The Spectral Density of a Markov ProcessAuthor(s): B. D. CravenSource: Advances in Applied Probability, Vol. 6, No. 2 (Jun., 1974), pp. 216-217Published by: Applied Probability TrustStable URL: http://www.jstor.org/stable/1426254 .

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216 3RD CONFERENCE ON STOCHASTIC PROCESSES AND APPLICATIONS

Corollary 4 (A generalization of the modified Poisson distribution). If the Xi are i.i.d. random variables with p.d.f. ;e-*x and Yj are non-random constants, then

oo 0 (t - Sn)'e-A(t-S") P(Wt 2-

n) = 1, = i!

if ln=i Yj = S, < t and zero otherwise.

Corollary 5. If the Xi are Gamma distributed with the same scale parameter A, and the Y, are Gamma distributed with the same scale parameter p, then from Theorem 3, the p.d.f. of Wt may be expressed in closed form in terms of incomplete Gamma functions I,(k), where these are defined by I,(k) = ftxk-1e-xdx.

The spectral density of a Markov process

B. D. CRAVEN, University of Melbourne

For a Markov process in discrete time, having finite autocovariances, both the time-dependent behaviour and the covariance structure may be summed up in a single generating function, here called the key function. If the process is stationary, and the key function possesses a suitable analytic extension, then the process possesses a continuous spectral density, which can be calculated from the key function. A converse result is also obtained, together with some related results for non-stationary processes. Some detailed results are obtained for the waiting-time process in the GI/G/1 queue.

For the process {Vt: t = 0, 1,2, ...}, whose state space may be discrete or continuous, let Vt have characteristic function ,t(s); then It+1 = Mot, where M is a linear operator characterizing the Markov process. The covariance struc- ture of the stationary process is calculable from the key function

00

O(s;0) = 1 0" g(s) (101 <1), n=O

satisfying the equation

Q -(- > = OMQ. Here O(s) is the characteristic function for the stationary process, and

E(VtVt+n) = ig'(O). Denote by m and yo the mean and variance of the stationary process. If the function

j{e-ism](s; 0)}I 0

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Sheffield, 13-17 August 1973 217

possesses a continuous extension from 10 < 1 to the circle 101 = 1, then the process possesses a continuous spectral density f (A), given by

27f(A) = - yo - mmT + 2 im 0,,(0; e'").

Here Q, = a0/as, and T denotes transpose. The condition is also necessary.

Infinite divisibility and stability of finite semi-Markov matrices

T. J. OTT, University of Rochester

In this paper the elements of a semi-Markov matrix A may have support anywhere on the real line, and A(+ co) may be a sub-Markov matrix.

The sub-characteristic matrix a of A is the matrix of Fourier-Stieltjes transforms corresponding to A.

If A(+ co) = a(O) is a Markov matrix, A is called a matrix distribution funotion and at a characteristic matrix.

Infinite divisibility and stability of semi-Markov and sub-characteristic matrices is defined as for distribution functions and characteristic functions.

Theorem 1. Let a be a finite sub-characteristic matrix. at is infinitely divisible if and only if there exists a continuous one-parameter semi-group {p, I t 0O} of sub-characteristic matrices with ix = a. The semi-group is, in general, not unique for a.

The semi-group {P, It ? 0} in Theorem 1 has the exponential form P,(z) = o(z) exp{tpt(z)} with p(z) continuous in z. The rank of P,(z) is equal to the number of nonzero eigenvalues of Pt(z) and is independent of t and z.

Theorem 2. Let at be a finite sub-characteristic matrix. at is stable if and only if it has the form

cz(z) = lim eibn(y(az))N (a, > 0, - co < b, < + co) n-- Wo

for some sub-characteristic matrix y. If a is stable it is infinitely divisible and the semi-group in Theorem 1 is unique

for a. If a is also irreducible it has one of the following two forms: I.

t(z) = eizxoe(z) (-co<xo < + co)

where e(z) is an idernpotent, irreducible characteristic matrix, i.e., of the form

ejk(z) = Pkexp{iz(k - j)} (Pk > O, Pk 1, < k <

+C). k

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