Central limit theorem under uncertain linear transformations

Abstract

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.

Asymptotic sequential Rademacher complexity of a finite function class

“…Our argumentation is based on a central limit theorem under model uncertainty (see [13]) which we now recall. Let (ξ i ) ∞ i=1 be a sequence of d-dimensional random variables with zero mean and identity covariance matrix:…”

Pde Approach to the Problem of Online Prediction With Expert Advice: A Construction of Potential-Based Strategies

“…Its application to the problems of online learning theory was initiated in [10], where an asymptotics of the sequential Rademacher complexity (the last notion was introduced in [7]) of a finite function class was related to the viscosity solution of a G-heat equation. In turn, the result of [10] is based on the central limit theorem under model uncertainty, studied within the same approach in [9].…”

Central limit theorem under variance uncertainty

“…and passing to the limit as n → ∞. This approach was proposed in [18] in the case of identically distributed (multidimensional) random variables ξ j . However, in the present context, it seems that this method requires hypotheses, which are stronger than the Lindeberg condition.…”

“…It is interesting to compare Theorem 1 with related results obtained in the framework of sublinear expectations theory. Besides the original result of Peng [13,14], which is discussed in [18], we mention the papers [11,20,8], where the random variables were not assumed to be identically distributed. We will discuss only the result of [20], which extends [11].…”

“…Let us briefly describe the construction of a sublinear expectation space (Ω, H, E), which allows to rewrite the expression (1.6) in terms of a sublinear expectation. This construction is, in fact, the same as in [18,Section 4], where some more details are given. Consider the space of sequences Ω = {(y i ) ∞ i=1 : y i ∈ R}, and introduce the space of random variables H as follows: H = ∪ ∞ n=1 H n , where H n is some linear space (we do not go into details) of functions Y = ψ(y 1 , .…”