ON A CLASS OF DISTRIBUTIONS STABLE UNDER RANDOM SUMMATION

Transcription

1 Applied Probability Trust (6 Deceber 200) ON A CLASS OF DISTRIBUTIONS STABLE UNDER RANDOM SUMMATION L.B. KLEBANOV, Departent of Probability and Statistics of Charles University A.V. KAKOSYAN, Yerevan State University S.T. RACHEV, University of Karlsruhe G. TEMNOV, School of Matheatical Sciences, University College Cork Abstract We investigate a faily of distributions having a property of stability-underaddition, provided that the nuber ν of added-up rando variables in the rando su is also a rando variable. We call the corresponding property a ν-stability and investigate the situation with the seigroup generated by the generating function of ν is coutative. Using results fro the theory of iterations of analytic functions, we show that the characteristic function of such a ν-stable distribution can be represented in ters of Chebyshev polynoials, and for the case of ν-noral distribution, the resulting characteristic function corresponds to the hyperbolic secant distribution. We discuss soe specific properties of the class and present particular exaples. Keywords: Stability, rando suation, characteristic function, hyperbolic secant distribution 2000 Matheatics Subject Classification: Priary 60E07 Secondary 60E0 Postal address: Prague Sokolovska 83, Prague-8, CZ 8675, Czech Republic; Eail: klebanov@chello.cz Postal address: Western Gateway Building, Western Road, Cork, Ireland ; Eail: g.tenov@ucc.ie

2 2 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov. Introduction In any applications of probability theory certain specific classes of distributions have becoe very useful, usually called fat tailed of heavy tailed distributions. The Stable distributions that originate fro the Central Liit proble, are probably ost popular aong the heavy tailed distributions, however there is a wide collection of classes of distributions, all related to Stable ones in any various ways, often these relations are not at all obvious. Besides, certain generalizations of stable distributions are known, using sus of rando nubers of rando variables (instead of sus with deterinistic nuber of suands), see e.g. Gnedenko [4], Klebanov, Mania, Melaed [9], for the exaples of such, including the so-called ν-stable distributions, introduced independently by Klebanov and Rachev [0] and Bunge []. In the present paper, we focuse on presenting further exaples of strictly ν-stable rando variables, that could be useful in practical applications, including applications in financial atheatics. 2. Definition of strictly ν-stable r.v. s, properties and exaples In the present section, we give a general insight on strictly ν-stable distributions and describe soe exaples. 2.. Basic definitions Let X, X, X 2,..., X n,... is a sequence of i.i.d. rando variables, and let { ν p, p } be a faily of soe discrete r.v. s taking values in the set of natural nubers N. Assue that this faily does not depend on the sequence {X j, j }, and that, for (0, ), Eν p = p, p. () Definition. We say that the r.v. X has a strictly ν stable distribution, if p it holds that X = d p /α ν p i= where α (0, 2] is called the index of stability. X j,

3 Distributions stable under rando suation 3 After this general definition, a narrower class is defined for α = /2. Definition 2. We call the r.v. X a strictly ν noral r.v., if EX = 0, EX 2 =, and the following holds: X d = p /2 ν p i= X j, p. Closely related to stability property is the property of infinite divisibility, so we also give the following definition. Definition 3. X has a strictly ν infinitely divisible distribution, if for any p, there exists a r.v. Y (p), s.t. X d = ν p j= Y (p) j, with Y (p), Y (p),..., Y n (p),... being iid r.v. s A powerful tool for investigating distributions properties is the generating function, so the generating function of the r.v. ν p, will be denoted by P p (z) := E [z νp ]. Moreover, we denote by A the seigroup generated by the faily { P p, p }, with the operation of the functions coposition Suary of the known results With regards to the definitions above, the following results are known (see e.g. [], [8], [0] for proofs and details). Theore 2.. For the faily { P p, p }, with E [ν p ] = p, there exists a strictly ν-noral distribution, iff the seigroup A is coutative. Suppose that we have a coutative seigroup A. Then the following stateents (that we refer to in the sequel as Properties) are known to be true (see e.g. [5] for proofs and details):. The syste ϕ(t) = P p (ϕ(pt)), p. (2) of functional equations has a solution that satisfies the initial conditions ϕ(0) =, ϕ (0) =. (3)

