asymptotic properties of near pfeifer records

Extremes (2011) 14:253–265DOI 10.1007/s10687-010-0108-4

Asymptotic properties of near Pfeifer records

Arup Bose ·Sreela Gangopadhyay

Received: 22 June 2009 / Revised: 18 February 2010 /Accepted: 26 February 2010 / Published online: 20 March 2010c© Springer Science+Business Media, LLC 2010

Abstract Asymptotic properties of the number of near records are known in the liter-ature. We generalize these results to the Pfeifer model which has a wider application.In particular we establish convergence in probability, in the almost sure sense and indistribution for the number of near records under the Pfeifer model.

Keywords Pfeifer records · Near Pfeifer records · Regularly varying functions ·Extended regularly varying functions · Limiting distribution

AMS 2000 Subject Classifications 60F05 · 60F15 · 60G70

1 Introduction

Insurance companies often change their policy when they receive a claim whichexceeds all previous claims, so that under the new policy such claims would be lessfrequent in probability. It is of interest to the company to study the total value andnumber of claims which are “very near” to the record claims. Balakrishnan et al.(2005) and Pakes (2007) have studied properties of near records in classical setup

Work done while Arup Bose was visiting Department of Economics, University of Cincinnati, USA.Research also supported by J.C.Bose Fellowship, Government of India.

Research of Sreela Gangopadhyay was supported by the Department of Science and Technology,Government of India, scheme no. SR/WOS-A/MS-02/2006.

A. Bose · S. Gangopadhyay (B)Statistics and Mathematics Unit, Indian Statistical Institute, 203 B. T. Road, Kolkata 700108, Indiae-mail: [email protected], [email protected]

A. Boseemail: [email protected]

254 A. Bose, S. Gangopadhyay

where observations are i.i.d. with a common continuous distribution. See also Li(1999), Li and Pakes (2001) and Pakes and Steutel (1997). In this article we studyproperties of the number of near records for the more general Pfeifer model (Pfeifer1984) which has been used in such situations. The setup is as follows:

Let {Xi j }i≥0, j≥1 be a double array of independent random variables. For eachfixed i , {Xi j , j ≥ 1} are i.i.d. with a common continuous cdf Fi where

1 − Fi = (1 − F)αi , αi > 0, ∀i = 1, 2, . . .

The Pfeifer records (precords in short) are defined as follows: The first precord R0 isby convention X01. Let

�(1) = inf{ j : X1 j > R0} and �(n + 1) = inf{ j : X(n+1) j > Xn�(n)}, n ≥ 1.

Then X01, X1�(1), . . . Xn�(n), . . . are precords denoted by R0, R1, . . . Rn+1, . . ..For further information on precords see Arnold et al. (1998), Villaseñor and

Arnold (2007), and Bose and Gangopadhyay (2009) etc.For any a > 0, the number of near precords {ξn(a)}n≥1, depending on a, are

defined as

ξn(a) := #{ j < �(n) : Rn−1 − a < Xnj < Rn−1}.

For any sequence {an}, {ξn(an)} will be called the number of near precords withvarying window width.

The following representation is important. Suppose Y1, Y2, . . . are independentrandom variables and

Yi ∼ Exp(αi ) (1.1)

and for any distribution function F ,

ψF (x) := F−1(1 − e−x ). (1.2)

For notational simplicity we will suppress the subscript of ψ . Then (see Arnold et al.1998),

(R0, R1, . . . Rn

) D= (ψ(Y1), ψ(Y1 + Y2), . . . ψ(Y1 + Y2 + . . . Yn)

)for all n.

Unless otherwise stated we will assume that

Assumption 1 F is a continuous, strictly increasing cdf with support, Supp(F) ⊂[0, ∞).

Assumption 2 {αn} is a non-decreasing positive real sequence, diverging toinf inity.

