Probability Theory on Vector Spaces III

Methods for proving functional limit laws are developed for sequences of stochastic processes which allow a recursive distributional decomposition either in time or space. Our approach is an extension of the so-called contraction method to the space C[0,1] of continuous functions endowed with uniform topology and the space D[0,1] of cadlag functions with the Skorokhod topology. The contraction method originated from the probabilistic analysis of algorithms and random trees where characteristics satisfy natural distributional recurrences. It is based on stochastic fixed-point equations, where probability metrics can be used to obtain contraction properties and allow the application of Banach’s fixed-point theorem. We develop the use of the Zolotarev metrics on the spaces C[0,1] and D[0,1] in this context. Applications are given, in particular, a short proof of Donsker’s functional limit theorem is derived and recurrences arising in the probabilistic analysis of algorithms are discussed.

/pdf/on-a-functional-contraction-method-31o7tcfvyj.pdf

On a functional contraction method

This monograph is devoted to some problems of the theory of probabilistic distributions in topological Abelian groups with sufficiently abundant dual groups.

Characteristic Functionals of Probabilistic Measures in DS-Groups and Related Topics

For a sequence of independent random elements belonging to an ideal of order-bounded elements of a Banach lattice, we investigate the asymptotic relative stability of extremal values, the law of large numbers for the pth powers, and the central limit theorem.

Limit Theorems for Random Elements in Ideals of Order-Bounded Elements of Functional Banach Lattices

A remark on the central limit theorem in Banach spaces

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A remark on the central limit theorem in Banach spaces