4 4 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov The solution is unique. In addition, there exists a distribution function (cdf) A(x) (with A(0) = 0) such that ϕ(t) = e tx da(x). (4) 0 2. The characteristic function (ch.f.) of the strictly ν-noral distribution has the for f(t) = ϕ(at 2 ), a > 0. (5) 3. A ch.f. g(t) is a ch.f. of a ν-infinitely divisible r.v., iff there exists a chf h(t) of an infinitely divisible (in the usual sense) r.v., such that f(t) = ϕ( ln h(t)). (6) The relation (6) allows obtaining explicit representations of ch.f. of strictly ν-stable distributions. Clearly, they are obtained through applying (6) to the ch.f. ( h(t) ) of strictly stable (in the usual sense) distributions. Moreover, note that the ch.f. ϕ(ait), a R, is the ch.f. of an analogue of the degenerate r.v., and that for the r.v. with such ch.f. the following analogue of the Law of Large Nubers exists. Theore 2.2. Let X, X 2,..., X n,... be a sequence of iid rando variables with the finite absolute value of the first oent, and { ν p, p } a faily of r.v. s taking values in N, independent of the sequence {X j, j =, 2,... }. Assue that E [ν p ] = p and that the seigroup A is coutative. Then the series p ν p X j j= convergence is a r.v. having the ch.f. is convergent is distribution, as p 0, and the liit of ϕ(ait). The proof of this theore follows straightforwardly fro the Property outlined above and fro the Transfer Theore of Gnedenko (see, [4]). In the following paragraph we discuss several particular exaples of strictly ν- noral and strictly ν-stable distributions Exaples and the outline of the proble

5 Distributions stable under rando suation 5 Exaple 2.. The usual stability. Assue the following setup : ν p = p with probability, where p = {, 2,..., n,... }, and so P p (z) = z /p. Clearly, here the seigroup A is coutative. Furtherore, ϕ(t) = exp{ t} = e tx da(x), where A(x) is a cdf with a single 0 unit-sized jup at x =. In this setup the strictly ν-noral ch.f. is the ch.f. of the noral (in the usual sense) r.v. with the zero ean. Exaple 2.2. The geoetric suation schee. Suppose, ν p is the r.v. having a geoetric distribution P{ν p = k} = p( p) k, k =, 2,..., p (0, ). Clearly, here E [ν p ] = p, and P p(z) = pz ( p)z, p (0, ). It is quite straightforward to check that A is coutative. Moreover, a direct calculation gives ϕ(t) = +t = e tx e x dx, i.e. A(x) is the cdf of the exponential distribution. So that a ν-analogue of the strictly noral distribution is the Laplace distribution with the ch.f. f(t) = +at 2. Exaple 2.3. Branching process schee. Let P(z) be soe generating function, with P () = p 0 notation is p 0 = /P (), with the condition p 0 < ). 0 > (so that the introduced Consider now a faily given by P 0 n (z) = P 0(n ) (P(z)), n =, 2,.... Related to { } that is another faily of the r.v. s ν p : P p (z) = P 0 n (z), p, n =, 2,... =:. Clearly, the seigroup A coincides with the faily {P p p }. The ch.f. ϕ(t) is a solution of the functional equation ϕ(t) = P (ϕ(p 0 t)). It can be noted that the content of the paper by Mallows and Shepp [2] is actually based on considering an exaple identical to the Exaple 2.3 above. Probably, neither the authors of that work nor its reviewers were failiar with the works by Klebanov and Rachev [0] and Bunge [], which had dealt with exactly the sae exaple a nuber of years earlier.bf Like entioned in Introduction, in the present work we ai in widening the collection of exaples that involve rando suation with the coutative seigroup A. For that reason, we address the description of pairs of certain coutative generating p n 0

