Limit theorems with rate of convergence under sublinear expectations

Abstract

Under the sublinear expectation [·] := sup θ∈Θ E θ [·] for a given set of linear expectations {E θ : θ ∈ Θ}, we establish a new law of large numbers and a new central limit theorem with rate of convergence. We present some interesting special cases and discuss a related statistical inference problem. We also give an approximation and a representation of the G-normal distribution, which was used as the limit in Peng (2007)'s central limit theorem, in a probability space.

Convergence rate of Peng’s law of large numbers under sublinear expectations

“…In this short note, we provide a simple and purely probabilistic proof of (2). We employ Chatterji's…”

Stein’s method for the law of large numbers under sublinear expectations

“…We say is distributed as , written as . A functional is a sublinear expectation on if and only if it can be represented as the supremum expectation of a weakly compact subset of probability measures on (see [1]),…”

A hypothesis-testing perspective on the G-normal distribution theory

“…The central limit theorem for sublinear expectation spaces, first developed by Peng [2008], has been formulated in various ways. In Theorem 1 we present the version given in Rokhlin [2015] [see also Fang et al, 2017], which can be seen as a generalization of the classical central limit theorem to controlled stochastic processes. It is an immediate corollary of Peng's original central limit theorem, but translated into the language of classical probability (see Appendix B).…”

“…A more detailed review of the G-normal distribution (and sublinear expectation spaces) will be given in Section 2. As noted in Fang et al [2017], to characterize the tail behavior of the G-normal distribution, equivalently we can consider the following stochastic control problem (see also Theorem 1 and Definition 1.) Problem 1.…”