛ ♥ ❞ ♦ ♠ ❧ ② ❞ ♦ ✐ ♥ ❣ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ❢ ♦ ✈ ❛ ✐ ♦ ✉ p(A) ❛ ♥ ❞ p(B) ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❞ ✐ ✛ ❡ ❡ ♥

❝ ❡ ♥ ❛ ✐ ♦ ✱ ✇ ❡ ❤ ♦ ✉ ❣ ❤ ❤ ❛ ✐ ✐ ♠ ♦ ❡ ❛ ❡ ❣ ✐ ❝ ♦ ✜ ❞ ❡ ✈ ❡ ❧ ♦ ♣ ❤ ❡ ❤ ❡ ♦ ② ❜ ② ♦ ❧ ✈ ✐ ♥ ❣ ❛ ✭ ❞ ✮ ✱ ✭ ❡ ✮ ❛ ♥ ❞ ✭ ❢ ✮ ❛ ♥ ❞ ❤ ❡ ♥ ✉ ❡

❤ ❡ ❤ ❡ ♦ ② ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❞ ✐ ✛ ❡ ❡ ♥ ❝ ❡ ♥ ❛ ✐ ♦ ❛ ♥ ❞ ❡ ✐ ✉ ✐ ♥ ❣ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ✳

✭ ❞ ✮ ▲ ❡ P XAn X

Bn

❜ ❡ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❉ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ❋ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✭ X An ✮ ❛ ♥ ❞ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇

❛ ❣ ❡ ♥ ✭ X Bn ❛ ✮ ✐ ♥ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤ ✳ ▲ ❡ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ● ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❋ ✉ ♥ ❝ ✐ ♦ ♥

✷

♦ ❢ P XAn X

Bn

❜ ❡ P XAn+1X

Bn+1

(u, v) ✳ ◆ ♦ ✇ ✱

P XAn+1X

Bn+1

(u, v)

= E

uXAn+1vX

Bn+1

= E

XAn

kA=1

uY (AA)n,kA

XBn

kB=1

uY (BA)n,kB

XAn

kA=1

vY (AB)n,kA

XBn

kB=1

vY (BB)n,kB

✭ ✸ ✷ ✮

= E XAn

kA=1 uY (AA)n,kA v

Y (AB)n,kA

XBn

kB=1 uY (BA)n,kB v

Y (BB)n,kB

=∞

xB=0

∞xA=0

E

xAkA=1

uY (AA)n,kA v

Y (AB)n,kA

xBkB=1

uY (BA)n,kB v

Y (BB)n,kB

|X An = xA, X Bn = xB

P

X An = xA, X Bn = xB

✭ ✸ ✸ ✮

=∞

xB=0

∞xA=0

E

xAkA=1

uY (AA)n,kA v

Y (AB)n,kA

xBkB=1

uY (BA)n,kB v

Y (BB)n,kB

P


✭ ✸ ✹ ✮

=∞

xB=0

∞xA=0

xAkA=1

E

uY (AA)n,kA v

Y (AB)n,kA

xBkB=1

E

uY (BA)n,kB v

Y (BB)n,kB

P


✭ ✸ ✺ ✮

=∞

xB=0

∞

xA=0 xA

kA=1

E uY (AA)kA v

Y (AB)kA

xB

kB=1

E uY (BA)kB v

Y (BB)kB P X A

n = xA, X B

n = xB ✭ ✸ ✻ ✮

=∞

xB=0

∞xA=0

E

uY (AA)

vY (AB)

xA E

uY (BA)

vY (BB)

xBP


✭ ✸ ✼ ✮

=∞

xB=0

∞xA=0

P (A) (u, v)

xA P (B) (u, v)

xBP


= P XAn X

Bn

P (A) (u, v) , P (B) (u, v)

✷

❙ ✐ ♥ ❝ ❡ ❤ ❡ ❡ ❛ ❡ ✇ ♦ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡

X An ❛ ♥ ❞

X Bn ✱ ❤ ❡ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ● ❡ ♥ ❡ ❛ ✐ ♥ ❣ ❋ ✉ ♥ ❝ ✐ ♦ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ❤ ❛ ❛ ❝ ❡ ✐ ③ ❡ ❞ ❜ ② ✇ ♦ ♣ ❛ ❛ ♠ ❡

u❛ ♥ ❞

v

✐ ♥ ❡ ❛ ❞ ♦ ❢ ❛ ✐ ♥ ❣ ❧ ❡ ♣ ❛ ❛ ♠ ❡ ❡ s ✭ ❧ ✐ ❦ ❡ ✇ ❡ ❞ ✐ ❞ ✐ ♥ ❝ ❧ ❛ ✮ ✳

✶ ✷



❊ ✉ ❛ ✐ ♦ ♥ ✸ ✷ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ② ✉ ❜ ✐ ✉ ✐ ♥ ❣ X An+1 ❛ ♥ ❞ X Bn+1 ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✶ ✳ ❚ ♦ ❣ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✸ ✱ ✇ ❡ ✉ ❡ ❤ ❡ ▲ ❛ ✇ ♦ ❢ ❚ ♦ ❛ ❧

❊ ① ♣ ❡ ❝ ❛ ✐ ♦ ♥ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❡ ❞ ♦ ♥ ❤ ❡ ❢ ❛ ❝ ❤ ❛ X An = xA ❛ ♥ ❞ X Bn = xB ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✹ ✇ ❡ ❝ ♦ ✉ ❧ ❞ ❡ ♠ ♦ ✈ ❡ ❤ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❜ ❡ ❝ ❛ ✉ ❡

❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ Y (AA)n,kA

✱ Y (AB)n,kA

✱ Y (BA)n,kB

❛ ♥ ❞ Y (BB)n,kB

❛ ❡ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ X An ❛ ♥ ❞ X Bn ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥

✸ ✺ ✇ ❡ ❝ ♦ ✉ ❧ ❞ ❡ ♣ ❡ ❛ ❡ ❤ ❡ ✇ ♦ ❡ ① ♣ ❡ ❝ ❛ ✐ ♦ ♥ ❜ ❡ ❝ ❛ ✉ ❡ ❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ Y (AA)n,kA

❛ ♥ ❞ Y (AB)n,kA

✐ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❛ ♥ ❞ ♦ ♠

✈ ❛ ✐ ❛ ❜ ❧ ❡ Y (BA)n,kB

❛ ♥ ❞ Y (BB)n,kB

✳ ❊ ✉ ❛ ✐ ♦ ♥ ✸ ✻ ❛ ♥ ❞ ✸ ✼ ✐ ❤ ❡ ❡ ✉ ❧ ♦ ❢ ❤ ❡ ❢ ❛ ❝ ❤ ❛ ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ✭ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❛ ✐ ♦ ♥ ✮ ♦ ❢

❤ ❡ ❛ ♥ ❞ ♦ ♠ ✈ ❛ ✐ ❛ ❜ ❧ ❡ Y (AA)n,kA

❛ ♥ ❞ Y (AB)n,kA

✭ Y (BA)n,kB

❛ ♥ ❞ Y (BB)n,kB

✮ ✐ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❤ ❡ ❡ ♣ ♦ ❝ ❤ n ❛ ♥ ❞ ❤ ❡ ❛ ❣ ❡ ♥ ♥ ✉ ♠ ❜ ❡ kA ✭ kB ✮ ✳

❋ ✐ ♥ ❛ ❧ ❧ ② ✇ ❡ ❤ ❛ ✈ ❡ ✱

P XAn+1X

Bn+1

(u, v) = P XAn X

BnP (A) (u, v) , P (B) (u, v) ✭ ✸ ✽ ✮

♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❤ ❡ ❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ✇ ✐ ❧ ❧ ❣ ♦ ❡ ① ✐ ♥ ❝ ❛ ❢ ❡ n ❡ ♣ ♦ ❝ ❤ ✐ P XAn X

Bn

(0, 0)✳ ❚ ♦ ✜ ♥ ❞ P XAn X

Bn

(0, 0) ✇ ❡ ❝ ❛ ♥ ❞ ♦ ❤ ❡

❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ❝ ✉ ✐ ♦ ♥

P XAn X

Bn

(0, 0) = P XAn−1X

Bn−1

P (A) (0, 0) , P (B) (0, 0)

P XA

n−1XBn−1

(u, v) = P XAn−2X

Bn−2

P (A) (u, v) , P (B) (u, v)

P XA

n XBn

(0, 0) = P XAn−2X

Bn−2

P (A)

P (A) (0, 0) , P (B) (0, 0)

, P (B)

P (A) (0, 0) , P (B) (0, 0)

✳

✳

✳

✳

✳

✳

✳

✳

✳

P XA

n XB

n

(0, 0) = P XA

0 XB

0

(α, β )

✇ ❤ ❡ ❡ α ❛ ♥ ❞ β ❛ ❡ ♦ ♠ ❡ ❝ ♦ ♥ ❛ ♥ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❛ ❢ ❡ n ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢ ❢ ✉ ♥ ❝ ✐ ♦ ♥ P (A) (u, v) ❛ ♥ ❞ P (B) (u, v) ✳ ❆ ❤ ✐ ♣ ♦ ✐ ♥ ✐

✐ ✐ ♥ ❡ ❡ ✐ ♥ ❣ ♦ ♥ ♦ ❡ ❛ ✈ ❡ ② ✐ ♠ ♣ ♦ ❛ ♥ ❡ ❧ ❛ ✐ ♦ ♥ ✇ ❤ ✐ ❝ ❤ ✇ ❡ ✇ ✐ ❧ ❧ ✉ ❡ ✐ ♠ ♠ ❡ ❞ ✐ ❛ ❡ ❧ ② ✳

P XAn X

Bn

(0, 0) = P XA0 X

B0

(α, β ) ⇒ P XAn+1X

Bn+1

(0, 0) = P XA1 X

B1

(α, β ) ✭ ✸ ✾ ✮

❚ ❤ ✐ ✐ ❜ ❡ ❝ ❛ ✉ ❡ ♦ ❣ ❡ ❢ ♦ ♠ ❡ ♣ ♦ ❝ ❤ n + 1 ♦ ❡ ♣ ♦ ❝ ❤ 1 ✇ ❡ ❤ ❛ ✈ ❡ ♦ ❞ ♦ n ❝ ♦ ♠ ♣ ♦ ✐ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ❛ ♠ ❡ ❢ ✉ ♥ ❝ ✐ ♦ ♥ P (A) (u, v) ❛ ♥ ❞

