[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

(Chebyshev) Assume

that [math]X_1[/math], [math]X_2[/math], \dots, [math]X_n[/math] are independent random variables with possibly different distributions and let [math]S_n[/math] be their sum. Let [math]m_k = E(X_k)[/math], [math]\sigma_k^2 = V(X_k)[/math], and [math]M_n = m_1 + m_2 +\cdots+ m_n[/math]. Assume that [math]\sigma_k^2 \lt R[/math] for all [math]k[/math]. Prove that, for any [math]\epsilon \gt 0[/math],

[[math]] P\left( \left| \frac {S_n}n - \frac {M_n}n \right| \lt \epsilon \right) \to 1 [[/math]]

as [math]n \rightarrow \infty[/math].

Notes