Abstract. We prove a variant of the central limit theorem (CLT) for a sequence of i.i.d. random variables ξ j , perturbed by a stochastic sequence of linear transformations A j , representing the model uncertainty. The limit, corresponding to a "worst" sequence A j , is expressed in terms of the viscosity solution of the G-heat equation. In the context of the CLT under sublinear expectations this nonlinear parabolic equation appeared previously in the papers of S. Peng. Our proof is based on the technique of half-relaxed limits from the theory of approximation schemes for fully nonlinear partial differential equations.
Problem formulation. Denote by ξ a random variable distributed as ξ j , and assume thatwhere I is the identity matrix. By the classical central limit theorem (CLT), for any bounded continuous function f :where η has the standard d-dimensional normal law. Note, that for given f the limit depends only on the covariance matrix AA T of Aξ.In this paper we consider the case where A is not known exactly, and can change dynamically within a prescribed set. This is a simple example of a probability model under uncertainty. The extension of the CLT, obtained below, looks similar to Peng's CLT under sublinear expectations: [12,14]. However, our problem formulation, as well as the proof, do not involve the nonlinear expectations theory in any way. On the other hand, similarly to Peng's approach, the key role is played by the viscosity solutions theory.Consider a filtered probability space (Ω, F , P, (F j ) ∞ j=0 ) and an adapted sequence (ξ j ) ∞ j=1 of d-dimensional random variables such that ξ j is independent from F j−1 and satisfy (1.1). Denote by M d (resp., S d ) the set of d × d matrices (resp., symmetric matrices). Let (A j ) ∞ j=0 be an adapted sequence with values in a compact set Λ ∈ M d .2010 Mathematics Subject Classification. 60F05, 35D40.