P (B) (u, v) ❧ ✐ ❦ ❡ ✇ ❡ ❞ ✐ ❞ ♦ ❣ ❡ ❢ ♦ ♠ ❡ ♣ ♦ ❝ ❤ n ♦ ❡ ♣ ♦ ❝ ❤ 0 ✳ ▲ ❡ ❞ ❡ ✜ ♥ ❡ ✇ ♦ ✈ ❛ ✐ ❛ ❜ ❧ ❡

an10 = P

X An = 0 , X Bn = 0|X A0 = 1, X B0 = 0

✭ ✹ ✵ ✮

an01 = P

X An = 0 , X Bn = 0|X A0 = 0, X B0 = 1

✭ ✹ ✶ ✮

◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ✱ an10 ✭ an01 ✮ ✐ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❤ ❛ ❤ ❡ ❞ ❡ ❝ ❡ ♥ ❞ ❛ ♥ ♣ ♦ ❞ ✉ ❝ ❡ ❞ ❜ ② ❛ ✐ ♥ ❣ ❧ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✭ ❚ ② ♣ ❡ ✲ ❇ ✮ ✇ ✐ ❧ ❧ ❜ ❡

❡ ① ✐ ♥ ❝ ❛ ❢ ❡ n ❡ ♣ ♦ ❝ ❤ ✳ ❧ ❡ ❛ ❡ ♥ ♦ ❡ ❤ ❛ an10 ❛ ♥ ❞ an01 ❛ ❡ ❡ ① ❛ ❝ ❧ ② ❤ ❡ ✉ ❛ ♥ ✐ ✐ ❡ ❛ ❦ ❡ ❞ ✐ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ◆ ♦ ✇ ✐ ❢

X A0 = 1 ❛ ♥ ❞ X B0 = 0 ✱ ✇ ❡ ❤ ❛ ✈ ❡

an10 = P XAn X

Bn

(0, 0)

= P XA0 X

B0

(α, β )

=xB

xA

αxAβ xBP

X A0 = xA, X B0 = xB

= α1β 0P

X A0 = 1, X B0 = 0

= α1β 0 · 1 = α ✭ ✹ ✷ ✮

an01 = β ♦ ♦ ❢ ✐ ✐ ♠ ✐ ❧ ❛ ♦ an10 = α ✭ ✹ ✸ ✮

❙ ✉ ❜ ✐ ✉ ✐ ♥ ❣

α❛ ♥ ❞

β ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✷ ❛ ♥ ❞ ✹ ✸ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✾ ✇ ❡ ❣ ❡ ✱

P XAn+1X

Bn+1

(0, 0) = P XA1 X

B1

(an10, an01)

P XA1 X

B1

(u, v) = P XA0 X

B0

P (A) (u, v) , P (B) (u, v)

P XA

n+1XBn+1

(0, 0) = P XA0 X

B0

P (A) (an10, an01) , P (B) (an10, an01)

✭ ✹ ✹ ✮

❆ ♣ ♣ ❧ ② ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✷ ❛ ♥ ❞ ✹ ✸ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✹ ✇ ❡ ❣ ❡

an+110 = P (A) (an10, an01) ✭ ✹ ✺ ✮

an+101 = P (B) (an10, an01) ✭ ✹ ✻ ✮

✶ ✸



❊ ✉ ❛ ✐ ♦ ♥ ✹ ✺ ❛ ♥ ❞ ✹ ✻ ❞ ❡ ✜ ♥ ❡ ❛ ❡ ❝ ✉ ✐ ✈ ❡ ❡ ❧ ❛ ✐ ♦ ♥ ❤ ✐ ♣ ✇ ❤ ✐ ❝ ❤ ✇ ❡ ❝ ❛ ♥ ✉ ❡ ♦ ❣ ❡ an10 ❛ ♥ ❞ an01 ✳ ❍ ♦ ✇ ❡ ✈ ❡ ♦ ❛ ❤ ❡ ❡ ❝ ✉ ✐ ♦ ♥

✇ ❡ ♥ ❡ ❡ ❞ a010 ❛ ♥ ❞ a001 ✳ ■ ✐ ✐ ✈ ✐ ❛ ❧ ♦ ♦ ❜ ❡ ✈ ❡ ❤ ❛ a010 = 0 ✭ a001 = 0 ✮ ❛ X A0 = 1 ❛ ♥ ❞ X B0 = 0 ✭ X A0 = 0 ❛ ♥ ❞ X B0 = 1✮ ✳

❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❉ ❝ ♦ ♥ ❛ ✐ ♥ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ✇ ❤ ✐ ❝ ❤ ✐ ♠ ♣ ❧ ❡ ♠ ❡ ♥ ❤ ✐ ❡ ❝ ✉ ✐ ♦ ♥ ♦ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ an10 ❛ ♥ ❞ an01 ✳

▲ ❊ ❆ ❙ ❊ ◆ ❖ ❚ ❊ ✿ ❚ ❤ ✐ ♣ ❛ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ❛ ❧ ♦ ❛ ❦ ✉ ♦ ✏ ❝ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ✐ ❡ ❢ ♦ ❤ ❡

❡ ① ❛ ♠ ♣ ❧ ❡ ✇ ❡ ✐ ♠ ✉ ❧ ❛ ❡ ❞ ✐ ♥ ❛ ✭ ❜ ✮ ✑ ✳ ❲ ❡ ✇ ✐ ❧ ❧ ❞ ❡ ❢ ❡ ❤ ✐ ♦ ❛ ✭ ❜ ✮ ✇ ❤ ✐ ❝ ❤ ✇ ❡ ✇ ✐ ❧ ❧ ❛ ♥ ✇ ❡ ❜ ❡ ❧ ♦ ✇ ✳

✭ ❡ ✮ ▲ ❡ ✜ ❞ ❡ ✜ ♥ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✱

a pq = P X An = 0 , X Bn = 0 ❢ ♦ ♦ ♠ ❡ n > 0|X A0 = p, X B0 = q ◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ♣ ❡ ❛ ❦ ✐ ♥ ❣ ✱ a pq ✐ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❛ ✐ ♥ ❣ ❢ ♦ ♠ p ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ q ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳ ❚ ❤ ❡ ❦ ❡ ②

❞ ✐ ✛ ❡ ❡ ♥ ❝ ❡ ❜ ❡ ✇ ❡ ❡ ♥ a10 ✭ a01 ✮ ❛ ♥ ❞ an10 ✭ an01 ✮ ✐ ❤ ❛ a10 ✭ a01 ✮ ❞ ♦ ❡ ♥ ♦ ♣ ❡ ❝ ✐ ❢ ② ❤ ❡ ✐ ♠ ❡ ♦ ❢ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ✳ ■ ♥ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✇ ❡ ✇ ✐ ❧ ❧

✜ ❤ ♦ ✇ ❤ ❛ a pq = (a10) p

(a01)q

✳

a pq = P

pi=1

{❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ✐ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❣ ♦ ❡ ① ✐ ♥ ❝ }

❛ ♥ ❞

qj=1

{❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ❥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❣ ♦ ❡ ① ✐ ♥ ❝ }

= P

p

i=1

{❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ✐ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❣ ♦ ❡ ① ✐ ♥ ❝ }· P

qj=1

{❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ❥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❣ ♦ ❡ ① ✐ ♥ ❝ }

✭ ✹ ✼ ✮

=

pi=1

P ( ❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ✐ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❣ ♦ ❡ ① ✐ ♥ ❝ )

·

qj=1

P ( ❉ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ✐ ♥ ✐ ✐ ❛ ❧ ❛ ❣ ❡ ♥ ★ ❥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❣ ♦ ❡ ① ✐ ♥ ❝ ) ✭ ✹ ✽ ✮

=

p

i=1

a10

q

j=1

a01

= (a10) p

(a01)q

✭ ✹ ✾ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✹ ✼ ❛ ♥ ❞ ✹ ✽ ✉ ❡ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ❝ ❡ ✱ ♠ ♦ ❡ ♣ ❡ ❝ ✐ ❡ ❧ ② ✱ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ✐ ✐ ♥ ❞ ❡ ♣ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❤ ❡

♥ ✉ ♠ ❜ ❡ ♦ ❢ ❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❛ ♥ ♦ ❤ ❡ ❛ ❣ ❡ ♥ ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✹ ✾ ✉ ❡ ❤ ❡ ❢ ❛ ❝ ❤ ❛ ❤ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❞ ✐ ✐ ❜ ✉ ✐ ♦ ♥ ♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢

❞ ❡ ❝ ❡ ♥ ❞ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✭ ❚ ② ♣ ❡ ✲ ❇ ✮ ❛ ❣ ❡ ♥ ❞ ♦ ❡ ♥ ♦ ✈ ❛ ② ✇ ✐ ❤ ❛ ❣ ❡ ♥ ♥ ✉ ♠ ❜ ❡ ✳ ◆ ♦ ✇ ❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✱

a10 = P

∞n=1

X An = 0, X Bn = 0

|X A0 = 1, X B0 = 0

✭ ✺ ✵ ✮

=∞j=0

∞i=0

P

∞n=1

X An = 0, X Bn = 0

|X A1 = i, X B1 = j, X A0 = 1, X B0 = 0

· P X A1 = i, X B1 = j |X A0 = 1, X B0 = 0 ✭ ✺ ✶ ✮

=∞j=0

∞i=0

P

∞n=1

X An = 0, X Bn = 0

|X A1 = i, X B1 = j, X A0 = 1, X B0 = 0

p(A) (i, j)

=∞j=0

∞i=0

P

∞n=1

X An = 0, X Bn = 0

|X A1 = i, X B1 = j

p(A) (i, j)

✭ ✺ ✷ ✮

=∞j=0

∞i=0

aij p

(A) (i, j)

✭ ✺ ✸ ✮

=∞j=0

∞i=0

(a10)

i(a01)

j p(A) (i, j)

= P (A) (a10, a01) ✭ ✺ ✹ ✮

✶ ✹



❚ ♦ ❣ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✶ ✱ ✇ ❡ ✉ ❡ ❤ ❡ ▲ ❛ ✇ ♦ ❢ ❚ ♦ ❛ ❧ ❊ ① ♣ ❡ ❝ ❛ ✐ ♦ ♥ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❡ ❞ ♦ ♥ ❤ ❡ ❢ ❛ ❝ ❤ ❛ X A1 = i ❛ ♥ ❞ X B1 = j ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✺ ✷