6 6 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov functions P and Q, i.e. the ones for which the balance equality P Q = Q P holds, but including only the case when there exists no such function H such that P = H 0k and Q = H 0 for soe k, N (which would be exactly the case of the Exaple 2.3). In a general setting, the proble of describing all such coutative pairs of generating functions appears, unfortunately, far involved to approach. However, certain special cases are rather straightforward for consideration. In order to approach the proble, we will use certain notions typical for the theory of iterations of analytic functions, that we outline in the separate section below. 3. Theoretic justification via iterations of analytic functions Let P be a rational function with (deg) 2. Denote by P 0n its nth iteration. The functions P and Q are called conjugates, if there exists a linear-fractional function R, such that P R = R Q. A subset E of the extended coplex plane C is called copletely invariant, if its coplete inverse iage P (E) coincides with E. The axial finite copletely invariant set E(P) exists and is called the exceptional set of the function P. It is always the case that card E(P) 2. Moreover, if card E(P) = then the function P is a conjugate to a polynoial, while for card E(P) = 2 the function P is a conjugate to Q(z) = z n, n Z\{0, }. Clearly, E(Q) = {0, }. If P is a rational function, then it is known (see e.g. [2]) that there us a finite nuber of open sets F i, i =,..., r, which are left invariant by the operator P and are such that (in the sequel, we will refer to the two points below as Conditions). the union r F i is dense on the plane i= 2. P behaves regularly on each of F i. The latter eans that the terini of the sequences of iterations generated by the points of F j are either precisely the sae set, which is then a finite cycle, or they are finite cycles of finite or annular shaped sets that are lying concentrically. In the first case the cycle is attracting, in the second one it is neutral. The sets F j are the Fatou doains of P, and their union is the Fatou set F (P) of

7 Distributions stable under rando suation 7 P. The copleent of F (P) is the Julia set J (P) of P. Note that J (P) is either a nowhere dense set (that is, without interior points) and an uncountable set (of the sae cardinality as the real nubers), or J (P) = C. Like F (P), J (P) is left invariant by P, and on this set the iteration is repelling, eaning that P(z) P(w) > z w for all eleents w in a neighborhood of z (within J (P)). This eans that P(z) behaves chaotically on the Julia set. Although there are points in the Julia set whose sequence of iterations is finite, there is only a countable nuber of such points (and they ake up an infinitely sall part of the Julia set). The sequences generated by points outside this set behave chaotically, a phenoenon called deterinistic chaos. Let z 0 be a repelling fixed point of the function P, and let λ = P (z 0 ). Define Λ : z λz. Then there exists a unique solution of the Poincaré equation F Λ = P F, F (0) = z 0, F (0) =, that is eroorphic in C. Now let I(P) = F (J (P)). If for two functions P and Q we have P Q = Q P, then they have the sae function F. There are the two following possibilities:. I(P) = C, in which case J (P) = C,. 2. I(P) is nowhere dense and consists of analytic cuvrves. Fatou [3], and Julia [6] investigated the case. It turned out that is this case P and Q can be reduced by a conjugancy either to the for P(z) = z and Q(z) = z n or to the for P(z) = T (z) and Q(z) = T n (z), where T k is the Chebyshev polynoial deterined by the equation cos(kζ) = T k (cos ζ).

8 8 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov 4. Main results 4.. A new exaple Let us return to the study of ν-noral and ν-stable rando variables. Recall that we deal with the faily {ν p, p } taking its values in N = {, 2,... }. As before, we work with the generating function P p (z) = E [ z ν p ] of ν p. The iportant result that we stressed says the a strictly ν-noral (resp. strictly ν-stable) r.v. exist iff the seigroup A generated by {P p, p } is coutative. If P p, p, is a rational function (with deg 2) satisfying Condition 2 of the above section, then either P p (z) is reduced to a for P p (z) = z /p, p { n, n =, 2,... }, and then we deal, in fact, with the classical (deterinistic) suation schee, or P p (z) is reduced to the for P p (z) = T / p (z), p { n 2, n =, 2,... }. Clearly, the polynoial T (z) is not a generating function itself, however a function to which it is a conjugate, specifically the function P p (z) = { } T / p (/z), p, n =, 2,... n2, (7) is indeed a generating function, the fact that we prove below. Moreover, below we consider in soe details a faily of r.v. s { ν p, p { n, n =, 2,... } } that have generating functions of the for (7), and investigate the corresponding strictly ν- noral and strictly ν-stable distributions. Lea. Let P n (x) be a polynoial with deg P n = n by to the even powers of x, and whose zeros are all within the interval (, ). Let P n () = and polynoial s coefficient with with x n be positive. Then for any natural nuber k, the function is a generating function. xk P(x) = P n ( x ) Proof. Represent P n (x) as P n (x) = b 0 + b x + + b n x n = b n n j= (x a j ), where a j (j =,..., n) are the zeros of the polynoial P n sorted in the order of ascendance. As P n is a polynoial by the even powers of x, then if a j is a zero of P n,

