CHAPTER 1 Introduction Let X be a random variable with values in a Banach space B. To avoid mea- surability problems, we assume for most of this chapter for simplicity that B is separable and X Borel measurable see Section 2.2 for measurability in the general case. Moreover, for definiteness, we consider real Banach spaces only the complex case is similar. If E }X} ă 8, then the mean E X exists as an element of B (e.g. as a Bochner integral ş X d P, see Section 2.4). Suppose now that we want to define the k:th moments of X for some k ě 2, assuming for simplicity E }X}k ă 8. If B is finite- dimensional, then the second moment of X is a matrix (the covariance matrix, if X is centred), and higher moments are described by higher-dimensional arrays of joint moments of the components. In general, it is natural to define the k:th moment of X using tensor products, see Section 2.3 for details: X bk is a random element of the projective tensor product B p b k, and we define the projective k:th moment of X as the expectation E X bk P B p b k (when this expectation exists, e.g. if E }X}k ă 8) we denote this moment by E X p b k. In particular, the second moment is E X p b 2 EpX b Xq P B p b B. An alternative is to consider the injective tensor product B q b k and the injective k:th moment E X q b k P B q b k. Another alternative is to consider weak moments, i.e., joint moments of the real-valued random variables x˚pXq for P (the dual space). The weak k:th moment thus can be defined as the function (1.1) px1 ˚ , . . . , xk ˚ q Þ Ñ E ` x1 ˚ pXq ¨ ¨ ¨ xk ˚ pXq ˘ P R, assuming that this expectation always exists (which holds, for example, if E }X}k ă 8). Note that the weak k:th moment is a k-linear form on B˚. The purpose of the present paper is to study these moments and their relations in detail, thus providing a platform for further work using moments of Banach space valued random variables. In particular, we shall give sufficient, and sometimes necessary, conditions for the existence of moments in various situations. One example of our results on relations between the different moments is that, at least in the separable case, the weak k:th moment is equivalent to the injective moment. (See Theorem 3.11 for a precise statement.) We study also the problem of moment equality: if Y is a second random variable with values in B, we may ask whether X and Y have the same k:th moments, for a given k and a given type of moment. (Assume for example that E }X}k , E }Y }k ă 8 so that the moments exist.) This problem, for the second moment, appears for example in connection with the central limit theorem for Banach space valued random variables, see e.g. [42, Chapter 10] where weak moments are used. (In 1
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