Max-analogues of N-infinite Divisibility and N-stability
Satheesh Sreedharan^{1}, Sandhya E.^{2}
^{1}Department of Applied Sciences, Vidya Academy of Science and Technology, Thalakkottukara, Thrissur, India
^{2}Department of Statistics, Prajyoti Niketan College, Pudukad, Thrissur, India
Email address:
To cite this article:
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
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 .
References