✐ ❤ ❡ ❡ ✉ ❧ ♦ ❢ ❤ ❡ ▼ ❛ ❦ ♦ ✈ ♦ ♣ ❡ ② ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✺ ✸ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ② ✐ ♠ ❡ ❤ ✐ ❢ ✐ ♥ ❣ n → n + 1 ✳ ❊ ✉ ❛ ✐ ♦ ♥ ✺ ✹ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ②

✉ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✾ ✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ② ✇ ❡ ❝ ❛ ♥ ♣ ♦ ✈ ❡ ❤ ❛ a01 = P (B) (a10, a01)✳ ❲ ❡ ✜ ♥ ❛ ❧ ❧ ② ❤ ❛ ✈ ❡ ✱

a10 = P (A) (a10, a01) ✭ ✺ ✺ ✮

a01 = P (B) (a10, a01) ✭ ✺ ✻ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✺ ✺ ❛ ♥ ❞ ✺ ✻ ✐ ❛ ② ❡ ♠ ♦ ❢ ♥ ♦ ♥ ❧ ✐ ♥ ❡ ❛ ❡ ✉ ❛ ✐ ♦ ♥ ✇ ✐ ❤ ✇ ♦ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✇ ❤ ✐ ❝ ❤ ❝ ❛ ♥ ❜ ❡ ♦ ❧ ✈ ❡ ❞ ♦ ♦ ❜ ❛ ✐ ♥ a10 ❛ ♥ ❞ a01 ✳

❧ ❡ ❛ ❡ ♥ ♦ ❡ ❤ ❛ a10 ❛ ♥ ❞ a01 ❛ ❡ ❡ ① ❛ ❝ ❧ ② ❤ ❡ ✉ ❛ ♥ ✐ ✐ ❡ ❛ ❦ ❡ ❞ ✐ ♥ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ❖ ♥ ❡ ✇ ❛ ② ♦ ♦ ❧ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✺

❛ ♥ ❞ ✺ ✻ ✐ ♦ ✉ ❡ ❤ ❡ ♦ ❧ ✈ ❡ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ▼ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ✳ ❍ ♦ ✇ ❡ ✈ ❡ ❤ ❡ ❡ ② ❡ ♠ ♦ ❢ ❡ ✉ ❛ ✐ ♦ ♥ ♠ ❛ ② ❤ ❛ ✈ ❡ ♠ ✉ ❧ ✐ ♣ ❧ ❡ ♦ ❧ ✉ ✐ ♦ ♥ ✳ ❖ ♥ ❡

✐ ✈ ✐ ❛ ❧ ♦ ❧ ✉ ✐ ♦ ♥ ✐ a10 = a01 = 1 ✳ ❚ ❤ ✐ ✐ ❜ ❡ ❝ ❛ ✉ ❡ P (A) (1, 1) = P (B) (1, 1) = 1 ✳ ❚ ❤ ❡ ❡ ❝ ❛ ♥ ❜ ❡ ♦ ❤ ❡ ♥ ♦ ♥ ✲ ✐ ✈ ✐ ❛ ❧ ♦ ❧ ✉ ✐ ♦ ♥

♦ ♦ ✳ ● ✐ ✈ ❡ ♥ ❤ ❛ ✐ ✐ ❛ ② ❡ ♠ ♦ ❢ ♥ ♦ ♥ ❧ ✐ ♥ ❡ ❛ ❡ ✉ ❛ ✐ ♦ ♥ ✐ ♥ ✇ ♦ ✈ ❛ ✐ ❛ ❜ ❧ ❡ ✱ ✐ ✐ ❞ ✐ ✣ ❝ ✉ ❧ ♦ ✉ ♥ ❞ ❡ ❛ ♥ ❞ ❤ ❡ ❣ ❡ ♦ ♠ ❡ ② ♦ ❢ ❤ ❡

❛ ♦ ❝ ✐ ❛ ❡ ❞ ❣ ❛ ♣ ❤ ❡ ✉ ✐ ❡ ❞ ♦ ♦ ❧ ✈ ❡ ❤ ❡ ♣ ♦ ❜ ❧ ❡ ♠ ✳ ■ ♥ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✇ ❡ ✇ ✐ ❧ ❧ ✉ ❣ ❣ ❡ ❛ ♥ ♦ ❤ ❡ ✇ ❛ ② ♦ ♦ ❧ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✺ ❛ ♥ ❞

✺ ✻ ❜ ② ❞ ❡ ✈ ❡ ❧ ♦ ♣ ✐ ♥ ❣ ❛ ❡ ❧ ❛ ✐ ♦ ♥ ❜ ❡ ✇ ❡ ❡ ♥ a10 ❛ ♥ ❞ an10 ✭ a01 ❛ ♥ ❞ an01 ✮ ✳

❊ ✉ ❛ ✐ ♦ ♥ ✺ ✵ ❝ ❛ ♥ ❛ ❧ ♦ ❜ ❡ ❡ ① ♣ ❡ ❡ ❞ ❛

a10 = limN →∞

P

N n=1

X An = 0, X Bn = 0

|X A0 = 1, X B0 = 0

✭ ✺ ✼ ✮

■ ❢ X A

n

= 0 ❛ ♥ ❞ X B

n

= 0 ❢ ♦ ❛ ❣ ✐ ✈ ❡ ♥ n✱ ❤ ❡ ♥ ❢ ♦ ❛ ♥ ② n ≥ n ✇ ❡ ❝ ❛ ♥ ✐ ♠ ♠ ❡ ❞ ✐ ❛ ❡ ❧ ② ❛ ② X A

n

= 0 ❛ ♥ ❞ X B

n

= 0 ✳ ❚ ❤ ✐ ✐ ❜ ❡ ❝ ❛ ✉ ❡

(0, 0) ✐ ❛ ♥ ❛ ❜ ♦ ❜ ✐ ♥ ❣ ❛ ❡ ✳ ❚ ❤ ✐ ✐ ♠ ❛ ❤ ❡ ♠ ❛ ✐ ❝ ❛ ❧ ❧ ② ❡ ① ♣ ❡ ❡ ❞ ❛

X An = 0, X Bn = 0

⊆

X An = 0, X Bn = 0

; n ≥ n

⇒N n=1

X An = 0, X Bn = 0

=

X AN = 0, X BN = 0

✭ ✺ ✽ ✮

❙ ✉ ❜ ✐ ✉ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✽ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✺ ✼ ✇ ❡ ❣ ❡

a10 = limN →∞

P

X AN = 0, X BN = 0

|X A0 = 1, X B0 = 0

= limN →∞ aN

10✭ ✺ ✾ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✺ ✾ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❢ ♦ ♠ ❤ ❡ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥ ♦ ❢ an10 ✭ ❡ ❢ ❡ ❡ ✉ ❛ ✐ ♦ ♥ ✹ ✵ ✮ ✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ② ✇ ❡ ❝ ❛ ♥ ♣ ♦ ✈ ❡ ❤ ❛

a01 = limN →∞

aN 01 ✭ ✻ ✵ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✺ ✾ ❛ ♥ ❞ ✻ ✵ ✉ ❣ ❣ ❡ ❤ ❛ ✇ ❡ ❝ ❛ ♥ ✜ ♥ ❞ a10 ❛ ♥ ❞ a01 ❜ ② ✐ ♠ ♣ ❧ ② ❝ ❛ ❧ ❝ ✉ ❧ ❛ ✐ ♥ ❣ aN 10 ❛ ♥ ❞ aN 01 ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ❢ ♦ ❧ ❛ ❣ ❡ n✳ ❲ ❡

❝ ❛ ♥ ❞ ♦ ❤ ✐ ❜ ② ✉ ✐ ♥ ❣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❡ ❝ ✉ ✐ ✈ ❡ ❡ ❧ ❛ ✐ ♦ ♥ ❞ ❡ ✈ ❡ ❧ ♦ ♣ ❡ ❞ ✐ ♥ ❛ ✭ ❞ ✮

an+110 = P (A) (an10, an01) ✭ ✻ ✶ ✮

an+101 = P (B) (an10, an01) ✭ ✻ ✷ ✮

❖ ❜ ❡ ✈ ❡ ❤ ❛ ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✶ ❛ ♥ ❞ ✻ ✷ ❞ ❡ ✜ ♥ ❡ ❛ ♥ ♦ ♥ ❧ ✐ ♥ ❡ ❛ ❞ ✐ ❝ ❡ ❡ ✐ ♠ ❡ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ✇ ✐ ❤ ❛ ❡ (an10, an01)✳ ❖ ♥ ❡ ❝ ❛ ♥

✉ ❡ ✐ ♦ ♥ ❤ ❡ ❛ ❜ ✐ ❧ ✐ ② ❛ ♥ ❞ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ♥ ❝ ❡ ♦ ❢ ❤ ✐ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ✳ ❚ ♦ ♣ ♦ ✈ ❡ ❛ ❜ ✐ ❧ ✐ ② ✇ ❡ ❝ ❛ ♥ ❤ ♦ ✇ ❤ ❛ ❤ ❡ ❡

S = {(x, y) : 0 ≤ x ≤ 1 ; 0 ≤ y ≤ 1}

✐ ✐ ♥ ✈ ❛ ✐ ❛ ♥ ✳ ❚ ♦ ♣ ♦ ✈ ❡ ❤ ✐ ❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❛ ❣ ✉ ♠ ❡ ♥ ✳ ❋ ✐ ♥ ♦ ❡ ❤ ❛ a010 = 0 ❛ ♥ ❞ a001 = 0 ✱ ✐ ✳ ❡

a010, a001

∈ S ✳ ❙ ❛ ②

❤ ❛ (an10, an01) ∈ S ✳ ◆ ♦ ✇

an+110 = P (A) (an10, an01)

= E

(an10)Y (AA)

(an01)Y (AB)

✭ ✻ ✸ ✮

✶ ✺



◆ ♦ ✇ 0 ≤ an10 ≤ 1 ❛ ♥ ❞ 0 ≤ an01 ≤ 1 ✐ ♥ ❝ ❡ (an10, an01) ∈ S ✳ ❆ ❧ ♦ Y (AA) ≥ 0 ❛ ♥ ❞ Y (AB) ≥ 0 ✳ ❚ ❤ ✐ ✐ ♠ ♣ ❧ ✐ ❡ ❤ ❛

0 ≤ (an10)Y (AA)

· (an01)Y (AB)

≤ 1 ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ 0 ≤ an+110 = E

(an10)

Y (AA)

(an01)Y (AB)

≤ 1 ✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ② ✇ ❡ ❝ ❛ ♥ ♣ ♦ ✈ ❡ ❤ ❛

