2023/11/18 by Zhang Li, Zhang, Li-Xin
Decision Sciences · Economics, Econometrics and Finance · #Probability and Risk Models #Stochastic processes and financial applications #Financial Risk and Volatility Modeling
paper · pdf · doi:10.48550/arxiv.2311.11100
Let \Xn;n≥ 1\ be a sequence of independent and identically distributed random variables in a regular sub-linear expectation space (Ω,\mathscrH,\widehat\mathbb E) with the finite Choquet expectation, upper mean μ and lower mean \underlineμ . Then for any Borel-measurable function φ(x1,…,xd) on \mathbb Rd or continuous function φ(x1,x2,…) on \mathbb R\mathbb N, ∑i=1n Xi/n converges to \underlineμ\wedge φ(X1,X2,…)\wedge μ with upper capacity 1. The limits of ∑i=1nXi/n can be with upper capacity 1 also a random set with boundaries being continuous functions or finite-dimensional Borel-measurable functions of (X1, X2,…).