Central limit theorem in a Banach space

Abstract

We describe the recent work by various authors on the central limit theorem in a Banach space E. Let {X_n} be a sequence of E-valued independent identically distributed random variables. X₁ (or its distribution in E) is said to obey the CLT if the distributions of {n^−1/2(X₁+...+X_n)} converge weakly to a probability measure ν on E (which is necessarily Gaussian). Hoffmann-Jørgensen and Pisier have recently shown that every E-valued random variable X₁ with ε[X₁]=0 and ε[‖X₁‖²]<∞ satisfies CLT ⇔ E is of “type 2”. Zinn has used this result to derive an earlier result of Jain and Marcus on CLT for C(S)-valued random variables. All these results and their proofs are included. Some recent work of Devary is also described where conditions are imposed on the distribution of X₁ rather than on its modulus of continuity (as Jain and Marcus did in their main result).

Among the new results presented are essentially the following: (1) If X₁ satisfies CLT and E is of “cotype 2” then ε[X₁]=0 and ε[‖X₁‖²]<∞, and (2) Kolmogorov's inequality holds in E⇔E is of type 2. Some open questions are mentioned in Section 5.

This work was partially supported by the NSF.