Asymptotic properties of near Pfeifer records 255

In Section 2, we derive the distributions of {ξn(a)}. In Section 3 we study asymp-totic properties of {ξn(a)} under various conditions on F and {αn}. In Section 4, westudy the limiting distributions of {ξn(an)} with varying window width.

Incidentally, in i.i.d. setup, Balakrishnan et al. (2005) obtained results similar toours. Our Theorem 1 is analogous to their Theorems 2.1 and 2.2. Our Theorem 2is similar to their Theorem 3.1(ii). Likewise, our Proposition 1 and Theorem 4are extensions of their Theorems 3.1(i) and 4.1. We imposed some extra condi-tions to deal with the Pfeifer model. The method of some of our proofs are alsodifferent.

2 Distribution of ξn(a)

Balakrishnan et al. (2005) have derived the distribution of {ξn(a)} in the i.i.d. model.Using similar arguments we may establish a basic formula for the joint distributionof the number of near records in the Pfeifer model. We need a few notation. For anycdf F ,

F̄(x) := 1 − F(x).

Note that

F̄n(x) = F̄αn (x) for all n ≥ 0.

Let

Rn(x) := P(Rn ≤ x) and ρn(x, a) := F̄n(x)

F̄n(x − a).

We will suppress a in ρn(x, a) whenever there is no chance of confusion. Let

Hi (x j+i , x j ) := P

(

0 < Zi < − logF̄j+i (x j+i )

F̄j (x j )

)

where ZiD= ∑ j+i

k= j+1 βk X∗k , βk = α−1

k and X∗k are i.i.d. Exp(1) r.v.s.

Theorem 1

(i) ξn given Rn = x is geometric with parameter ρn(x) and P(ξn(a) = k) =∫R ρn(x)(1 − ρn(x))kd Rn(x).

(ii) The joint distribution of (ξn(a1), ξn+1(a2), . . . ξn+k−1(ak)) is given by

P(ξn(a1) = r1, . . . ξn+k−1(ak) = rk)

=∫ ∞

−∞

∫ ∞

xn

· · ·∫ ∞

xn+k−2

k∏

j=1

ρn+ j−1(xn+ j−1, a j

)

× (1 − ρn+ j−1

(xn+ j−1, a j

))r j

×k∏

j=2

d H1(xn+ j−1, xn+ j−2

)d Rn(xn).


(iii) The joint distribution of (ξn(a1), ξn+k(ak+1)) is given by

P(ξn(a1) = r1, ξn+k(ak+1) = rk+1)

=∫ ∞

−∞

∫ ∞

xn

ρn(xn, a1)(1 − ρn(xn, a1))r1ρn+k(xn+k, ak+1)

× (1 − ρn+k(xn+k, ak+1))rk+1 d Hk(xn+k, xn)d Rn(xn).

Proof (i) It is enough to show that

P(ξn(a) = k|Rn = x) = ρn(x)(1 − ρn(x))k . (2.1)

The proof is given in Pakes (2007). For the sake of completeness, we presentit below. If Rn = x then �(n + 1) = i if and only if X(n+1)i > x andX(n+1)1, X(n+1)2, . . . , X(n+1)(i−1) ≤ x . ξn(a) = k if and only if k of the i − 1above listed observations fall in (x −a, x) and others fall below x −a. Hence the lefthand side of Eq. 2.1 equals

∑

i>k

(i − 1

k

)[F(x) − F(x − a)]k Fi−1−k(x − a)F̄(x)

which after simplification gives the right hand side. The proofs of (ii) and (iii) followfrom the two simple observations:

(a) For any finite increasing sequence n1 < n2 < ..... < ni and any positivereals a1, a2, . . . , ai , ξn1(a1), ξn2(a2), . . . , ξni (ai ) are conditionally independentgiven Rn1 , Rn2 , . . . , Rni .

