aa r X i v : . [ m a t h . P R ] S e p Journal of the Kerala Statistical Association , Vol. 10, December 1999, p. 01-07.
On Geometric Infinite Divisibility
E Sandhya
Department of Statistics, Prajyoti Niketan College, Pudukkad, Thrissur - 680 301, India.and
R N Pillai
Valiavilakam, Ookode P.O., Vellayani, Trivandrum, India.
Abstract.
The notion of geometric version of an infinitely divisible law is in-troduced. Concepts parallel to attraction and partial attraction are developedand studied in the setup of geometric summing of random variables. Inroduction
Klebanov, et al. [5] dened;
Definition 1.1.
A random variable (r.v) X is geometrically infinitely divisible(GID) if for every p ∈ (0 , X d = X ( p )1 + . . . + X ( p ) N p , where N p and { X ( p ) i } areindependent, { X ( p ) i } are i.i.d and N p is a geometric r.v with mean 1 /p .Equivalently, a r.v X with characteristic function (c.f) φ ( t ) is GID if and only ifexp { − /φ ( t ) } is an infinitely divisible (ID) c.f. They also introduced geometricallystrictly stable (GSS) laws as: Definition 1.2.
A r.v Y is GSS if for every p ∈ (0 , c ( p ) > Y d = c ( p ) { X + . . . + X N p } , where N p and { X i } are independent, { X i } are i.i.d, Y d = X and N p is a geometric r.v with mean 1 /p .Pillai [9] introduced semi- α -Laplace laws as: Definition 1.3.
A distribution function (d.f) F with c.f φ ( t ) = 1 / (1 + g ( t )) issemi- α -Laplace of exponent α , 0 < α ≤
2, if g ( t ) = ag ( bt ) for some constants a and b , 0 < b < < a and α is the unique solution of ab α = 1, b is called the order ofthe semi- α -Laplace law. Note. If b and b are orders of F such that log b / log b is irrational, then g ( t ) = c | t | α for some constant c > Remark 1.1.
The result and discussion in Kagan et al. [4, p.323,324] is relevantin this context.This study is motivated by the one-to-one correspondence between ID and GIDlaws (see the equvalent of definition 1.1) and extend concepts of attraction andpartial attraction to the class of GID laws.It is known that the class of GID laws is a proper subclass of the class of IDlaws, Sandhya [12]. The idea of geometric compounding is related to p -thinning ofpoint processes which has applications in the study of patterns of occurrences ofcrimes and accidents as many of them go unreported. When we consider p -thinningof renewal processes for every p ∈ (0 , p -thinning, asthose with semi-Mittag-Leffler laws as the interval distribution and observed that,a renewal process is Cox if and only if its interval distribution is GID. Aspects ofgeometric compounding and GID laws in system reliability studies are explored inPillai and Sandhya [11]. Since exponential mixtures are GID (Pillai and Sandhya[10]), construction of distributions useful in situations mentioned above is easier.In section 2 the ideas of geometric version of an ID law and geometric attractionare introduced and studied. The notion of partial geometric attraction is developedin section 3.A detailed presentation of these ideas and a generalization of GID laws to ν -IDlaws are available in Sandhya [12]. For a different approach to geometric attractionsee Mohan et al. [8]. Klebanov and Rachev [6] develop ν -infinite divisibility, definegeometric attraction for random vectors and the domains of attraction of ν -stablerandom vectors. Kozubowski and Rachev [7] discuss multivariate geometric stablelaws. Semigroup related to random stability was studied by Bunge [1].2. Geometric Version and Geometric AttractionDefinition 2.1.
Let F be an ID d.f with c.f exp {− g ( t ) } . Then the d.f G is thegeometric version (g.v) of F if and only if G has c.f 1 / (1 + g ( t )). Remark 2.1.
Definition 2.1 leads to a one-to-one correspondence between thec.fs of an ID law and its g.v. It may also be noticed that the relation between an
N GEOMETRIC INFINITE DIVISIBILITY 3
ID law and its g.v basically is the relation between a compound Poisson law andthe corresponding compound geometric law.
Remark 2.2.
The g.v of an ID law is GID and hence ID.
Theorem 2.1.
A d.f G is the g.v of an ID law F with c.f exp {− g ( t ) } , if and onlyif G is the limit law of geometric sums of the form U n = X ,n + . . . + X N n ,n where N n a geometric r.v with mean n , is independent of { X i,n } and { X i,n } is i.i.d withc.f exp {− g ( t ) /n } .Proof. If φ U n ( t ) denotes the c.f of U n , then we have, φ U n ( t ) → / (1 + g ( t )) as n → ∞ . The converse is obtained by retracing the steps. (cid:3)
Definition 2.2.
A d.f F is said to be geometrically attracted to another d.f G , if G is the limit law of a sum Y n = B − n { X + . . . + X Np n } (2.1)where N p n , which is geometric with mean 1 /p n , is independent of { X i } which arei.i.d as F, B n > n → ∞ , p n → B n → ∞ . Definition 2.3.
The set of all d.fs that are geometrically attracted to G is calledthe domain of geometric attraction (d.g.a) of G . Theorem 2.2.