0 ≤ an+101 ≤ 1 ✳ ❍ ❡ ♥ ❝ ❡

an+110 , an+1

01

∈ S ✳ ❲ ❡ ✜ ♥ ❛ ❧ ❧ ② ❝ ♦ ♥ ❝ ❧ ✉ ❞ ❡ ❤ ❛ ✐ ❢

a010, a001

∈ S ❤ ❡ ♥ (an10, an01) ∈ S ; ∀n ≥ 0✳ ❚ ❤ ✐

♣ ♦ ✈ ❡ ❤ ❛ S ✐ ❛ ♥ ✐ ♥ ✈ ❛ ✐ ❛ ♥ ❡ ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ an10 ❛ ♥ ❞ an01 ✐ ❧ ♦ ✇ ❡ ❛ ♥ ❞ ✉ ♣ ♣ ❡ ❜ ♦ ✉ ♥ ❞ ❡ ❞ ❜ ② 0 ❛ ♥ ❞ 1 ❡ ♣ ❡ ❝ ✐ ✈ ❡ ❧ ② ✳ ❚ ❤ ✐ ❝ ♦ ♠ ♣ ❧ ❡ ❡

❤ ❡ ♣ ♦ ♦ ❢ ❤ ❛ ❤ ❡ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ❞ ❡ ✜ ♥ ❡ ❞ ❜ ② ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✶ ❛ ♥ ❞ ✻ ✷ ✐ ❛ ❜ ❧ ❡ ✳

❍ ♦ ✇ ❡ ✈ ❡ ✱ ❡ ✈ ❡ ♥ ✐ ❢ ❤ ❡ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ✐ ❛ ❜ ❧ ❡ ✐ ♠ ❛ ② ♥ ♦ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ✳ ❘ ❛ ❤ ❡ ✱ ✐ ♠ ❛ ② ❣ ♦ ✐ ♥ ♦ ❛ ❧ ✐ ♠ ✐ ❝ ② ❝ ❧ ❡ ✳ ■ ❤ ♦ ✉ ❧ ❞ ❜ ❡

♣ ♦ ✐ ❜ ❧ ❡ ♦ ♣ ♦ ✈ ❡ ❤ ❛ ❤ ✐ ❞ ② ♥ ❛ ♠ ✐ ❝ ❛ ❧ ② ❡ ♠ ❞ ♦ ❡ ❝ ♦ ♥ ✈ ❡ ❣ ❡ ✱ ❜ ✉ ✇ ❡ ❝ ♦ ✉ ❧ ❞ ♥ ♦ ♣ ♦ ✈ ❡ ✐ ✳

❚ ❤ ❡ ▼ ❆ ❚ ▲ ❆ ❇ ❝ ♦ ❞ ❡ ♦ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ a10 ❛ ♥ ❞ a01 ✐ ✐ ♥ ❝ ❧ ✉ ❞ ❡ ❞ ✐ ♥ ❆ ♣ ♣ ❡ ♥ ❞ ✐ ① ✲ ❉ ✳

✭ ❢ ✮ ■ ♥ ▲ ❡ ❝ ✉ ❡ ✶ ✻ ✱ ✇ ❡ ❧ ❡ ❛ ♥ ❡ ❞ ❛ ❜ ♦ ✉ ❤ ♦ ✇ ♦ ✐ ♥ ❢ ❡ ✱ ✐ ❢ ❛ ❜ ❛ ♥ ❝ ❤ ✐ ♥ ❣ ♣ ♦ ❝ ❡ ✐ ❣ ♦ ✐ ♥ ❣ ♦ ❡ ① ✐ ♥ ❝ ✱ ❜ ② ❝ ❤ ❡ ❝ ❦ ✐ ♥ ❣ ❤ ❡ ❡ ① ♣ ❡ ❝ ❛ ♥ ❝ ②

♦ ❢ ❤ ❡ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✛ ♣ ✐ ♥ ❣ ✳ ❲ ❡ ✇ ♦ ✉ ❧ ❞ ❧ ✐ ❦ ❡ ♦ ❞ ♦ ❤ ❡ ❛ ♠ ❡ ❤ ❡ ❡ ✳ ❙ ♦ ✇ ❡ ❝ ❛ ❧ ❝ ✉ ❧ ❛ ❡ ❤ ❡ ❡ ① ♣ ❡ ❝ ❛ ♥ ❝ ② ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇

❛ ❣ ❡ ♥ ❛ ❤ ❡ nth ❡ ♣ ♦ ❝ ❤

E

X An+1

=

u

d

du

P XA

n+1XBn+1

(u, 1)u=1

= u d

du

P XA

n XBn

P (A) (u, 1) P (A) (1, 1) , P (B) (u, 1) P (B) (1, 1)

u=1

✭ ✻ ✹ ✮

= u

d

du P XAn X

Bn P

(A)

(u, 1) , P

(B)

(u, 1)u=1

= u d

du

P XA

n XBn

(X (u) , Y (u))u=1

✇ ❤ ❡ ❡ X (u) = P (A) (u, 1) , Y (u) = P (B) (u, 1)

=

u

dX

du · P X XA

n XBn

(X (u) , Y (u)) + udY

du · P Y

XAn X

Bn

(X (u) , Y (u))

u=1

✭ ✻ ✺ ✮

= E

Y (AA)

· P X XAn X

Bn

(1, 1) + E

Y (BA)

· P Y XAn X

Bn

(1, 1)

= E

Y (AA)

· E

X An

+ E

Y (BA)

· E

X Bn

✭ ✻ ✻ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✻ ✹ ✐ ♦ ❜ ❛ ✐ ♥ ❡ ❞ ❜ ② ✉ ❜ ✐ ✉ ✐ ♥ ❣ (u, v) = (u, 1) ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✸ ✽ ✳ ■ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✺ P X XAn X

Bn

(X (u) , Y (u)) ✭ P Y XAn X

Bn

(X (u) ,

♠ ❡ ❛ ♥ ♣ ❛ ✐ ❛ ❧ ❞ ✐ ✛ ❡ ❡ ♥ ✐ ❛ ✐ ♦ ♥ ♦ ❢ P XAn X

Bn

✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ X ✭ Y ✮ ✳ ❙ ✐ ♠ ✐ ❧ ❛ ❧ ② ✇ ❡ ❝ ❛ ♥ ♣ ♦ ✈ ❡

E X Bn+1 = E Y (AB)E X An + E Y (BB)E X Bn ✭ ✻ ✼ ✮

❈ ♦ ♠ ❜ ✐ ♥ ✐ ♥ ❣ ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✻ ❛ ♥ ❞ ✻ ✼ ✇ ❡ ❣ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♠ ❛ ✐ ① ❡ ✉ ❛ ✐ ♦ ♥ E

X An+1

E

X Bn+1

=

E

Y (AA)

E

Y (BA)

E

Y (AB)

E

Y (BB) E

X An

E

X Bn

▲ ❡ µAA = E

Y (AA)

✱ µAB = E

Y (AB)

✱ µBA = E

Y (BA)

❛ ♥ ❞ µBB = E

Y (BB)

✳ ❙ ✉ ❜ ✐ ✉ ✐ ♥ ❣ ✐ ♥ ❤ ❡ ❛ ❜ ♦ ✈ ❡ ❡ ✉ ❛ ✐ ♦ ♥

✇ ❡ ❣ ❡ E

X An+1

E

X Bn+1

=

µAA µBA

µAB µBB

E

X An

E

X Bn

✭ ✻ ✽ ✮

❊ ✉ ❛ ✐ ♦ ♥ ✻ ✽ ❞ ❡ ✜ ♥ ❡ ❛ ▲ ❚ ■ ② ❡ ♠ ✇ ✐ ❤ ✇ ♦ ❛ ❡ ✳ ❚ ❤ ❡ ❡ ✐ ❣ ❡ ♥ ✈ ❛ ❧ ✉ ❡ ♦ ❢ ❤ ❡ ② ❡ ♠ ❛ ❡

λM =

µAA + µBB

+

(µAA − µBB)

2+ 4µBAµAB

2✭ ✻ ✾ ✮

λm =

µAA + µBB

−

(µAA − µBB)

2+ 4µBAµAB

2✭ ✼ ✵ ✮

λM ❛ ♥ ❞ λm ❞ ❡ ✜ ♥ ❡ ❞ ❜ ② ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✾ ❛ ♥ ❞ ✼ ✵ ❤ ❛ ✈ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ♣ ♦ ♣ ❡ ✐ ❡ ✿

✶ ✮ λM ❛ ♥ ❞ λm ❛ ❡ ❛ ❧ ✇ ❛ ② ❡ ❛ ❧ ✳

✷ ✮ λM ≥ 0✸ ✮ λM ≥ |λm| ✳

✶ ✻



▲ ❡ X A0 = p ❛ ♥ ❞ X B0 = q ✐ ♠ ♣ ❧ ② ✐ ♥ ❣ E

X A0

= p ❛ ♥ ❞ E

X B0

= q ✳ ❖ ♥ ❡ ❝ ❛ ♥ ❤ ♦ ✇ ❤ ❛ ❤ ❡ ❣ ❡ ♥ ❡ ❛ ❧ ✉ ❝ ✉ ❡ ♦ ❢ ❤ ❡ ✐ ♠ ❡

❡ ✈ ♦ ❧ ✉ ✐ ♦ ♥ ♦ ❢ E

X An

❛ ♥ ❞ E

X Bn

✐

E

X An

= α1 (λM )n

p + α2 (λm)n

q ✭ ✼ ✶ ✮

E

X Bn

= β 1 (λM )n

p + β 2 (λm)n

q ✭ ✼ ✷ ✮

❚ ❛ ❦ ✐ ♥ ❣ ❤ ❡ ❧ ♣ ❢ ♦ ♠ ❧ ❡ ❝ ✉ ❡ ✶ ✻ ✭ ♣ ❛ ❣ ❡ ✸ ✮ ✇ ❡ ❞ ❡ ✈ ❡ ❧ ♦ ♣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✉ ❧ ❡ ✿

✶ ✮ ■ ❢ limn→∞

E X An = 0 ✭ limn→∞

E X Bn = 0 ✮ ❤ ❡ ♥ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✭ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✮ ❲ ■ ▲ ▲ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳

✷ ✮ ■ ❢ limn→∞

E

X An

= θ ✭ limn→∞

E

X Bn

= θ ✮ ✱ ✇ ❤ ❡ ❡ θ ✐ ❛ ✜ ♥ ✐ ❡ ❝ ❛ ❧ ❛ ✉ ❛ ♥ ✐ ② ✱ ❤ ❡ ♥ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✭ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✮ ❲ ■ ▲ ▲

❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳

✸ ✮ ■ ❢ limn→∞

E

X An

= ∞ ✭ limn→∞

E

X Bn

= ∞ ✮ ❤ ❡ ♥ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✭ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✮ ▼ ❆ ❨ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳ ■ ✐ ✏ ▼ ❆ ❨ ✑ ❜ ❡ ❝ ❛ ✉ ❡

✐ ♥ ❤ ✐ ❝ ❛ ❡ ❤ ❡ ❡ ✐ ❛ ✜ ♥ ✐ ❡ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ♦ ❢ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ✳

❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❜ ♦ ❤ ❤ ❡ ❛ ❣ ❡ ♥ ❲ ■ ▲ ▲ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦ ♥ ✿

❚ ❤ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦ ♥

λM ≤ 1 ⇒µAA + µBB+ (µAA − µBB)2 + 4µBAµAB

2 ≤ 1 ✭ ✼ ✸ ✮

■ ❢ λM < 1 ❤ ❡ ♥ ❢ ♦ ♠ ❡ ✉ ❛ ✐ ♦ ♥ ✼ ✶ ❛ ♥ ❞ ✼ ✷ ✇ ❡ ❝ ❛ ♥ ❡ ❡ ❤ ❛ limn→∞

E

X An

= 0 ❛ ♥ ❞ limn→∞

E

X Bn

= 0✳ ❍ ❡ ♥ ❝ ❡ ❜ ♦ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞

❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳

■ ❢ λM = 1 ❤ ❡ ♥ limn→∞

E

X An

= θ ❛ ♥ ❞ limn→∞

E

X Bn

= φ ✱ ✇ ❤ ❡ ❡ θ ❛ ♥ ❞ φ ❛ ❡ ✜ ♥ ✐ ❡ ❝ ❛ ❧ ❛ ✉ ❛ ♥ ✐ ② ✳ ❍ ❡ ♥ ❝ ❡ ❜ ♦ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞

❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳

❧ ❡ ❛ ❡ ♥ ♦ ❡ ❤ ❛ ✐ ♥ ❡ ✉ ❛ ❧ ✐ ② ✼ ✸ ✐ ❤ ❡ ❝ ✐ ❡ ✐ ❛ ❢ ♦ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ✇ ❤ ✐ ❝ ❤ ✐ ❛ ❦ ❡ ❞ ✐ ♥ ❛ ✭ ❢ ✮ ✳ ❲ ❡ ❛ ❧ ♦ ❞ ❡ ✐ ✈ ❡ ❢ ❡ ✇

♦ ❤ ❡ ❝ ✐ ❡ ✐ ❛ ❜ ❡ ❧ ♦ ✇ ♦ ❤ ❡ ❧ ♣ ✉ ✇ ✐ ❤ ❤ ❡ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ✐ ♥ ❛ ✭ ❜ ✮ ✳

❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❜ ♦ ❤ ❤ ❡ ❛ ❣ ❡ ♥ ▼ ❆ ❨ ✢ ♦ ✉ ✐ ❤ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦ ♥ ✿

❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❤ ❛ ❜ ♦ ❤ ❛ ❣ ❡ ♥ ▼ ❆ ❨ ✢ ♦ ✉ ✐ ❤ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦ ♥ ✐

λM > 1 or |λm| > 1 ✭ ✼ ✹ ✮

❯ ♥ ❞ ❡ ❛ ♥ ② ♦ ❢ ❤ ❡ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ limn→∞

E

X An

= ∞ ❛ ♥ ❞ limn→∞

E

X Bn

= ∞ ✳ ❍ ❡ ♥ ❝ ❡ ❜ ♦ ❤ ❛ ❣ ❡ ♥ ♠ ❛ ② ✢ ♦ ✉ ✐ ❤ ✳

❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ❲ ■ ▲ ▲ ❡ ① ✐ ♥ ❝ ❜ ✉ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ▼ ❆ ❨ ✢ ♦ ✉ ✐ ❤ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦

❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❤ ❛ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❡ ① ✐ ♥ ❝ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ♠ ❛ ② ✢ ♦ ✉ ✐ ❤ ✐

µBA = 0 and µAA ≤ 1 and µBB > 1 ✭ ✼ ✺ ✮

❇ ② ❛ ♣ ♣ ❧ ② ✐ ♥ ❣ ❤ ❡ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✻ ✽ ✇ ❡ ❝ ❛ ♥ ❝ ♦ ♥ ❝ ❧ ✉ ❞ ❡ ❤ ❛ ✿

✶ ✮ limn→∞

E

X An

= 0 ✐ ❢ µAA < 1 ✐ ♠ ♣ ❧ ② ✐ ♥ ❣ ❤ ❛ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳ ■ ❢ µAA = 1 ❤ ❡ ♥ limn→∞

E

X An

= θ ❛ ❣ ❛ ✐ ♥

✐ ♠ ♣ ② ❧ ✐ ♥ ❣ ❤ ❛ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✇ ✐ ❧ ❧ ❜ ❡ ❝ ♦ ♠ ❡ ❡ ① ✐ ♥ ❝ ✳

✷ ✮ limn→∞

E

X Bn

= ∞ ✐ ♠ ♣ ❧ ② ✐ ♥ ❣ ❤ ❛ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ♠ ❛ ② ✢ ♦ ✉ ✐ ❤ ✳

◆ ❖ ❚ ❊ ✿ µBA = 0 ✐ ♠ ♣ ❧ ✐ ❡ ❤ ❛ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❞ ♦ ❡ ♥ ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✳

❈ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❢ ♦ ✇ ❤ ✐ ❝ ❤ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❲ ■ ▲ ▲ ❡ ① ✐ ♥ ❝ ❜ ✉ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ▼ ❆ ❨ ✢ ♦ ✉ ✐ ❤ ❢ ♦ ❛ ♥ ② ✐ ♥ ✐ ✐ ❛ ❧ ♣ ♦ ♣ ✉ ❧ ❛ ✐ ♦

❚ ❤ ✐ ✐ ✐ ♠ ✐ ❧ ❛ ♦ ❤ ❡ ♣ ❡ ✈ ✐ ♦ ✉ ❝ ❛ ❡ ✳ ❚ ❤ ❡ ❡ ✉ ✐ ❡ ❞ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ✐

✶ ✼



µAB = 0 and µBB ≤ 1 and µAA > 1 ✭ ✼ ✻ ✮

◆ ❖ ❚ ❊ ✿ µAB = 0 ✐ ♠ ♣ ❧ ✐ ❡ ❤ ❛ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ❞ ♦ ❡ ♥ ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✳

✭ ❜ ✮ ■ ♥ ❤ ✐ ♣ ❛ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ✇ ❡ ❞ ♦ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✿

• ❉ ♦ ▼ ♦ ♥ ❡ ❈ ❛ ❧ ♦ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❢ ♦ ✉ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ❞ ✐ ✛ ❡ ❡ ♥ ❝ ❛ ❡ ✿ ✐ ✮ ❇ ♦ ❤ ❆ ❣ ❡ ♥ ❡ ① ✐ ♥ ❝ ✳ ✐ ✐ ✮ ❇ ♦ ❤ ❛ ❣ ❡ ♥

✢ ♦ ✉ ✐ ❤ ❡ ✳ ✐ ✐ ✐ ✮ ❚ ② ♣ ❡ ✲ ❆ ❡ ① ✐ ♥ ❝ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ✢ ♦ ✉ ✐ ❤ ❡ ✳ ✐ ✈ ✮ ❚ ② ♣ ❡ ✲ ❇ ❡ ① ✐ ♥ ❝ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❆ ✢ ♦ ✉ ✐ ❤ ❡ ✳

• ❈ ♦ ♠ ♠ ❡ ♥ ❤ ♦ ✇ ❤ ❡ ♠ ❡ ❛ ♥ ♥ ✉ ♠ ❜ ❡ ♦ ❢ ♦ ✛ ♣ ✐ ♥ ❣ ♦ ❢ ❡ ❛ ❝ ❤ ② ♣ ❡ ✇ ✐ ❧ ❧ ❛ ✛ ❡ ❝ ✇ ❤ ✐ ❝ ❤ ❝ ❛ ❡ ✇ ✐ ❧ ❧ ❜ ❡ ♦ ❜ ❡ ✈ ❡ ❞ ✳ ❲ ❡ ❤ ❛ ✈ ❡ ❛ ❧ ❡ ❛ ❞ ②

❛ ♥ ✇ ❡ ❡ ❞ ❤ ✐ ✐ ♥ ❛ ✭ ❢ ✮ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ✳ ■ ♥ ❞ ❡ ❡ ❞ ✱ ✇ ❡ ✉ ❡ ❤ ❡ ❡ ✉ ❧ ❢ ♦ ♠ ♣ ❛ ✭ ❢ ✮ ♦ ❣ ❡ ♥ ❡ ❛ ❡ ❤ ❡ ❢ ♦ ✉ ❞ ✐ ✛ ❡ ❡ ♥

❝ ❛ ❡ ✳

• ❈ ♦ ♠ ♣ ✉ ❡ ❤ ❡ ❡ ① ✐ ♥ ❝ ✐ ♦ ♥ ♣ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ✐ ❡ ❢ ♦ ❤ ❡ ❢ ♦ ✉ ❞ ✐ ✛ ❡ ❡ ♥ ❝ ❛ ❡ ✳ ❚ ❤ ✐ ✇ ❛ ❛ ❦ ❡ ❞ ✐ ♥ ❛ ✭ ❞ ✮ ♦ ❢ ❤ ❡ ✉ ❡ ✐ ♦ ♥ ✳

❲ ❡ ❛ ♥ ❞ ♦ ♠ ❧ ② ❣ ❡ ♥ ❡ ❛ ❡ ❞ ❢ ♦ ✉ ❡ ♦ ❢ p(A) ❛ ♥ ❞ p(B)❡ ❛ ❝ ❤ ❝ ♦ ❡ ♣ ♦ ♥ ❞ ✐ ♥ ❣ ♦ ♦ ♥ ❡ ♦ ❢ ❤ ❡ ❢ ♦ ✉ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ✇ ❤ ✐ ❝ ❤ ✇ ❡ ❞ ❡ ✐ ✈ ❡ ❞ ✐ ♥

