International Journal of Statistical Distributions and Applications
Volume 1, Issue 2, December 2015, Pages: 33-36

Max-analogues of N-infinite Divisibility and N-stability

Satheesh Sreedharan1, Sandhya E.2

1Department of Applied Sciences, Vidya Academy of Science and Technology, Thalakkottukara, Thrissur, India

2Department of Statistics, Prajyoti Niketan College, Pudukad, Thrissur, India

(S. Sreedharan)
(Sandhya E.)

Satheesh Sreedharan, Sandhya E. Max-analogues of N-infinite Divisibility and N-stability. International Journal of Statistical Distributions and Applications. Vol. 1, No. 2, 2015, pp. 33-36. doi: 10.11648/j.ijsd.20150102.11

Abstract: Here we discuss the max-analogues of random infinite divisibility and random stability developed by Gnedenko and Korolev [5]. We give a necessary and sufficient condition for the weak convergence to a random max-infinitely divisible law from that to a max-infinitely divisible law. Introducing random max-stable laws we show that they are indeed invariant under random maximum. We then discuss their domain of max-attraction.

Keywords: Max-infinite Divisibility, Max-stability, Domain of Max-attraction, Extremal Processes

Contents

1. Introduction

In the classical summation scheme a characteristic function (CF)  is infinitely divisible (ID) if for every  integer there exists a CF  such that . The classical de-Finetti theorem for ID laws states that  is ID iff  where  are some positive constants and  are CFs.

Klebanov, et al. [1] extended the notion of ID laws to geometrically ID (GID) laws using geometric (with mean ) sums. According to this,  is GID if for every  there exists a CF  such that , the geometric law being independent of the distribution of  for every . They also proved an analogue of the de-Finetti theorem in the context, viz.  is GID iff , where  and  are as above. Consequently,  is GID iff  where  is a CF that is ID. Subsequently [2] (also reported in [3]), [4], [5] and [6] have discussed attraction and the first three works that of partial attraction for GID laws.

Later [2] (also reported in [7]), [4], [5], [6] and [8] extended the notion of GID to random (N) ID laws based on -sums. [2] and [7] defined N-ID laws as: a CF  is N-ID, where  is a positive integer-valued random variable (r.v) having finite mean with probability generating function (p.g.f)  if there exists a CF  such that  for every . We need the distributions of  and  to be independent for every . She noticed that when  and  are of the same type, the above is an Abel (Poincare) equation. She also gave two examples of non-geometric laws for . [5] (section 4.6) and [6] went further by proving the de-Finetti analogue for N-ID laws viz. a CF  is N-ID iff  where  is a Laplace transform (LT) that is also a solution to the Poincare (Abel) equation. They then concluded that a CF  is N-ID iff  where  is CF that is ID. In this description  and  are related by , , , where  is the p.g.f of the r.v  that is positive integer-valued having finite mean. [8] also arrived at the same conclusion under the same assumptions but the arguments were based on Levy processes instead of proving the de-Finetti analogue enroute. Poincare equation is given by P being a p.g.f. [5], [6] and [9] discussed Poincare equation and examples of deriving a p.g.f from .

To circumvent the main constraints in the development of N-ID laws viz. that  is a positive integer-valued r.v having finite mean,  is a LT that is also a solution to the Poincare equation, [10] introduced -ID laws for any LT  and  a non-negative integer-valued r.v derived from . The important case of compound Poisson distributions was thus brought under random-ID laws. [5], [6] and [10] also discussed attraction and partial attraction for N-ID/ -ID laws. The discrete analogue of this was developed in [9]. The r.v  in N-ID laws has the following property.

Lemma 1.1  as , and the LT of U is , see [5], p.138.

Coming to the max-analogue, [11] introduced the notion of max infinitely divisible (MID) laws. A distribution function (d.f)  is MID if  is a d.f for each integer . Since  is always a d.f in the univariate case all d.fs in R are MID, see [13]. Hence a discussion of MID laws is relevant for d.f s in  integer and the max operations are to be taken component wise. Thus in this paper all d.fs are assumed to be in  integer, unless stated otherwise. Later [12] introduced geometric max infinitely divisible (GMID) laws and geometric max stable (GMS) laws, see also [13]. [12] also discussed certain connections between GMID/ GMS laws and extremal processes. From [11] we have the max-analogue of the classical de Finetti's theorem.

Theorem 1.1 A d.f  is MID iff for some d.fs  and constants

.

Using the transfer theorem for maximums in [14] we can study the limit distributions of random maximums. [15] briefly discussed the max-analogue of N-ID laws to obtain stationary solutions to a generalized max-AR(1) scheme. However, there was an inadvertent omission, as the discussion did not stress that the LT  should also be a standard solution to the Poincare equation.

Proceeding from [15], we discuss random (N) MID (N-MID) laws that is the max-analogue of N-ID laws, in section 2. In section 3 we discuss random (N) max-stable laws, generalise certain results on GMS laws in [12] to N-max-stable laws and their domain of max-attraction. The convergence discussed here is weak convergence of d.fs, unless stated otherwise.

2. Random MID Laws

We begin by defining N-MID laws analogous to the N-ID laws in [5] correcting the omission mentioned above.