(b) Hi (x j+i , x j ) is the transition probability function P(R j+i ≤ x j+i |R j = x j ), as

P(R j+i ≤ x j+i |R j = x j )

= P

⎛

⎝j+i∑

k=1

βk X∗k ≤ ψ−1(x j+i )|

j∑

k=1

βk X∗k = ψ−1(x j )

⎞

⎠

= P

⎛

⎝j+i∑

k= j+1

βk X∗k ≤ ψ−1(x j+i ) − ψ−1(x j )

⎞

⎠

= P

⎛

⎝j+i∑

k= j+1

βk X∗k ≤ − log

F̄(x j+i )

F̄(x j )

⎞

⎠ . �

3 Asymptotic behaviour of {ξn(a)}

Let

rF := sup{supp(F)}, lF := inf{supp(F)} and βn := 1

αn.


Note that rF may be equal to ∞. Balakrishnan et al. (2005) have shown that inthe i.i.d. model, if rF < ∞, then ξn(a) → ∞ almost surely (a.s.) as n → ∞.The fact that the records Rn in the classical set-up, have the closed form density[(n − 1)!]−1(− log F̄(x))n−1 with respect to the basic cdf F , plays a crucial rolein the above work. Though in the Pfeifer model this density exists, it is not knownin a closed form. Hence their arguments cannot be used. We need the followingLemma.

Lemma 1

(i) If∑∞

n=1 βn = ∞ then Rn → rF a.s..(ii) If

∑∞n=1 βn < ∞ then Rn → W a.s. where W is a continuous random variable

with the same support as F and with a cdf which is strictly increasing on itssupport.

Proof

(i) Suppose X∗i are i.i.d. Exp(1). By the Kolmogorov Three Series Theorem,

∑ni=1 βi X∗

i → ∞ a.s. Therefore, ψ( ∑n

i=1 βi X∗i

) → rF a.s. Hence RnD=

ψ( ∑n

i=1 βi X∗i

) → rF in distribution and hence in probability. Since {Rn} isincreasing, this convergence holds a.s..

(ii) If∑n

i=1 βi < ∞ then∑n

i=1 β2i < ∞. Therefore,

∑ni=1 V ar

(βi X∗

i − βi) =∑n

i=1 β2i < ∞. Khinchine–Kolmogorov’s one-series Theorem implies that∑n

i=1 βi X∗i → V a.s. where V is a finite random variable.

The limit V is the sum of weighted exponential random variables and henceis infinitely divisible with a Lévy measure supported on the positive reals.Hence V has the same support. See Bondesson (1992). Now

P(W ≤ w) = P(ψ(V ) ≤ w) = P(1 − e−V ≤ F(w)

). (3.1)

But 1 − e−V is supported on [0,1], so the left hand side of Eq. 3.1 is con-stant in an interval if and only if F(w) is constant on the same interval. Hencesupp(W ) = supp(F) �

From now on, we will assume without loss of generality, lF = 0.

3.1 Case 1. rF < ∞

Proposition 1

(i) ξn(a) → ∞ in probability.(ii) Suppose, (a)

∑∞n=1 λαn < ∞, if 1 > λ > 0 and (b) αn

n → 0. Then ξn(a) →∞ a.s.


Proof (i) It is enough to show that for any fixed k > 0, P(ξn(a) ≤ k) → 0, asn → ∞. Observe that

P(ξn(a) ≤ k) =k∑

j=0

∫ ∞

−∞ρn(x)(1 − ρn(x)) j d Rn(x)

=∫ ∞

−∞[1 − (1 − ρn(x))k+1]d Rn(x)

≤ (k + 1)

∫ ∞

−∞ρn(x)d Rn(x) = (k + 1)E(ρn(Rn)).

Recall that

ρ(x) = F̄(x)

F̄(x − a)and ρn(x) = (ρ(x))αn .

