On the Convergence of Types for Radon Probability Measures in Banach Spaces

Abstract

A Radon probability measure v on a Banach space F belongs to the same type as a measure μ on a Banach space E provided that v = T(μ) * δ y for some linear operator T from E into F and some y ∈ F. Assume that the probability measures v n belong to the same type as the measures μ n for each n = 1, 2,..., and, moreover, {μ n } and {v n } converge weakly to μ and v, respectively. Then the present paper is concerned with the problem whether or not the limit v belongs to the same type as the limit μ. A classical result due to Khinchin (cf. [6, Ch. VIII, §2, Lemma 1]) asserts that this is so on the real line provided that μ is non-degenerated. Similar results under weaker assumptions can be found in [12, Section 2.3]. Expansions of such results to n-dimensional spaces were studied in Billingsley [1] and Sharpe [13]. In arbitrary Banach or even more general spaces this problem was treated in Parthasarathy [11, p.58], and Csiszar/Rajput [3] in the special case E = F and \(E=Fand{{v}_{n}}=({{\alpha }_{n}}I)({{\mu }_{n}})*{{\delta }_{{{y}_{n}}}}\) where I denotes the identity operator of E and an are some real numbers. The case of arbitrary linear oparators T n (instead of α n I) was investigated in Jouandet [7]; but these results are false. The basic aim of the present paper is to derive general convergence of types theorems in arbitrary Banach spaces. It turns out that one has to assume the boundedness of the sequence of operators in order to obtain the classical results in the infinite dimensional setting.