A Weak Law of Large Numbers for Dependent Random Variables

2022/04/22 by Ioannis Karatzas, Karatzas, Ioannis, Walter Schachermayer +1 · 1 citation
Decision Sciences · Economics, Econometrics and Finance · #Probability and Risk Models #Credit Risk and Financial Regulations

paper · pdf · doi:10.48550/arxiv.2204.10681

Abstract

Every sequence f₁, f₂, ⋯ of random variables with lim_{M → ∞} ( M sup_{k ∈ ℕ} ℙ ( |f_k| > M ) )=0 contains a subsequence f_k1, f_k2 , ⋯ that satisfies, along with all its subsequences, the weak law of large numbers: lim_{N → ∞} ( (1/N) ∑_n=1^N f_kn - D_N ) =0 , in probability. Here D_N is a "corrector" random variable with values in [-N,N], for each N ∈ ℕ ; these correctors are all equal to zero if, in addition, \liminf_{k → ∞} 𝔼 ( f_k² 1_ \ |f_k| ≤ M \ ) =0 holds for every M ∈ (0, ∞) .

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