Case 1. If∑∞

n=1 βn = ∞, then ρ(Rn)a.s.→ 0, by Lemma 1. Hence 1 ≥ ρn(Rn)

a.s.→ 0,

so dominated convergence theorem implies that E[ρn(Rn)] → 0.Case 2. If

∑∞n=1 βn < ∞, then ρ(Rn)

a.s.→ ρ(W ) < 1 from the above supportproperty. Since αn → ∞ by Assumption 2 in Section 1, it follows thatE[ρn(Rn)] → 0.

For (ii), choose ε ∈ (rF − a, rF ).

P(ξn(a) ≤ k) ≤ (k + 1)E(ρn(Rn))

= (k + 1)E(ρn(Rn)IRn<ε

) + (k + 1)E(ρn(Rn)IRn≥ε

)

= (k + 1)[E1,n,ε + E2,n,ε] (say).

Now, if ε ≤ x ≤ rF , then

ρ(x) ≤ λ := F̄(ε)

F̄(rF − a)< 1.

So condition (a) ensures∑

E2,n,ε < ∞.

Now observe that, since {αi } is non-decreasing (Assumption 2), we get,

E1,n,ε ≤ P(Rn < ε) = P

(n∑

i=1

βi X∗i < ε′

)

≤ P

(n∑

i=1

X∗i < ε′αn

)

, where ε′ := ψ−1(ε) < ∞.


As∑n

i=1 X∗i ∼ (n), the last probability above has an upper bound as follows:

P

(n∑

i=1

X∗i < ε′αn

)

= 1

(n − 1)!∫ ε′αn

0e−x xn−1dx

≤ 1

(n − 1)!∫ ε′αn

0xn−1dx = (ε′αn)n

n! .

Using Stirling’s formula n! = (2πn)12( n

e

)n(1 + O( 1n )

)and applying root test to the

above upper bound we see that it is summable if (b) holds. Now using Borel–CantelliLemma we have the desired result. �

Example Let αn = nδ, 0 < δ < 1. Then conditions (a) and (b) of Proposition 1 aresatisfied. The following result is analogous to Theorem 3.1(ii) in Balakrishnan et al.(2005) and the proof is similar.

Theorem 2 ρn(Rn)ξn(a)D→ E where E ∼ Exp(1).

Proof Simple calculation shows that the Laplace–Stieltjes transform of ρn(Rn, a)×ξn(a) is given by

E[e−θρn(Rn)ξn(a)

]= ERn

[ρn(Rn)

1 − e−θρn(Rn)(1 − ρn(Rn))

]. (3.2)

From Proposition 1, ρn(Rn)a.s→ 0. Therefore, e−θρn(Rn) = 1 − θρn(Rn)+ o(ρn(Rn)).

Hence from (3.2),

E[e−θρn(Rn)ξn(a)

]= E

[1

1 + θ + o(1)

].

As 0 ≤ ρn(Rn) ≤ 1, the integrand is bounded for any fixed θ , |θ | < 1. Hence by thebounded convergence theorem the above expression converges to (1 + θ)−1 which isthe Laplace–Stieltjes transform of E(1). �

Remark It is clear from the above Theorem that {Rn} and {ρn(Rn)ξn(a)} are asymp-totically independent, since the limit of the conditional distribution of ρn(Rn)ξn(a),given Rn is independent of Rn .

Corollary 1 For some continuous random variables V1 and V2 with densities,

(i) If∑n

i=1 β2i → ∞, then

log ξn(a) − αn∑n

i=1 βi

αn

√∑ni=1 β2

i

D→ N (0, 1).


(ii) If∑∞

i=1 β2i < ∞ and

∑ni=1 βi → ∞, then

log ξn(a) − αn∑n

i=1 βi

αn

D→ V1.

(iii) If∑∞

i=1 βi < ∞, thenlog ξn(a)

αn

D→ V2.