9 Distributions stable under rando suation 9 then a j is also a zero of P n. Therefore, Obviously, P n ( x ) = = b n j= n/2 b n n ( x a j) j= = n/2 b n j= ( x a ) ( j x + a j) ( x a ) ( j x + a j) = n/2 x 2 b n a 2 (8) j= j x2 x 2 a 2 = j x2 k=0 a 2k j x 2k+2 is a series with positive (non-negative) coefficients, converging when x. Fro (8), it now follows that P(x) = xk P n ( x ) is a series also convergent when x, having non-negative coefficients, and P() =. Hence, P(x) is a generating function of soe rando variable. Corollary. Let T n (x) be a Chebyshev polynoial of degree n. Then P(x) = P n ( x ) is a generating function of soe r.v. which takes values in N. Proof. When n is an even nuber, the result follows directly fro Lea and fro the properties of Chebyshev polynoials. For odd n, consider the representation T n (x) = xp n (x), where P n (x) is a polynoial by the even degrees of x, satisfying the conditions of Lea. Let us now set := { n 2, n =, 2,... }. Consider the faily of generating functions P p (z) = T / p (/z), p. Clearly, P p P p2 = P p2 P p for all p, p 2, due to the well known property of Chebyshev polynoials stating that T n (T (x)) = T n (x). In other words, seigroup

10 0 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov generated by the faily { P p, p } is coutative. It follows (see e.g. [8]) that there exists a solution to the syste of equations ϕ(t) = P p (ϕ(pt)), p, (9) satisfying initial conditions ϕ(0) =, ϕ (0) =, (0) and the solution is unique. Since T n (x) = cos(n arccos x) = cosh(n arccosh x), the direct plugging gives that the function ( ) ϕ(t) = / cosh 2t () satisfies the syste (9), as well as the conditions (0). Hence, the function f(t) = cosh(at), a > 0 (2) is actually a ch.f. of a strictly ν-noral r.v.. The ch.f. (2) is, in fact, well known it is the ch.f. of the hyperbolic secant distribution. Clearly, a here is the scale paraeter. When a =, it is the case of the standard hyperbolic secant distribution, whose pdf has the for while the cdf is p(x) = ( πx ) 2 sech, 2 F (x) = 2 π arctan [exp ( πx )]. 2 Furtherore, in order to obtain the expression for the ch.f. of strictly ν-stable distributions, one just needs to apply the relation (6) to the strictly stable (in the usual sense) ch.f. h An interesting property Note that the function ϕ, as represented by (), can be viewed soewhat interesting on its own, and so we shall address its properties and consider its cdf A(x) (which corresponds to ϕ(t) via (4) ). Let W (t) and W 2 (t), t 0, be two independent Wiener processes. Consider a r.v. ξ = 0 W 2 (t)dt + 0 W 2 2 (t)dt. (3)

11 Distributions stable under rando suation This r.v. is well studied, and it is known that its Laplace transfor equals to E [ e tξ ] = which coincides with ϕ(t) as given by (). cosh ( 2t ), Hence A(x) is the cdf of the r.v. ξ. On the other hand, as follows fro Gnedenko s Transfer Theore, A(x) = li p 0 P { p ν p < x }. Consequently, the following theore is valid. Theore 4.. Let { ν p < x, p } be is faily of r.v. s having generating functions Then P p (z) = { } T / p ( z ), p =, n =, 2,.... n2 li P { p ν p < x } = P{ξ < x}, p 0 where the r.v. ξ is the one defined via (3). Theore 4. ay be reforulated in the following way. Let Then T n ( z ) = p k (n)z k. k=0 [n 2 x] li p k (n) = P{ξ < x}. n k=0 On Figure, the plot of the [n 2 x] k=0 p k(n) is given as a function of n starting with n = 2 until n = 50. We see that the functions attains the constant level rather quickly, and therefore it is possible to use the asyptotic result for n > 25. Corollary 2. Let X be a r.v. having the standard hyperbolic secant distribution. Then its distribution can be represented in the for of a scale ixture of noral distributions with zero ean and standard deviation ξ, where ξ is defined via (3).