❛ ✭ ❢ ✮ ♦ ❢ ❤ ✐ ✉ ❡ ✐ ♦ ♥ ✳ ❍ ♦ ✇ ❡ ✈ ❡ ✇ ❡ ❞ ✐ ❞ ✐ ♠ ♣ ♦ ❡ ♦ ♠ ❡ ❡ ✐ ❝ ✐ ♦ ♥ ♦ ♥ ❤ ❡ ✉ ❝ ✉ ❡ ♦ ❢ p(A) ❛ ♥ ❞ p(B)✱ ✐ ✳ ❡ ✳ ♦ ♥ ❡ ❚ ② ♣ ❡ ✲ ❆

❛ ❣ ❡ ♥ ❝ ❛ ♥ ♣ ♦ ❞ ✉ ❝ ❡ ❛ ♠ ❛ ① ✐ ♠ ✉ ♠ ♦ ❢ ✷ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ❛ ♥ ❞ ✶ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❆ ◆ ❉ ❙ ■ ▼ ■ ▲ ❆ ❘ ▲ ❨ ♦ ♥ ❡ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❝ ❛ ♥ ♣ ♦ ❞ ✉ ❝ ❡ ❛

♠ ❛ ① ✐ ♠ ✉ ♠ ♦ ❢ ✷ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❛ ♥ ❞ ✶ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ✳ ❲ ❡ ❤ ❛ ❞ ♦ ✐ ♠ ♣ ♦ ❡ ❤ ❡ ❡ ❝ ♦ ♥ ❞ ✐ ✐ ♦ ♥ ❜ ❡ ❝ ❛ ✉ ❡ ♦ ❤ ❡ ✇ ✐ ❡ ✐ ✇ ❛ ❜ ❡ ❝ ♦ ♠ ✐ ♥ ❣

✈ ❡ ② ❞ ✐ ✣ ❝ ✉ ❧ ♦ ❛ ♥ ❞ ♦ ♠ ❧ ② ❣ ❡ ♥ ❡ ❛ ❡ p(A) ❛ ♥ ❞ p(B)✇ ❤ ✐ ❝ ❤ ❤ ❛ ✈ ❡ ♦ ♠ ❡ ❞ ❡ ✐ ❡ ❞ ♣ ♦ ♣ ❡ ✐ ❡ ✳ ❉ ✉ ❡ ♦ ❤ ✐ ❡ ✐ ❝ ✐ ♦ ♥ ♦ ♥ ❤ ❡

✉ ❝ ✉ ❡ ♦ ❢ p(A) ❛ ♥ ❞ p(B)✱ ❤ ❡ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ❡ ✉ ❧ ❛ ❡ ♥ ♦ ✈ ❡ ② ✐ ♥ ❡ ❡ ✐ ♥ ❣ ✳

❚ ❤ ❡ ✐ ♠ ✉ ❧ ❛ ✐ ♦ ♥ ❡ ✉ ❧ ❛ ❡ ❤ ♦ ✇ ♥ ✐ ♥ ❤ ❡ ♥ ❡ ① ♣ ❛ ❣ ❡ ✳ ✳

✶ ✽



Epoch

0 2 4 6 8 10 12 14 16 18 20

N o . o f T y p e - A a g e n t

0

0.5

1

1.5

2A

B

A & B

Epoch

0 2 4 6 8 10 12 14 16 18 20

N o . o f T y p e - B a g e n t

0

0.5

1

1.5

2A

B

A & B

Epoch

0 2 4 6 8 10 12 14 16 18 20


0

0.2

0.4

0.6

0.8

1Extinction probability starting with one Type-A agent

Epoch

0 2 4 6 8 10 12 14 16 18 20


0

0.2

0.4

0.6

0.8

1Extinction probability starting with one Type-B agent

Both Type-A and Type-B agent goes extinct

❋ ✐ ❣ ✉ ❡ ✸ ✿ ❇ ♦ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❣ ♦ ❡ ❡ ① ✐ ♥ ❝ ✳ ❆ ❢ ♦ ❤ ❡ ❧ ❡ ❣ ❡ ♥ ❞ ✬ ❆ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✱

✬ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❛ ♥ ❞ ✬ ❆ ✫ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ♦ ♥ ❡ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳

Epoch

0 2 4 6 8 10 12 14 16 18 20


0

100

200

300

400

500A

B

A & B

Epoch0 2 4 6 8 10 12 14 16 18 20


0

200

400

600A

B

A & B

Epoch

0 2 4 6 8 10 12 14 16 18 20


0

0.1

0.2

0.3

0.4Extinction probability starting with one Type-A agent

Epoch

0 5 10 15 20


0

0.1

0.2

0.3

0.4

0.5Extinction probability starting with one Type-B agent

Both Type-A and Type-B agent flourishes

❋ ✐ ❣ ✉ ❡ ✹ ✿ ❇ ♦ ❤ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ✢ ♦ ✉ ✐ ❤ ❡ ✳ ❆ ❢ ♦ ❤ ❡ ❧ ❡ ❣ ❡ ♥ ❞ ✬ ❆ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ✱

✬ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❛ ♥ ❞ ✬ ❆ ✫ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ♦ ♥ ❡ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳

✶ ✾



Epoch0 2 4 6 8 10 12 14 16 18 20


0

0.2

0.4

0.6

0.8

1A

B

A & B

Epoch0 2 4 6 8 10 12 14 16 18 20


0

500

1000

1500A

B

A & B

Epoch

0 2 4 6 8 10 12 14 16 18 20


0

0.05

0.1

0.15


Epoch

0 2 4 6 8 10 12 14 16 18 20


0

0.05

0.1


Type-A extincts and Type-B flourishes

❋ ✐ ❣ ✉ ❡ ✺ ✿ ❚ ② ♣ ❡ ✲ ❆ ❛ ❣ ❡ ♥ ❣ ♦ ❡ ❡ ① ✐ ♥ ❝ ❜ ✉ ❚ ② ♣ ❡ ✲ ❇ ✢ ♦ ✉ ✐ ❤ ❡ ✳ ❆ ❢ ♦ ❤ ❡ ❧ ❡ ❣ ❡ ♥ ❞ ✬ ❆ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢

❚ ② ♣ ❡ ✲ ❆ ✱ ✬ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❛ ♥ ❞ ✬ ❆ ✫ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ♦ ♥ ❡

♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳

Epoch

0 2 4 6 8 10 12 14 16 18 20


0

500

1000

1500

2000A

B

A & B

Epoch

0 2 4 6 8 10 12 14 16 18 20

N o . o

f T y p e - B a g e n t

0

0.2

0.4

0.6

0.8

1A

B

A & B

Epoch0 2 4 6 8 10 12 14 16 18 20


0

0.1

0.2

0.3


Epoch0 2 4 6 8 10 12 14 16 18 20


0

0.1

0.2

0.3


Type-B extincts and Type-A flourishes

❋ ✐ ❣ ✉ ❡ ✻ ✿ ❚ ② ♣ ❡ ✲ ❇ ❛ ❣ ❡ ♥ ❣ ♦ ❡ ❡ ① ✐ ♥ ❝ ❜ ✉ ❚ ② ♣ ❡ ✲ ❆ ✢ ♦ ✉ ✐ ❤ ❡ ✳ ❆ ❢ ♦ ❤ ❡ ❧ ❡ ❣ ❡ ♥ ❞ ✬ ❆ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢

❚ ② ♣ ❡ ✲ ❆ ✱ ✬ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❧ ② ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ❛ ♥ ❞ ✬ ❆ ✫ ❇ ✬ ♠ ❡ ❛ ♥ ❛ ✐ ♥ ❣ ✇ ✐ ❤ ♦ ♥ ❡ ❛ ❣ ❡ ♥ ♦ ❢ ❚ ② ♣ ❡ ✲ ❆ ❛ ♥ ❞ ♦ ♥ ❡

♦ ❢ ❚ ② ♣ ❡ ✲ ❇ ✳

✷ ✵



✸ ❚ ❤ ❡ ♦ ❡ ✐ ❝ ❛ ❧ ♦ ❜ ❧ ❡ ♠

✸ ✳ ✶ ❆ ♥ ♦ ❤ ❡ ❈ ✐ ❡ ✐ ♦ ♥ ❢ ♦ ❚ ❛ ♥ ✐ ❡ ♥ ❝ ❡

✭ ❛ ✮ ♦ ✈ ✐ ♥ ❣ ❤ ❛ f ∗ ∈ A✱ ♠ ❡ ❧ ❞ ♦ ✇ ♥ ♦ ♣ ♦ ✈ ✐ ♥ ❣ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ✿

♦ ♦ ❢ ❤ ❛ f ∗ (k) = 0 ✿

f ∗ (k) = supf ∈A

f (k) = supf ∈A

0 ∵ f (k)=0

❛

f ∈A

= 0

♦ ♦ ❢ ❤ ❛ 0 ≤ f ∗ ( j) ≤ 1 ; ∀ j ∈ S ✿

f ∗ ( j) = supf ∈A

f ( j) ≤ supf ∈A

1 ∵ f (j)≤1 ; ∀j ❛

f ∈A

= 1

f ∗ ( j) = supf ∈A

f ( j) ≥ supf ∈A

0 ∵ f (j)≥0 ; ∀j

❛

f ∈A

= 0

❍ ❡ ♥ ❝ ❡ 0 ≤ f ∗ ( j) ≤ 1 ; ∀ j ∈ S ✳

♦ ♦ ❢ ❤ ❛ f ∗ ✐ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✿

❈ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥

f i ≡ arg supf ∈A

f (i) ✭ ✼ ✼ ✮

◗ ✉ ❛ ❧ ✐ ❛ ✐ ✈ ❡ ❧ ② ♣ ❡ ❛ ❦ ✐ ♥ ❣ ✱ f i ✐ ❛ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ❜ ❡ ❧ ♦ ♥ ❣ ✐ ♥ ❣ ♦ ❤ ❡ ❡ A ✇ ❤ ✐ ❝ ❤ ♠ ❛ ① ✐ ♠ ✐ ③ ❡ f (i) ✳ ■ ♥ ♦ ❤ ❡ ✇ ♦ ❞ f ∗ (i) = f i (i) ✳ ◆ ♦ ✇

❝ ♦ ♥ ✐ ❞ ❡ ❤ ❡ ❢ ♦ ❧ ❧ ♦ ✇ ✐ ♥ ❣

j∈S P ijf ∗ ( j) ≥

j∈S P ijf i ( j) ✭ ✼ ✽ ✮

≥ f i (i) ✭ ✼ ✾ ✮

= f ∗ (i)