Proof (i) By Theorem 2, ρn(Rn)ξn(a)D→ E . Taking the log on both sides we have

αn log(F̄(Rn)) − αn log(F̄(Rn − a)) + log ξn(a)D→ log E . (3.3)

Now, − log(F̄(Rn))D= ∑n

i=1 βi X∗i , where X∗

i ∼ Exp(1). Since∑n

i=1 β2i → ∞, by

the central limit theorem∑n

i=1 βi X∗i − ∑n

i=1 βi√∑ni=1 β2

i

D→ N (0, 1).

Note that F̄(Rn − a) → F̄(rF − a), a constant (> 0) a.s. Therefore, addingand subtracting αn

∑ni=1 βi on left hand side of (3.3) and dividing both sides by

αn

√∑ni=1 β2

i we get (i).For (ii) and (iii), we use similar arguments as in (i) and use the fact that∑ni=1 βi (X∗

i − 1) converges a.s. to S = ∑∞i=1 βi (X∗

i − 1), if∑∞

i=1 β2i < ∞. Since

S is a convolution of β1(X∗1 − 1) and

∑∞i=2 βi (X∗

i − 1) and the former has a density,we infer that S has a density and consequently V1 and V2 have densities. �

3.2 Case 2. rF = ∞

In this case, the asymptotic behaviour of ξn(a) depends on β(a) := limx→∞ ρ(x, a),when it exists. If β(a) < 1 or Rn → W , then ρn(Rn) → 0 a.s. and the arguments ofProposition 1 and Theorem 2 go through and we obtain the following Theorem. Weomit the details of the arguments.

Theorem 3 Suppose β(a) = limx→∞ ρ(x, a) exists. If β(a) < 1 or, β(a) = 1 butRn → W , then

(i) ξn(a) → ∞ in probability

(ii) ρn(Rn)ξn(a)D→ E .

Unfortunately, when β(a) = 1 and Rn → ∞, i.e.∑∞

i=1 βi = ∞, the situa-tion becomes quite complicated. We deal with a special case below. Let F̄(x) ∼x−γ , as x → ∞ (where γ > 0 is a constant), then clearly β(a) = 1. Further,

ρn(x) = ρ(x)αn =(

1 − a

x

)γαn. (3.4)


Theorem 4 Assume F̄(x) ∼ x−γ , as x → ∞ and∑∞

i=1 βi = ∞. If(log αn)−1 ∑n

j=1 β j → ∞ then P(ξn(a) = 0) → 1.

Proof Fix q > 1. Observe that, if x > a[1 − (

1 − βqn)1/γ ]−1

(=: xn), (1 − a/x)γ >

1 − βqn . By the mean value theorem,

xn = aγβ−qn ζ

1−1/γn , (3.5)

where 1−βqn < ζn < 1, so that ζn → 1. Now from (3.4) it follows that, for all x > xn ,

ρn(x) >(1−β

qn)αn → 1 as αn ↑ ∞ by Assumption 2. Therefore, given ε > 0, there

exists n0, depending on q and ε, such that for all n > n0, ρn(x) > 1 − ε, ∀ x > xn .

Fix any ε > 0. Then for all n > n0,

P(ξn(a) = 0) =∫ ∞

0ρn(x)d Rn(x) >

∫ ∞

xn

ρn(x)d Rn(x) > (1 − ε)P(Rn > xn).

(3.6)Also,

P(Rn > xn) = P[∑n

i=1 βi X∗i > ψ−1(xn)

] = P[∑n

i=1 βi X∗i > − log F̄(xn)

].

(3.7)

Using the expression (3.5) for xn , we have

− log F̄(xn) ∼ −qγ log βn + γ log γ − γ

(1

γ− 1

)log ζn + γ log a

= qγ log αn + k + o(1), as n → ∞,

where k(:= γ log(γ a)) is a constant. Therefore, denoting∑n

i=1 βi X∗i by Sn , the

probability on the right hand side of (3.7) can be written as

P[(log αn)−1Sn > qγ + o(1)

]. (3.8)

This converges to 1 if Sn/ log αnp→ ∞. But this follows from the Laplace–Stieltjes

transform condition∏n

j=1(1 + θβ j/ log αn) → 0, and taking logs we see that this isimplied by the condition of the Theorem. �

Example If αn = nδ and δ < 1 then the condition of Theorem 4 is satisfied for allγ > 0.

