A Central Limit Theorem for Sets of Probability Measures

TL;DR

This paper establishes a central limit theorem for sequences of random variables with ambiguous means, where the limit distribution is characterized by a backward stochastic differential equation, modeling an ambiguous continuous-time random walk.

Contribution

It introduces a novel CLT for ambiguous measures with limits described by backward stochastic differential equations, extending classical probability theory.

Findings

Limit distribution is not normal but defined by a backward SDE.

Provides a new framework for ambiguous stochastic processes.

Extends CLT to unstructured mean ambiguity.

Abstract

We prove a central limit theorem for a sequence of random variables whose means are ambiguous and vary in an unstructured way. Their joint distribution is described by a set of measures. The limit is (not the normal distribution and is) defined by a backward stochastic differential equation that can be interpreted as modeling an ambiguous continuous-time random walk.