On the Convergence in Distribution of Measurable Multifunctions (Random Sets) Normal Integrands, Stochastic Processes and Stochastic Infima

Abstract

The concept of the distribution function of a closed-valued measurable multifunction is introduced and used to study the convergence in distribution of sequences of multifunctions and the epi-convergence in distribution of normal integrands and stochastic processes; in particular various compactness criteria are exhibited. The connections with the classical convergence theory for stochastic processes are analyzed and for purposes of illustration we apply the theory to sketch out a modified derivation of Donsker's Theorem (Brownian motion as a limit of normalized random walks). We also suggest the potential application of the theory to the study of the convergence of stochastic infima.

Variational Theory for Optimization under Stochastic Ambiguity

“…Moreover, it can easily handle any finite dimension, i.e., any number of uncertain parameters. In fact, its extension to infinite-dimensional spaces appears clear (see the ideas in [36]), but such possibilities are beyond the scope of the present paper.…”

“…The distribution functions are then said to also converge weakly. The following result is implicit in [37,36], where the development is more abstract dealing with probability semicontinuous measures on closed sets. Here, we provide for the first time an explicit statement for the present context and give a new simplified proof that only relies on Proposition 3.1.…”

“…This results in a novel treatment of distribution functions on IR m , including the definition of convexity on this metric space and convenient estimates of the hypo-distance. Moreover, we provide an explicit statement of the fact that convergence of distribution functions in the hypo-distance is equivalent to weak convergence, with a new proof exclusive relying on elementary concepts from variational analysis; the result is implicit in [37,36]. Of course, there are numerous metrics available for spaces of probability measures including the Levy-Prokhorov metric, which also characterizes weak convergence in the present context, the Wasserstein metrics, which require finite moments and is stronger than weak convergence, and the total variation metric, which is also stronger than weak convergence.…”

Delta Method, Infinite Dimensional

“…In that case the convergence (2) is equivalent to the convergence in distribution of {d(y, r n ( (X n ) − x))} y∈F as stochastic processes to {d(y, (θ ,x) (X))} y∈F if F is finitedimensional [28,Theorem 2.5].…”

Random Set Theory and Problems of Modeling

“…Ripley (1981,9.1) also discusses the problem of choice of test sets. The strongest result so far available is due to Salinetti and Wets (1986), who prove that {T(U): U set of all finite unions of closed balls in E} determines the probability measure of a RACS.…”