Every GSS law is geometrically attracted to itself. Conversely, a d.f F belongs to the d.g.a of a GSS law provided the d.f of Y n in (2.1) with B n = p − n/α , < α ≤ , and p n = p n tends to a limit for two values of p say p and p , suchthat log p / log p is irrational.Proof. Let F be GSS. By a little algebra it follows from definition 1.2 that its c.f φ ( t ) satisfies, for every p ∈ (0 , φ ( t ) = pφ ( ct )1 − qφ ( ct ) , q = 1 − p, with c = p /α , 0 < α ≤
2. And on iteration, φ ( t ) = p n φ ( p n/α t )1 − (1 − p n ) φ ( p n/α t ) . E Sandhya and R N Pillai
The right hand side corresponds to the c.f of Y n in (2.1) with B n = p − n/α and p n = p n , for all n > F is geometrically attracted to itself. Conversely,if Y n = p n/α { X + . . . + X Np n } , then its c.f is φ Y n ( t ) = 11 + p − n g ( p n/α t )where g = (1 /φ ) − φ being the c.f of X . Suppose φ Y n ( t ) → / (1 + g ( t )), then φ Y n +1 ( t ) →
11 + p − g ( p /α t ) = 11 + g ( t ) . Hence, g ( t ) = p − g ( p /α t ) . (2.2)If for two values of p , say p and p , such that log p / log p is irrational, (2.2) issatised then by Kagan et al. [4, p.324], the converse follows. (cid:3) With [ x ] denoting the integer part of x , we have; Theorem 2.3.
A necessary condition that F is in the d.g.a of G which is GSS, isthat [1 /p n ] = n . Conversely, if F is in the d.g.a of another law G and [1 /p n ] = n ,then G is GSS.Proof. Let F belong to the d.g.a of G . In terms of the corresponding c.fs we have,11 + p − n g ( t/B n ) →
11 + g ( t ) . That is, { [1 /p n ] + θ n } g ( t/B n ) → g ( t )where 0 ≤ θ n = p − n − [ p − n ] ≤
1. Hence[1 /p n ] g ( t/B n ) → g ( t ) . This implies that [1 /p n ] = n , since nowexp {− [1 /p n ] g ( t/B n ) } → exp {− g ( t ) } which is strictly stable. Converse follows by retracing the steps. (cid:3) N GEOMETRIC INFINITE DIVISIBILITY 5 Partial Geometric AttractionDefinition 3.1.
A d.f F is said to be partially geometrically attracted to anotherd.f G if there exists a subsequence k < k < . . . < k n < . . . of the set of positiveintegers such that Y k n → Y in law, where Y n is defined in (2.1) and G is the d.f of Y . Definition 3.2.
The set of all d.fs that are partially geometrically attracted to ad.f G is called the domain of partial geometric attraction (d.p.g.a) of G . Theorem 3.1.
Corresponding to every semi- α -Laplace law there exists an ID lawthat is partially geometrically attracted to it. Conversely, if an ID law is partiallygeometrically attracted to some law G with k n +1 /k n → a , and B k n +1 /B k n → /b ,then G is semi- α -Laplace of order b and ab α = 1 .Proof. Suppose Y is semi- α -Laplace with d.f G . Hence its c.f φ satisfies φ ( t ) = 11 + g ( t ) = 11 + a n g ( b n t ) , < b < < a. Choose a subsequence { k n } such that k n = [ a n ] and B k n = b − n for all k and n ,positive integers. Let φ k n ( t ) = { a n ] g ( b n t ) } − and put θ n = a n − [ a n ]. Now, | φ k n ( t ) − φ ( t ) | = | θ n g ( b n t ) || φ k n ( t ) || φ ( t ) | . Since φ k n ( t ) and φ ( t ) are c.fs | φ k n ( t ) || φ ( t ) | ≤ g ( b n t ) → n → ∞ and hence φ k n ( t ) → φ ( t ). But φ k n ( t ) is the g.v ofexp {− [ a n ] g ( b n t ) } and it corresponds to the r.v Y k n = b n { X + . . . + X N / [ an ] } and Y k n → Y in law.Conversely, suppose that an ID law F is in the d.p.g.a of G with k n +1 /k n → a and B k n +1 /B k n → /b . Choose k n = [ a n ] and B k n = 1 /b n for all k and n positiveintegers. That is, { a n ] g ( b n t )) } − → { g ( t ) } − , E Sandhya and R N Pillai where the L.H.S is the c.f of Y k n and the R.H.S that of G . Now the c.f of Y k n +1 is { a n +1 ] g ( b n +1 t ) } − andlim n →∞ [ a n +1 ] g ( b n +1 t ) = lim n →∞ [ a n ] g ( b n +1 t ) + lim n →∞ { a n ( a −
1) + θ n − θ n +1 } g ( b n +1 t )= g ( bt ) + ( a − g ( bt )= ag ( bt ) . That is, { g ( t ) } − = { ag ( bt ) } − , implying G is semi- α -Laplace, and theproof is complete. (cid:3) Theorem 3.2.
An ID law F is in the domain of partial attraction of a semi-stablelaw G having order b > , if and only if its g.v is in the d.p.g.a of a semi- α -Laplacelaw which is the g.v of G .Proof. Follows from the definitions of partial attraction, partial geometric attrac-tion and g.v of an ID law. (cid:3)
Gnedenko and Kolmogorov [2] gave a transitive relation for a law to belong tothe domain of partial attraction of another law. An analogous result for d.p.g.a isgiven below, the proof of which is straight forward.
Theorem 3.3.
If a law F ′ is in the d.p.g.a of the law G ′ , and G ′ is in the d.p.g.aof the law H ′ , then F ′ is in the d.p.g.a of the law H ′ , if F ′ , G ′ and H ′ are the g.vsof some ID laws F, G and H respectively. Acknowledgements.
The authors thank the referee and the editor for some usefulsuggestions. The first author’s work was supported by a research fellowship fromC.S.I.R, India.
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N GEOMETRIC INFINITE DIVISIBILITY 7 [5] Klebanov, L.B, Maniya, G.M. and Melamed, I.A (1984); A problem of Zolotarev and analogsof infinitely divisible and stable distributions in the scheme for summing a random numberof random variables,
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