Definition 2.1 Let  be a standard solution to the Poincare equation and , a positive integer-valued r.v having finite mean with p.g.f . A d.f  in  is N-MID if for each , there exists a d.f  that is independent of , such that  for all .

Theorem 2.1 A d.f  which is the weak limit of a sequence  of N-MID d.fs is itself N-MID.

Proof. By virtue of the continuity of p.g.fs, for every , we have

.

We now have an analogue of theorem 1.1, a de Finetti type theorem, for N-MID laws.

Theorem 2.2 Let  be a standard solution to the Poincare equation. A d.f  in  is N-MID iff for some d.fs  and constants , .

Proof. See the proof of theorem 3.5 in [15].

Notice that for a LT  is a p.g.f. Hence the above representation is essentially the weak limit of random-maximums under the transfer theorem for maximums. The next result facilitates the construction and/ or identification of N-MID d.fs.

Theorem 2.3 A d.f  is N-MID iff , where  is a standard solution to the Poincare equation and  is a MID d.f.

Proof. We have seen that an N-MID d.f  admits the representation for some d.fs ,

Since  is continuous we can proceed as

Where  is MID. Note the fact that every Poisson maximum is MID and every MID d.f is the weak limit of Poisson maximums [11]. Conversely, consider

where  is MID and  is the LT of the d.f . Now  is N-MID since the above is the integral representation of a d.f that is the weak limit under the transfer theorem for maximums. This completes the proof.

Corollary 2.1 A d.f is N-MID iff it is the limit distribution, as , of a random maximum of i.i.d r.vs.

Now we proceed to prove the max-analogue of theorem 4.6.5 in [5]. Let, for every  with d.f  be i.i.d random vectors in  and  a positive integer-valued r.v having finite mean with p.g.f that is independent of  for every  and . Let  denote the integer part of .

Theorem 2.4 Let  be N-MID. Then

(1)

iff there exists a d.f  that is MID and

(2)

Proof. The sufficiency of the condition (2) follows from the transfer theorem for maximums by invoking the relation  and  as . Conversely (1) implies

(3)

Since  is a LT we have;

Again, since , this implies that

(4)

Since  is a d.f that is N-MID for every ,  is also N-MID by theorem 2.1. Hence there exists a d.f  that is MID such that

(5)

On the other hand for  we have

(6)

Hence by (4) and (5) we get from (6)

(7)

Now applying the transfer theorem for maximums it follows that

.

Hence by (1) . That is, by (7), (2) is true with  being MID, completing the proof.

3. Random Max-stable Laws

Theorem 2.4 identifies the weak limit of partial -maximums of certain component r.vs as a function of the weak limit of partial maximums of the same component r.vs and vice-versa. This description thus enables us to define random max-stable (N-max-stable) laws analogous to the N-stable laws in [5] and their domains of N-max-attraction. This is facilitated by prescribing in theorem 2.4. Notice also that here the discussion can be for d.fs in R.

Definition 3.1 A d.f  is N-max-stable iff , where  a max-stable d.f and  is a standard solution to the Poincare equation.

Theorem 3.1 An N-max-stable d.f can be represented as , for every , where  and  are d.fs of the same type. Here  and  are independent for each ,  is the p.g.f of , a positive integer-valued r.v having finite mean.

Proof. Since  is N-max-stable we have the following representation for every .
.

Notice that  and  are d.fs of the same type, [16]. Since  is max-stable,  also is max-stable. Thus the above representation describes an N-max-stable d.f as an -sum of d.fs of the same type for every , proving the result.

We now generalise proposition 3.2 on GMS laws in [12] to N-max-stable laws.

Theorem 3.2 For a d.f  on  the following statements are equivalent.

(i)  is N-max-stable

(ii)  is max-stable

(iii) There exists an  and an exponent measure  concentrated on  such that for

.

(iv) There exists a multivariate extremal process  governed by a max-stable law and an independent r.v  with d.f  and LT  such that .

Proof.  is N-max-stable implies , where  is max-stable. This implies  is max-stable.

From the representation of a max-stable d.f by an exponent measure  and from (ii) we have  This implies  or .

By (iii) we have the exponent measure  corresponding to the max-stable law identified in (ii). Let  be the extremal process governed by this max-stable law. That is .

Hence

.

is now obvious. Thus the proof is complete.

A notion that is closely associated with max-stable laws is their domain of max-attraction. The notion of geometric max-attraction for GMS laws was discussed in [12] and [13]. We now briefly discuss this for N-max-stable laws.

Definition 3.2 A d.f  belongs to the domain of N-max-attraction (DNMA) of the d.f  (with non-degenerate marginals) if there exists constants  and  such that meaning that

, for each  where and .

Recalling that  is continuous and that max-attraction of  to  is equivalently specified by , we have the following result as an immediate consequence of theorem 2.2.

Theorem 3.3 Let  be a standard solution to the Poincare equation. A d.f  is N-max-stable iff for some d.f  and constants  and ,

.

Again, from theorem 2.4, choosing  and  such that , where  and , from the classical results on max-stable laws and their domains of attraction, we have

Theorem 3.4 A d.f  belongs to the DNMA of the d.f  iff it belongs to the DMA of .

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