ON THE CONVERGENCE OF STABLE MEASURES IN A BANACH SPACE

Abstract. In this paper a space of functions generating stable measures on a Banach space is introduced. Some properties of ths space are investigated. Then as applications, necessary and sufficient conditions for the weak convergence of stable measures on a Banach space are given. The convergence of sums of independent stable random variables wit'h Banach values is investigated as well. 1. NOTATFQNS AND DEFINITIONS I Let (U, Z, m) be a fixed finite measure space and let X be a separable 1, Banach space with norm I I- 1 1 and topological dual X*. Denote by L,(X; U, Z, m) (L,(X) for short) the set of all X-valued functions f such that Ilf l lp = {J Ilf tu)llpdmtu))l'P < +-U In the case X is the set of real numbers we write L, instead of L,(X). sL,(X) denotes the set of all X-valued functions f such that < f, X*)E L, for each x * EX*. In the sequel, if not otherwise stated, p is a real number with 0 < p < 2. A Radon probability measure p on X is called p-stable iff for given a> 0, b> 0 we have i; (ax*) f i (bx*). = ji ((ap + b3 ' I p x*) for a11 x * E X*, where ji is the characteristic functional (ch.f.) of p, i.e. fi(x*) = jexp {i (x, x*)) dp(x). X Thus, in this paper we consider only symmetric stable measures. Let (6jp1) be a sequence of independent identically distributed real random variables 138 Nguyen Zui Tien with ch.f. exp (-lr(P). A Banach space X is said to be of stabk type p if for some (each) r ~ ( 0, p) there exists a constant c> 0 such that for all x,,..., x, in X. For properties of Banach spaces of stable type we refer the reader to [53, [lo]. Let now (a, A, P) be a probability space. By an X-valued. random variable (r.v.) we mean a strongly measurable mapping r: 62 +X. 9((e) denotes the distribution of 5, where g(X) is the set of all Bore1 subsets of X. If Y(5) is p-stable, then we say that 5 is p-stable r.v. 2. THE SPACE S,(X) Let f be an element of sL,(X). Put xf ( x * ) = exp {- jj<f (4, x*)tPdm(u)), x * € X C