12 2 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov Figure Figure : Plot of the [n 2 x] k=0 p k(n) as the function of n = 2,..., 50 To prove the above, one just needs to write the ch.f. of X in the for e t2x da(x), and note that e t2x is actually the ch.f. of the standard Noral r.v. N(0, σ 2 ) (σ 2 = x), while A(x) is the cdf of ξ. Note that there is a certain analogy between the representation A(x) as the cdf of the r.v. ξ fro (3) and the corresponding result in the schee of the rando suation with geoetric distribution. Specifically, considering the faily {ν p, p (0, )} having the geoetric distribution P { ν p function ϕ turns into ϕ(t) = + t = = k } = p( p) k, k =, 2,..., the 0 e tx da (x), where A (x) is the cdf of the exponential distribution, i.e. A (x) = e x for x > 0 and A = 0 for x 0. It can be checked that if η and η 2 are two independent standard Noral r.v. s, then A is a cdf of the r.v. related to (3) Characterizations 0 ξ = η 2 + η 2 2, which is, in a way, Let us now turn to the characterizations of the distribution of the r.v. (3) and of the hyperbolic secant distribution. Theore 4.2. Let X,..., X n,... be a sequence of non-negative iid rando variables, and ν p, p { n 2, n = 2,... }, is a faily of the r.v. s having the generating function

13 Distributions stable under rando suation 3 P p (z) = T / p ( z ), independent of the sequence {X j, j }. If, for soe fixed p, ν p d X = p X j, (4) (where d = is the equality in distribution), then X has the distribution whose Laplace j= transfor is E e tx = cosh ( at ), a > 0. (5) Proof. The equality (4), in ters of the Laplace transfor Ψ(t) = E e tx, can be represented as Ψ(t) = P p (Ψ(pt)). (6) Clearly, the function Ψ a (t) = cosh ( at ) satisfies (6) for any a > 0 and, oreover, it is analytic in the strip t < r ( r > 0 ). In the following, we use the results of the book by Kakosyan, Klebanov and Melaed [7]. Exaple.3.2 of this book shows that { Ψ a, a > 0 } fors a strongly E-positive faily on the set C of restrictions of Laplace transfors of probability distributions given in R + on an interval [0, T ] (0 < T < r). Clearly, the operator A : f P p (f(pt)) on C is intensively onotone. The result follows fro Theore.. of the above entioned book (page 2). Theore 4.3. Let X,..., X n,... be a sequence of non-negative iid rando variables, having a syetric distribution, while {ν p, p } is the sae faily as in the previous Theore. If, for soe fixed p, X d = p /2 ν p j= then X has the hyperbolic secant distribution whose ch.f. is f(t) = X j, (7) cosh (at), a > 0. (8) Proof. Quite analogous to the proof of the previous Theore, with the difference that instead of Exaple.3.2, the use of the Exaple.3. fro [7] is sufficient.

14 4 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov 5. On other rando sus of rando nuber of suands with rational generating functions In Section 3 it was entioned that in the case described there, if two functions P and Q satisfy P Q = Q P, then P and Q can be reduced by a conjugancy either to the for P(z) = z and Q(z) = z or to the for P(z) = T (z) and Q(z) = T (z). Therefore, the following question arises: Let R be a fraction-linear function. Put P = R S R, where S is either z ( > ), or T (z). Is there a function R(z) a z for which P is a generating function? Here we will show the for the case S(z) = z the answer is negative, while for the case S(z) = T n () the answer is yes. 5.. Case S(z) = z Consider linear-fractional function Because P = R S R, we have R(z) = az + b, c 0. (9) cz + d P (z) = P (z) = d(az + b) b(cz + d) a(cz + d) c(az + b). (20) However, P has to be a generating function of an integer-valued rando variable ν, and therefore we ust have P () =, P (0) = 0, i.e. db = bd, (a + b) (c + d) = (a + b)(c + d). (2) The syste (2) leads to six sub-cases: a + b = 0, { d = 0, (22) a + b = 0, b = d (23)

15 Distributions stable under rando suation 5 c + d = 0, b = 0, c + d = 0, b = d, a + b = c + d, b = 0, a + b = c + d, d = 0. (24) (25) (26) (27) All the sub-cases have to be considered separately, but the ethod of consideration is siilar for all of the, therefore we consider here one of the only. Let it be the case (26). In the case (26) the generating function P has the for P (z) = We ay suppose that cd(c + d) 0. following for d(c + d) z (c + d)(cz + d) c(c + d) z (28) P (z) = Denoting p = c/(c + d) rewrite (28) in the q z (p z + q ) p z, (29) where q = p. It is clear that P is a generating function if and only if Q (z) = is also a generating function. However, q (p z + q ) p z q Q (z) = p p k= (z z k), where z k (k=,2,...,) are the zeros of the polynoial (p z + q ) p z. it is easy to find these zeros. We consider two cases: a) p > 0 b) p < 0.