Remark If δ = 1 in the above example, then (log αn)−1 ∑nj=1 β j → 1 and hence

the condition of the Theorem is not satisfied. However, if γ < 1, we can find q > 1such that qγ < 1 and the probability in (3.8) converges to unity as n → ∞, since[ ∑n

j=1 β j]−1

Sna.s.→ 1 (see Lemma 2 below), and consequently P(ξn(a) = 0) → 1.

So the condition is not necessary.


4 Limiting distribution of {ξn(an)}

Pakes (2007) obtained several interesting limit theorems for number of near recordswith varying window width in classical setup. We investigate whether analogousresults can be derived for the Pfeifer model. In particular, if we allow the windowwidth to vary, the number of near records converges to a limit in distribution withoutany normalization. This result might be of interest in insurance studies. Throughoutwe will assume rF = ∞. The following notions are well known.

A function g(x) is said to be regularly varying (at ∞) of index ν if for somefunction C(·) such that C(x) → C > 0 as x → ∞,

g(x) = C(x) exp

{∫ x

1(ε(t)/t)dt

}, where ε(t) → ν as t → ∞. (4.1)

If in (4.1), ε(t) is assumed to be only bounded, then G is said to be extended regularlyvarying (at ∞). See Bingham et al. (1987).

We need the following lemma.

Lemma 2 Suppose ψ has a density ψ ′ which is extended regularly varying.

(i) Ifn∑

i=1

β2i → ∞ then

Rn − ψ( ∑n

i=1 βi)

(√∑ni=1 β2

i

)ψ ′( ∑n

i=1 βi)

D→ N (0, 1).

(ii) If∞∑

i=1

β2i < ∞ and

∞∑

i=1

βi = ∞, thenRn − ψ

( ∑ni=1 βi

)

ψ ′( ∑ni=1 βi

) D→ Z , where Z is

continuous.

Proof Observe that by the mean value theorem

Rn − ψ

(n∑

i=1

βi

)D=

(n∑

i=1

βi X∗i −

n∑

i=1

βi

)

ψ ′(V ∗n ) (4.2)

where V ∗n is a random variable lying almost surely between

∑ni=1 βi X∗

i and∑n

i=1 βi .

Also since∑n

i=1 βi → ∞, and βn is monotonically decreasing by our assumption,the strong law for weighted averages of i.i.d. random variables is applicable (seeEtemadi 2006). Hence

∑ni=1 βi X∗

i∑ni=1 βi

→ 1 a.s.

Note that V ∗n lies between

∑ni=1 βi X∗

i and∑n

i=1 βi . Therefore setting∑n

i=1 βi = un ,

ψ ′(V ∗n )

ψ ′( ∑ni=1 βi

) = ψ ′(unδn)

ψ ′(un), where δn → 1 a.s. (4.3)


Using (4.1) for ψ ′, and routine calculations, the right hand side of Eq. 4.3 con-verges to 1. Now, (i) follows by (4.2), (4.3) and the central limit theorem for[ ∑n

i=1 β2i

]−1/2][ ∑ni=1 βi (X∗

i − 1)].

(ii) follows from (4.2), (4.3) and Khinchine–Kolmogorov’s one-series Theoremfor

∑ni=1 βi (X∗

i − 1). See p. 201 in Arnold et al. (1998). �

The condition on ψ ′ in the above Lemma is fairly general. We impose a slightlystronger condition on ψ ′ for our next result. Let ψ−1 be denoted by �. Observe thatif ψ ′ exists then �′ exists, as �′(x) = [ψ ′(ψ−1(x))]−1, (recall that ψ is continuousand strictly increasing by Assumption 1 and hence ψ ′ > 0 on (0, ∞), if it exists).

