Introduction to Probability · Problem 8.1.12

Law of Large Numbers — Grinstead & Snell

Mathematics — Probability · chebyshev-law-of-large-numbers

Problem

(Chebyshev) Assume that X1,X2,…,XnX_1, X_2, \ldots, X_n are independent random variables with possibly different distributions and let SnS_n be their sum. Let mk=E(Xk)m_k = E(X_k), σk2=V(Xk)\sigma_k^2 = V(X_k), and Mn=m1+m2+⋯+mnM_n = m_1 + m_2 + \cdots + m_n. Assume that σk2<R\sigma_k^2 < R for all kk. Prove that, for any ϵ>0\epsilon > 0, P(∣Snn−Mnn∣<ϵ)→1P\left(\left|\frac{S_n}{n} - \frac{M_n}{n}\right| < \epsilon\right) \to 1 as n→∞n \to \infty.

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