TAILIEUCHUNG - Independent And Stationary Sequences Of Random Variables - Chapter 2

Chapter 2 STABLE DISTRIBUTIONS ; ANALYTICAL PROPERTIES AND DOMAINS OF ATTRACTION § 1. Stable distributions Definition . A distribution function F is called stable if, for any a 1 , a2 0 and any b1 , b2 , there exist constants a 0 and b such that F(a l x+b l ) * F(a 2 x+b2) = F(ax+b) . ( .1) | Chapter 2 STABLE DISTRIBUTIONS ANALYTICAL PROPERTIES AND DOMAINS OF ATTRACTION 1. Stable distributions Definition. A distribution function F is called stable if for any ar a2 0 and any b1 b2 there exist constants a Q and b such that F a1x b1 F a2x b2 F ax b . It clearly suffices to take b2 b2 0. Then in terms of the characteristic function of F becomes t a1 t a2 f t a e ibt. Interest in the stable distributions is motivated by the fact that under weak assumptions they are the only possible limiting distributions of normed sums Zn X2 Xn - An of stationarily dependent random variables. In this section we establish this result for independent random variables the general case is dealt with in Theorem . Theorem . In order that a distribution function F be the weak limit of the distribution of Zn for some sequence of independent identically distributed random variables it is necessary and sufficient that F be stable. If this is so then unless F is degenerate the constants Bn in must take the form Bn nll h n where 0 a 2 and h n is a slowly varying function in the sense of Karamata. 38 STABLE DISTRIBUTIONS Chap. 2 Proof. Let be the common characteristic function of the Xh and let be the characteristic function corresponding to the distribution F. Since a degenerate distribution is trivially stable we exclude this case and prove that necessarily lim Bn oo lim Bn 1 Bn 1 . n oo n oo Suppose that the first condition in does not hold so that there is a subsequence B k with limit B oo. Then i i Bjr i t i i o i so that for all t i tBj 1 l o l . This is possible only if t 1 for all t which implies that F is degenerate. Thus the first part of is proved so that lim t B 1 1 . n oo Thus and ir t B 1 i i i0 t i i O i . Substituting Bnt Bn x for t in the former and then Bn j t B for t in the latter we deduce that as n- oo lim lim 1 . If B JBn-f l we can find a subsequence of either Bn JB or B Bn J converging to some B 1.

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