\(\displaystyle \mbox{Let }\, X_1,\, X_2,...\,\mbox{ be a sequence of independent r.v.'s}\)
\(\displaystyle \mbox{with }\, EX_i\, =\, \mu_i,\, Var(X_i)\, =\, {\sigma_i}^2,\, i\, =\, 1,\, 2,....\)
\(\displaystyle \mbox{Assume that }\)
\(\displaystyle \mbox{i) }\, \lim_{n\rightarrow \infty}\, \frac{1}{n^2}\, \sigma_{i=1}^n\, {\sigma_i}^2\, =\, 0\)
\(\displaystyle \mbox{ii) }\, \lim_{n\rightarrow \infty}\, \frac{1}{n}\, \sigma_{i=1}^n\, \mu_i\, =\, \mu\)
\(\displaystyle \mbox{Show that for every }\, \epsilon\, >\, 0,\)
\(\displaystyle \lim_{n\rightarrow \infty}\, P\left(\left|\frac{\sigma_{i=1}^n\, X_i}{n}\, -\, \mu\right|\, >\, \epsilon\right)\, =\, 0\)