■ ♥ ❡ ✉ ❛ ❧ ✐ ② ✼ ✽ ❡ ✉ ❧ ❢ ♦ ♠ ❤ ❡ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥ ♦ ❢ f ∗ ( j) ✱ ✐ ✳ ❡ ✳ f ∗ ( j) = supf ∈A

f ( j) ≥ h ( j) ; ∀h ∈ A✳ ■ ♥ ✐ ♥ ❡ ✉ ❛ ❧ ✐ ② ✼ ✽ ✱ h = f i ✳

■ ♥ ❡ ✉ ❛ ❧ ✐ ② ✼ ✾ ❡ ✉ ❧ ❢ ♦ ♠ ❤ ❡ ❢ ❛ ❝ ❤ ❛ f i ∈ A ❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ f i ✐ ❛ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ❢ ✉ ♥ ❝ ✐ ♦ ♥ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✳ ❚ ❤ ❡ ❛ ❜ ♦ ✈ ❡

♣ ♦ ♦ ❢ ❤ ♦ ✇ ❤ ❛ f ∗ ✐ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✳

✭ ❜ ✮ ■ ♥ ❛ ✭ ❛ ✮ ✇ ❡ ❛ ❧ ❡ ❛ ❞ ② ♣ ♦ ✈ ❡ ❞ ❤ ❛ f ∗ ✐ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✱ ✐ ✳ ❡ ✳j∈S

P ijf ∗ ( j) ≥ f ∗ (i) ; ∀i = k

◆ ♦ ✇ ✇ ❡ ❤ ❛ ✈ ❡ ♦ ♣ ♦ ✈ ❡ ❤ ❛ f ∗ ✐ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✳ ❚ ❤ ✐ ♠ ❡ ❧ ❞ ♦ ✇ ♥ ♦ ♣ ♦ ✈ ✐ ♥ ❣ ❤ ❛ ❤ ❡ ❡ ❡ ① ✐ ♥ ♦ i ∈ S ❢ ♦

✇ ❤ ✐ ❝ ❤ j∈S

P ijf ∗ ( j) > f ∗ (i) ✭ ✽ ✵ ✮

❛ ♥ ❞ ❤ ❡ ♥ ❝ ❡ ❤ ❡ ♦ ♥ ❧ ② ♣ ♦ ✐ ❜ ✐ ❧ ✐ ② ✐

j∈S

P ijf ∗ ( j) = f ∗ (i) ; ∀i = k ✳ ❲ ❡ ✇ ✐ ❧ ❧ ♣ ♦ ✈ ❡ ❤ ✐ ❜ ② ✉ ✐ ♥ ❣ ❝ ♦ ♥ ❛ ❞ ✐ ❝ ✐ ♦ ♥ ✳ ▲ ❡ ❞ ❡ ✜ ♥ ❡ ❛

❢ ✉ ♥ ❝ ✐ ♦ ♥ g ✉ ❝ ❤ ❤ ❛

g (i) =j∈S

P ijf ∗ ( j) ; ∀i ∈ S ✭ ✽ ✶ ✮

✷ ✶



❆ ❝ ❝ ♦ ❞ ✐ ♥ ❣ ♦ ✐ ♥ ❡ ✉ ❛ ❧ ✐ ② ✽ ✵ ✱ g (i) > f ∗ (i) ✳ ■ ❢ ✇ ❡ ❝ ❛ ♥ ❤ ♦ ✇ ❤ ❛ g ∈ A✱ ❤ ❡ ♥ ✐ ✇ ✐ ❧ ❧ ❝ ♦ ♥ ❛ ❞ ✐ ❝ ❤ ❡ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥ ♦ ❢ f ∗ ✱ ❤ ❡ ❡ ❜ ②

♣ ♦ ✈ ✐ ♥ ❣ ❤ ❛ ✐ ♥ ❡ ✉ ❛ ❧ ✐ ② ✽ ✵ ✐ ♥ ♦ ♣ ♦ ✐ ❜ ❧ ❡ ✳ ◆ ♦ ✇ ❛ ❧ ❧ ✇ ❡ ❤ ❛ ✈ ❡ ♦ ♣ ♦ ✈ ❡ ✐ g ∈ A✳

♦ ♦ ❢ ❤ ❛ g (k) = 0 ✿

g (k) =j∈S

P kjf ∗ ( j) = f ∗ (k) = 0 ∵ f ∗ ∈ A

♦ ♦ ❢ ❤ ❛ 0 ≤ g ( j) ≤ 1 ; ∀ j ∈ S ✿

g ( j) = u∈S

P juf ∗ (u) ≤ u∈S

P ju · 1 ∵ f ∗(u)≤1 ❛

f ∗∈A

= 1 ✭ ❘ ♦ ✇ ✉ ♠ ♦ ❢ ♦ ❜ ❛ ❜ ✐ ❧ ✐ ② ❚ ❛ ♥ ✐ ✐ ♦ ♥ ▼ ❛ ✐ ① ✐ 1)

g ( j) =u∈S

P juf ∗ (u) ≥u∈S

P ju · 0 ∵ f ∗(u)≥0 ❛ f ∗∈A

= 0

❍ ❡ ♥ ❝ ❡ 0 ≤ g ( j) ≤ 1 ; ∀ j ∈ S ✳

♦ ♦ ❢ ❤ ❛ g ✐ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✿

j∈S

P ijg ( j)

=j∈S

P ij

u∈S

P juf ∗ (u)

=j∈S

P ij

u∈S

P juf u (u)

✭ ✽ ✷ ✮

≥j∈S

P ij

u∈S

P juf j (u)

✭ ✽ ✸ ✮

≥ j∈S

P ijf j ( j) ✭ ✽ ✹ ✮

=j∈S

P ijf ∗ ( j) ✭ ✽ ✺ ✮

= g (i)

❊ ✉ ❛ ✐ ♦ ♥ ✽ ✷ ❛ ♥ ❞ ✽ ✺ ✐ ❛ ❞ ✐ ❡ ❝ ❝ ♦ ♥ ❡ ✉ ❡ ♥ ❝ ❡ ♦ ❢ ❤ ❡ ❞ ❡ ✜ ♥ ✐ ✐ ♦ ♥ ♦ ❢ f ✐ ♥ ❡ ✉ ❛ ✐ ♦ ♥ ✼ ✼ ✳ ■ ♥ ❡ ✉ ❛ ❧ ✐ ② ✽ ✸ ❡ ✉ ❧ ❢ ♦ ♠ ❤ ❡ ❢ ❛ ❝ ❤ ❛

f u (u) ≥ f j (u) ; ∀ j ∈ S ✳ ■ ♥ ❡ ✉ ❛ ❧ ✐ ② ✽ ✹ ✐ ✈ ❛ ❧ ✐ ❞ ❜ ❡ ❝ ❛ ✉ ❡ f ∗ ✐ ❛ ✉ ❜ ❤ ❛ ♠ ♦ ♥ ✐ ❝ ✇ ✐ ❤ ❡ ♣ ❡ ❝ ♦ P ✳

❚ ❤ ❡ ❛ ❜ ♦ ✈ ❡ ❤ ❡ ❡ ♣ ♦ ♦ ❢ ❤ ♦ ✇ ❤ ❛ g ∈ A✳ ❚ ❤ ✐ ❝ ♦ ♥ ❝ ❧ ✉ ❞ ❡ ❤ ❡ ♣ ♦ ♦ ❢ ✳

✷ ✷



11/25/15 10:50 AM C:\Users\sahah_000\Desk...\Markov_Maze.m 1 of 4

clear

clc

load(!ro"a"#l#$%_&ra's#$#o'_Ma$r#.ma$)*

+++++++++++++++++++++++++ Code for 1.1.". : ,&A-& ++++++++++++++++++++++

!_$#lde!*

!_$#lde(:)0*

!_$#lde()1*

!_$#lde*

(:)3*

(:)3*

fo'es(151)*

m_$a#'v(e%e(15))6f*

s$rAvera7e &#me "efore -a$h"er$ reaches cheese 'm2s$r(m_$a(15))3*

d#s8(s$r)*

$em8m_$a(15)*

++++++++++++++++++++++++++ Code for 1.1.". : 9D +++++++++++++++++++++++

+++++++++++++++++++++++++ Code for 1.1.c. : ,&A-& ++++++++++++++++++++++

clearvars ece8$ ! ; $em8

!_$#lde!*

!_$#lde(:)0*

!_$#lde()1*

!_$#lde*

(:)3*

(:)3*

fzeros(151)*

f(<1)1*

f(=1)1*

m_shock#'v(e%e(15))6f*

s$rAvera7e 'm"er of shock "efore reach#'7 cheese 'm2s$r(m_shock(15))3*

d#s8(s$r)*

++++++++++++++++++++++++++ Code for 1.1.c. : 9D +++++++++++++++++++++++

+++++++++++++++++++++++++ Code for 1.1.d. : ,&A-& ++++++++++++++++++++++


!_$#lde!*

!_$#lde(:)0*

Appendix A: Markov Maze




!_$#lde(=:)0*

!_$#lde(>:)0*

!_$#lde()1*

!_$#lde(==)1*

!_$#lde(>>)1*

ro?_dele$e = >3*

col_dele$e = >3*

!_$#lde*

(ro?_dele$e:)3*

(:col_dele$e)3*

clear ro?_dele$e col_dele$e

ro?_dele$e = >3*

col_dele$e1 2 4 5 @ < 10 11 12 1 14 15 1@3*

-!_$#lde*

-(ro?_dele$e:)3*

-(:col_dele$e)3*

U#'v(e%e(1))6-*

s$r!ro"a"#l#$% of reach#'7 cheese ?#$ho$ 7e$$#'7 shocked 'm2s$r(U(11))3*

d#s8(s$r)*

++++++++++++++++++++++++++ Code for 1.1.d. : 9D +++++++++++++++++++++++

+++++++++++++++++++++++++ Code for 1.1.e. : ,&A-& ++++++++++++++++++++++


Mzeros(25@25@)*

for #0:1:255

for 0:1:255

s_r_#floor(#/1@)B1*

s_s_##1@6(s_r_#1)B1*

s_r_floor(/1@)B1*

s_s_1@6(s_r_1)B1*

M(#B1B1)!(s_r_#s_r_)6;(s_s_#s_s_)*

e'd

e'd

M_$#ldeM*

for #0:1<:255

M_$#lde(#B1:)zeros(125@)*

M_$#lde(#B1#B1)1*

e'd

M_$#lde*

ro?_dele$e0:1<:255*

col_dele$e0:1<:255*

ro?_dele$ero?_dele$eB1*

col_dele$ecol_dele$eB1*



11/25/15 10:50 AM C:\Users\sahah_000\Desk...\Markov_Maze.m of 4

(ro?_dele$e:)3*

(:col_dele$e)3*

fo'es(2401)*

m_$a#'v(e%e(240))6f*

s$rAvera7e &#me "efore -a$"er$ a'd ,am mee$ each o$her 'm2s$r(m_$a(22@))3*

d#s8(s$r)*

++++++++++++++++++++++++++ Code for 1.1.e. : 9D +++++++++++++++++++++++

+++++++++++++++++++++++++ Code for 1.1.f. : ,&A-& ++++++++++++++++++++++

clearvars ece8$ ! $em8

8_s$'r%o'es(11@)6#'v(e%e(1@)!Bo'es(1@1@))* + -es'#ck ormla

$a1/8_s$'r%()*

s$rAvera7e &#me "efore -a$"er$ ca' res$ 'm2s$r($em8)B'm2s$r($a)(C1)3*

d#s8(s$r)*

++++++++++++++++++++++++++ Code for 1.1.f. : 9D +++++++++++++++++++++++

+++++++++++++++++++++++++ Code for 1.1.7. : ,&A-& ++++++++++++++++++++++

clearvars ece8$ !