16 6 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov Let us start with the case a). In this case the zeros of the polynoial (p z + q ) p z have the for where ε (k) z k = q p / ε (k) p, k =, 2,...,, (30) (k =, 2,..., ) are roots of order fro. In other words, ε (k) 2(k )π = cos + i sin 2(k )π, k =, 2,...,. Using partial fraction decoposition let us write the function Q in the for where A k = / j k (z k z j ). q Q (z) = p p k= A k z z k, (3) Now it is easy to find the expression of Q (z) in the for of power series. Naely, q Q (z) = p p ( j k (z k z j ) s=0 k= z s+ k ) z s. (32) Because Q is a generating function, the series in (32) ust converge for all coplex z under condition z, the nearest to zero singular point of the series has to lie on positive sei-axes, and the coefficients of the series have to be non-negative. Moreover, because Q Q = Q 2, the sae properties have to hold not only for one fixed, but for the sequence l, l =, 2,..., i.e. for all functions Q l, l =, 2,.... The property of the convergence of the series inside the closed unit circle iplies, that z k > for all k =, 2,...,, i.e. we ust have cos 2 2πn + p+/ cos 2πn 2p + p 2/ cos 2 2πn, n = 0,,.... The last inequality has to hold not only for one fixed value of, but for a sequence l as l (for exaple, for l = l ). Passing to liit as in the case of n = 0 we get + p, which is wrong. So, in the case a), the function P cannot be a generation function. Let us now consider the case b) p < 0. Denote p 2 = p > 0, q = p = + p 2. the polynoial (p z + q ) p z = ( p 2 z + ( + p 2 )) + p 2 z has the for + p 2 z k =, k =, 2,...,, (33) p / 2 δ (k) + p 2

17 Distributions stable under rando suation 7 where It is easy to calculate that Therefore, δ (k) (2k )π = cos + i sin (2k )π, k =, 2,...,. z k = ( + p 2)(p / 2 cos (2k )π + p i sin (2k )π ) p 2/ 2 + 2p +/ 2 cos (2k )π + p 2 2 k =, 2,...,. z k = ( + p 2 ), k =, 2,...,. (34) p 2/ + 2p +/ 2 cos (2k )π + p 2 2 If the function Q would be a generating function, then z k with inial absolute value has to lie on positive sei-axis, but, as (34) shows, it is not true. Therefore in the case b), the function P cannot be a generation function either. To suarize: as we have seen, P cannot be a generation function in the case (26). The cases (22)-(25) and (27) can be considered in siilar way. Finally, we see that there is no generating functions, which are conjugate to power z and to equal to this power Case S = T For any a [/2, ] and any integer > let us define the following function Hypothesis P (z) = at (/(az) ( a)/a) + ( a). (35) For any a (/2, ] and any integer > the function P (z) defined by (35) is a generating function. Unfortunately, we cannot prove this Hypothesis in full. However, we will give the proof for the case of even. Theore 5.. For any a (/2, ] and any integer even > the function P (z)definedby(35) is a generating function. Proof. Consider the equation at (x) + ( a) = 0.

18 8 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov Its roots are ( x k = cos arccos a a + 2πk ), k = 0,,...,, and x k <. In view of the fact that is an even nuber, the roots are syetric around zero. Therefore at (x) + ( a) = a2 (x x k ) = a2 (x 2 x 2 k), k=0 where the latter product is taken over such k for which x k > 0. Now we see, that ( P (z) = az ( a)z ) λ ( x 2 k (az/( ( a)z))2 ), where λ = a2. The stateent now follows fro the fact, that x 2 k (az/( ( a)z))2 ) = (x k ) 2j (az/( ( a)z)) 2j, and az/( ( a)z) is the su of a geoetric progression with denoinator ( a). j=0 The faily of functions (35) for any fixed a [/2, ] is coutative with respect to convolution, i.e. P (P n (z)) = P n (P (z)) = P n (z), and consequently, there exists ν-gaussian distribution, where the faily {ν p, p } is defined by the faily of corresponding generating functions P (z), = 2, 4, 6,..., and the paraeter p is defined by through the relation p = p() = P (). We will not study here corresponding characteristic functions. It can be done using the general approach described above. 6. Exaples with non-rational generating functions There exist exaples of the pairs of coutative functions, which are not rational. Here we refer to the two classes of such functions, the first of which was investigated by Melaed [] and the second appears at first in the present work.