Theorem 5 Suppose ψ has a density ψ ′. Let a > 0 be given. Then there exists asequence of positive reals {an} such that ξn(an) → ξ in distribution, where

(i) ξ has geometric distribution with parameter e−a, if∑n

i=1 βi → ∞ and ψ ′ isregularly varying with index ν > −1.

(ii) ξ has mixed geometric distribution, with E(sξ ) = E[ e−a�′(W )

1−s(1−e−a�′(W ))

]where W

is the almost sure limit of {Rn}, if∑∞

i=1 βi < ∞ and �′ has at most countablymany discontinuities.

Proof Let {an} be a sequence of positive reals to be chosen later. Easy computationyields

E(sξn(an)) = E

[ρn(Rn, an)

1 − s(1 − ρn(Rn, an)))

].

Observe that for any fixed s with |s| < 1 the integrand is bounded since 0 ≤ρn(Rn, an) ≤ 1. Hence {ρn(Rn, an)[1 − s(1 − ρn(Rn(an)))]−1} is uniformly inte-grable. So it is enough to show that ρn(Rn, an) converges in distribution to theappropriate limit. First we will prove (i). Here two cases can arise.

Case 1.∑n

i=1 β2i → ∞. For this case, define An := ψ

( ∑ni=1 βi

), Bn :=(√∑n

i=1 β2i

)ψ ′( ∑n

i=1 βi)

and Zn := B−1n (Rn − An). Let gn(x) :=

Bn x + An so that gn(Zn) = Rn . Recall �(x) = − log F̄(x) andρn(Rn, an) = (

F̄(Rn)/F̄(Rn − an))αn . Now using the mean value theorem

we have,

− log ρn(Rn, an) = αn�(gn(Zn)) − αn�

(gn

(Zn − an

Bn

))

= αnan

Bn�′(gn(Z∗

n))g′n(Z∗

n)

= αnan

Bn�′(Bn Z∗

n + An)Bn

= αnan�′(

An

(1 + Bn

AnZ∗

n

))(4.4)


where Z∗n is a random variable lying almost surely between Zn and Zn −

B−1n an . By Karamata’s Theorem (see Resnick 1987, page 17),

limx→∞

xψ ′(x)

ψ(x)= ν + 1. (4.5)

Therefore,

limn→∞

Bn

An= lim

n→∞

(√∑ni=1 β2

i

)ψ ′( ∑n

i=1 βi)

ψ( ∑n

i=1 βi)

= limn→∞

√∑ni=1 β2

i∑n

i=1 βilim

n→∞

( ∑ni=1 βi

)ψ ′( ∑n

i=1 βi)

ψ( ∑n

i=1 βi) = 0 (4.6)

since the first limit in (4.6) is 0 as βn converges to 0 monotonically by

Assumption 2 and second limit is ν+1 by (4.5). Since by Lemma 2(i) ZnD→

N (0, 1) and since Z∗n lie between Zn and Zn − B−1

n an a.s., Z∗n

D→ N (0, 1)

if B−1n an converges to 0. This together with (4.6) gives 1 + A−1

n Bn Z∗n =

1 + op(1). Since by our assumption on ψ ′, �′ exists and is a regularlyvarying function

�′(An(1 + BnAn

Z∗n)

)

�′(An)= 1 + op(1). (4.7)

Now choose

an := aψ ′( ∑ni=1 βi

)

αn.

By Assumption 2, B−1n an → 0. Putting this value of an in (4.4) and using

(4.7) we have

− log ρn(Rn, an) = a(1 + op(1))�′(An)ψ ′(

n∑

i=1

βi

)

= a(1 + op(1)),

(4.8)

since �′(An)ψ ′( ∑ni=1 βi

) = �′(ψ( ∑n

i=1 βi))

ψ ′( ∑ni=1 βi

) = 1. There-fore, − limn→∞ log ρn(Rn, an) = a in probability. This proves (i) forCase 1.

