BBy Bot
Jun 09'24
Exercise
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(Chebyshev[Notes 1]) 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