19 Distributions stable under rando suation 9 Exaple 6.. (See Melaed [] for detailed study) Consider the faily of generating functions P p (z) = p / z, (36) ( ( p) z ) / where p (0, ), and is a fixed positive integer. Obviously, in the case =, P p (z) reduces to the generating function of the geoetric distribution, and has already been entioned this case above. Hence, assue that 2. In that case, it is easy to check that ϕ(t) =, (37) ( + t) / and therefore the ch.f. of the strictly ν-noral distribution (for the faily {ν p, p } having the generating function (36) ) has the for with a paraeter a > 0. ϕ(t) = ( + at 2 ) /, Exaple 6.2. Consider the faily of functions where p { n 2, n = 2,... }, and (an integer). P p (z) = ( ( T/ )) /, (38) p z Using a slightly odified version of the proof of Lea, it is easy to check that P p (z) is a generating function of soe r.v. ν p, p { n 2, n = 2,... } for any fixed whole nuber (surely, both P p dependence in the notation). and ν p both depend on, but we oit this The case = has already been considered above. For 2 analogous ethods are applicable, and so will shall only refer to the results. Specifically, ϕ(t) = ( cosh 2t ) /, (39) while the ch.f. of the corresponding strictly ν-noral distribution has the for f(t) =, (40) / (cosh at)

20 20 L.B. Klebanov, A.V. Kakosyan, S.T. Rachev, G. Tenov where a > 0. Note that in the case = 2, we have the following expressions for the distributions whose Laplace transfors are (37) and (39). For = 2, the forula (37) gives ϕ(t) = + 2t. This function is the Laplace transfor of the distribution of the r.v. X 2, with X being the standard Noral r.v. In a siilar way, (39) gives for = 2 ϕ(t) = cosh 4t. This function is the Laplace transfor of the distribution of the r.v. I = X 2 (t)dt, where X(t) is the standard Wiener process. 0 References [] Bunge, J. (996). Copositions seigroups and rando stability. Annals of Probab., 24, [2] Ereenko, A.E. (989). Soe functional equations connected with the iteration of rational functions, Algebra i Analiz, (Translated in Leningrad Math. J. (990), ) [3] Fatou, P. (923). Sur l iteration analitique et les substitutions perutables, J. Math. 2, 343. [4] Gnedenko, B.V. (983). On soe stability theores. Lecture Notes in Math., 982, 24-3, Springer, Berlin. [5] Gnedenko, B.V. and Korolev, V.Yu. (996). Rando Suation: Liit Theores and Applications. CRC Press, Boca Raton. [6] Julia, G. (922). Méoire sur la perutabilité des fractions rationneles, Ann. Sci. École Nor. Sup. 39, [7] Kakosyan, A.V., Klebanov, L.B. and Melaed, I.A. (984) Characterization of Distributions by the Method of Intensively Monotone Operators, Springer, Berlin- Heidelberg.

21 Distributions stable under rando suation 2 [8] Klebanov, L.B. (2003). Heavy Tailed Distributions. Matfyz-press, Prague. [9] Klebanov, L.B., Maniya, G.M. and Melaed, I.A.(984). A proble of Zolotarev and analogs of infinitely divisible and stable distributions in a schee for suing a rando nuber of rando variables. Theory Probab. Appl., 29, [0] Klebanov, L.B. and Rachev, S.T.(996). Sus of a rando nuber of rando variables and their approxiations with ν-accopanying infinitely divisible laws. Serdica, 22, [] Melaed, I.A. (989). Liit theores in the set-up of suation of a rando nuber of independent and identically distributed rando variables. Lecture Notes in Math., 42, Springer, Berlin, [2] Mallows, C.L. and Shepp, L.A. (2005). B-stability. J. Applied Probability, 42, No 2,