Case 2.∑∞

i=1 β2i < ∞. We define An , Bn , gn and Zn as in Case 1. In this case

Z∗n

D→ Z , where Z is a random variable which is not standard normal. Therest of the argument is as before. This completes the proof of (i).


Now to prove (ii). Since∑∞

i=1 βi < ∞ now, Rn → W a.s., where W is acontinuous random variable (see Lemma 1). Using the mean value theorem,we have

− log ρn(Rn, an) = αn�(Rn) − αn�(Rn − an) = αnan�′(R∗

n), (4.9)

where R∗n is a random variable lying almost surely between Rn −an and Rn .

Then by Assumption 2, an := aαn

→ 0, and hence limn→∞ Rn − an = Wa.s. which implies R∗

n → W a.s. Since by our hypothesis, �′ has at most

countably many discontinuities, �′(R∗n)

D→ �′(W ). Thus (4.9) implies,

ρn(Rn, an)D→ e−a�′

0(W ) which completes the proof of (ii). �

Example

(i) If αn = nδ , with 0 < δ < 1 and F(x) = 1−e−x1μ

with μ > 0 then ψ(x) = xμ

and all the conditions of Theorem 5 (i) are satisfied.(ii) If αn = nδ , δ > 1 and ψ(·) is any strictly increasing continuous function with

monotone density, then all conditions of Theorem 5 (ii) are satisfied.

Acknowledgments The authors thank the referees for their careful reading of the manuscript anddetailed comments. Their suggestions have led to a substantial improvement. The research of Arup Bosehas been supported by the J.C.Bose Fellowship, Department of Science and Technology, Govt. of India.His work was done while he was visiting Department of Economics, University of Cincinnati, USA. SreelaGangopadhyay thanks Department of Science and Technology, Govt. of India for funding this research(scheme no. SR/WOS-A/MS-02/2006) and Indian Statistical Institute, Kolkata for providing researchfacilities.

References

Arnold, B.C., Balakrishnan, N., Nagaraja, H.N.: Records. Wiley, New York (1998)Balakrishnan, N., Pakes, A.G., Stepanov, A.: On the number and sum of near-record observations. Adv.

Appl. Probab. 37, 765–780 (2005)Bingham, N., Goldie, C., Teugels, J.: Regular Variation. Cambridge University Press, Cambridge (1987)Bondesson, L.: Generalized Gamma Convolutions and Related Classes of Distributions and Densities.

Lecture Notes Statist., vol. 76. Springer, New York (1992)Bose, A., Gangopadhyay, S.: Convergence of linear functions of Pfeifer records. Extremes 13, 89–106

(2009)Etemadi, N.: Convergence of weighted averages of random variables revisited. Proc. Am. Math. Soc.

134(9), 2739–2744 (2006)Li, Y.: A note on the number of records near the maximum. Stat. Probab. Lett. 43, 153–158. (1999)Li, Y., Pakes, A.G.: On the number of near-maximum insurance claims. Insur. Math. Econ. 28, 309–318

(2001)Pakes, A.G.: Limit theorems for numbers of near records. Extremes 10, 207–224 (2007)Pakes, A.G., Steutel, F.W.: On the number of records near the maximum. Aust. J. Stat. 39, 179–193 (1997)Pfeifer, D.: Limit laws for inter-record times from nonhomogeneous record values. J. Organ. Behav. Stat.

1, 69–74 (1984)Resnick, S.I.: Extreme Values, Regular Variation, and Point Processes. Springer, New York (1987)Villaseñor, J.A., Arnold, B.C.: On limit laws for sums of Pfeifer records. Extremes 10(1), 235–248 (2007)

asymptotic properties of near pfeifer records

Documents