!_$#lde!*

!_$#lde(:)0*

!_$#lde(=:)0*

!_$#lde(>:)0*

!_$#lde()1*

!_$#lde(==)1*

!_$#lde(>>)1*

ro?_dele$e = >3*

col_dele$e = >3*

!_$#lde*

(ro?_dele$e:)3*

(:col_dele$e)3*

clear ro?_dele$e col_dele$e

ro?_dele$e = >3*

col_dele$e1 2 4 5 @ < 10 11 12 1 14 15 1@3*

-!_$#lde*

-(ro?_dele$e:)3*

-(:col_dele$e)3*

U#'v(e%e(1))6-*

8=U(12)*

8>U(1)*




!_$#lde(=:)!(=:)*

!_$#lde(>:)!(>:)*

clear

!_$#lde*

(:)3*

(:)3*

fzeros(151)*

f(=1)1*


=m_shock(<)*

fzeros(151)*

f(<1)1*


>m_shock(=)*

s$r98ec$ed 'm"er of shock (mod#f#ed) 'm2s$r(8=6(=B1)B8>6(>B1))3*

d#s8(s$r)*

++++++++++++++++++++++++++ Code for 1.1.7. : 9D +++++++++++++++++++++++

clear



11/25/15 10:50 AM C:\Users\sah...\Email_Sharing_Analysis.m 1 of 2

funcion !mean"mean_a##iional"$ariance"$ariance_a##iional"E%incion&ro'"

E%incion&ro'_a##iional( ) Email_Sharing_Analysis*m"+",-",",

ecor Creaion: S+A3+

mean)4eros*1"+16

mean_a##iional)4eros*1"+16

$ariance)4eros*1"+16

$ariance_a##iional)4eros*1"+16

E%incion&ro')4eros*1"+16

E%incion&ro'_a##iional)4eros*1"+16

ecor Creaion: E7

-niiali4aion for 0h E,och: S+A3+

mean*1)m6

mean_a##iional*1)m6

$ariance*1)06

$ariance_a##iional*1)06

E%incion&ro'*1)06

E%incion&ro'_a##iional*1)06

-niiali4aion for 0h E,och: E7

&reliminary Calculaion: S+A3+

i)numel*,-6

mu_-)sum**0:1:*i819,-6

$ar_-)sum****0:1:*i818mu_-.9**0:1:*i818mu_-.9,-6

o)numel*,6

mu_)sum**0:1:*o819,6

$ar_)sum****0:1:*o818mu_.9**0:1:*o818mu_.9,6

a##)numel*,6

mu_)sum**0:1:*a##819,6

$ar_)sum****0:1:*a##818mu_.9**0:1:*a##818mu_.9,6

A)!mu_9mu_- 06 $ar_9mu_-$ar_-9*mu_;2 *mu_9mu_-;2(6

A_a##)A6

A_a##*2"1)A_a##*2"129mu_9mu_9mu_-6

<_a##)!mu_6 $ar_mu_;2(6

&reliminary Calculaion: E7

al,ha)06

'ea)16

for n)1:1:+

Appendix B: Email Sharing



11/25/15 10:50 AM C:\Users\sah...\Email_Sharing_Analysis.m 2 of 2

=)!mean*n6mean*n;2$ariance*n(6

=)A9=6

mean*n1)=*16

$ariance*n1)=*28=*1;26

=a##)!mean_a##iional*n6mean_a##iional*n;2$ariance_a##iional*n(6

=a##)A_a##9=a##<_a##6

mean_a##iional*n1)=a##*16

$ariance_a##iional*n1)=a##*28=a##*1;26

'ea)'ea9&ro'_>en_?unc*,"al,ha6

al,ha)&ro'_>en_?unc*,-"&ro'_>en_?unc*,"al,ha6

E%incion&ro'*n1)al,ha;m6

E%incion&ro'_a##iional*n1)*al,ha;m9'ea6

en#

en#

funcion !$al( ) &ro'_>en_?unc*,#f"s

$al)06

)numel*,#f6

for i)0:1:*81

$al)$al,#f*i19*s;i6

en#

en#



11/25/15 10:51 AM C:\Use...\Branching_TwoWays_MonteCarlo. 1 o! 2

!"nction #$_A%$_B& ' Branching_TwoWays_MonteCarlo(T%)_A%)_B%$_A_0%$_B_0*

+ ,n)"ts:

+ T -- "er o! i"lation te)s (incl"ing the 0th e)och*

+ )_A -- 34 o! o!!s)rings o! Ty)e-A agent

+ )_B -- 34 o! o!!s)rings o! Ty)e-B agent

+ $_A_0 -- 6 o! Ty)e-A agents in the 0th e)och

+ $_B_0 -- 6 o! Ty)e-B agents in the 0th e)och

+ 7"t)"ts:

+ $_A -- 6 o! Ty)e-A agents at i!!erent e)ochs

+ $_B -- 6 o! Ty)e-B agents at i!!erent e)ochs

$_A'8eros(1%T91*

$_B'8eros(1%T91*

$_A(1*'$_A_0 + 6 o! Ty)e-A agents in the 0th e)och

$_B(1*'$_B_0 + 6 o! Ty)e-B agents in the 0th e)och

!or n'1:1:T

+ Contri"tion o! Ty)e-A agent !or the ne;t e)och: TA<T

!or =A'1:1:$_A(n*

#>_AA%>_AB&'<an"?en_24()_A* + Ty)e-A an Ty)e-B agent )ro"ce y =Ath agent

o! Ty)e-A in the c"rrent e)och

$_A(n91*'$_A(n91*9>_AA + U)ation o! Ty)e-A agent !or the ne;t e)och

$_B(n91*'$_B(n91*9>_AB + U)ation o! Ty)e-B agent !or the ne;t e)och

en

+ Contri"tion o! Ty)e-A agent !or the ne;t e)och: @4

+ Contri"tion o! Ty)e-B agent !or the ne;t e)och: TA<T

!or =B'1:1:$_B(n*

#>_BA%>_BB&'<an"?en_24()_B* + Ty)e-A an Ty)e-B agent )ro"ce y =Bth agent

o! Ty)e-B in the c"rrent e)och

$_A(n91*'$_A(n91*9>_BA + U)ation o! Ty)e-A agent !or the ne;t e)och

$_B(n91*'$_B(n91*9>_BB + U)ation o! Ty)e-B agent !or the ne;t e)och

en

+ Contri"tion o! Ty)e-B agent !or the ne;t e)och: @4

en

en

+ 24 <ano "er ?enerator: TA<T

!"nction #col%row&'<an"?en_24()*

+ col is the A agent

+ row is the B agent

_col'n"el()(1%:**

_row'n"el()(:%1**

ppendix C: Monte Carlo Simulation of 2-Way Branching Proce



11/25/15 10:51 AM C:\Use...\Branching_TwoWays_MonteCarlo. 2 o! 2

te)'ran

acc')(1%1*

i! (te)''1*

col'_col

row'_row

else

col'1

row'1

while(te)'acc*

col'col91

i! (col_col*

col'1

row'row91

en

acc'acc9)(row%col*

en

en

col'col-1

row'row-1

en

+ 24 <ano "er ?enerator: @4



11/25/15 10:51 AM C:\Users...\Branching_TwoWays_Analysis.m 1 o 1

!nc"ion #$ro%_A&$ro%_B' ( Branching_TwoWays_Analysis)$A&$B&T*

+ ,n$!":

+ $A -- ro%a%ili"y is"ri%!"ion o s$rings o Ty$e-A agen"

+ $B -- ro%a%ili"y is"ri%!"ion o s$rings o Ty$e-B agen"

+ T -- im!la"ion Time

+ !"$!":

+ $ro%_A -- 34"inc"ion ro%a%ili"y s"ar"ing wi"h one Ty$e-A agen"

+ $ro%_B -- 34"inc"ion ro%a%ili"y s"ar"ing wi"h one Ty$e-B agen"

$ro%_A(eros)1&T61*7

$ro%_B(eros)1&T61*7

$ro%_A)1*(07 + 34"inc"ion ro%a%ili"y a" 0"h e$och is 0.

$ro%_B)1*(07 + 34"inc"ion ro%a%ili"y a" 0"h e$och is 0.

or n(1:1:T

$ro%_A)n61*(ro%_8en_9!nc)$A&$ro%_A)n*&$ro%_B)n**7 + ec!rsion 9orm!la

$ro%_B)n61*(ro%_8en_9!nc)$B&$ro%_A)n*&$ro%_B)n**7 + ec!rsion 9orm!la

en;

en;

+ ro%a%ili"y 8enera"ing 9!nc"ion o Two <aria%les: TAT

!nc"ion #=al' ( ro%_8en_9!nc)$;&!&=*

>a(n!mel)$;)1&:**7

>%(n!mel)$;):&1**7

=al(07

or ?(0:1:)>%-1*

or i(0:1:)>a-1*

=al(=al6$;)?61&i61*@)!i*@)=?*7

en;

en;

en;

+ ro%a%ili"y 8enera"ing 9!nc"ion o Two <aria%les: 3>

Appendix D: Extinction Probability of 2-Way Branching Process

solution of stochastic process and modelling

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