Question

Let Xn be a sequence of Rvs with common finite variance σ∧2. Suppose that the correlation coefficient between Xi and Xj is<0 for all i ≠j. Show that the WLLN holds for the sequence {Xn}.

Answer 1

$P\left(|\bar{X}_n-\mu|\leq \epsilon \right )\geq 1-\frac{Var(\bar{X}_n)}{\epsilon^2}..................................................(1)\\ where,~\mu=E(\bar{X}_n)=\frac{1}{n}\sum_{i=1}^nE(X_i)\\ Var(\bar{X}_n)=\frac{1}{n^2}\left(\sum_{i=1}^nVar(X_i)&plus;2\sum_{i<j}^ncov(X_i,X_j) \right )\leq \frac{\sum_{i=1}^nVar(X_i)}{n^2}=\frac{\sigma^2}{n},...................(2)\\ ~since~Cov(X_i,X_j)<0~\forall~i\neq j,~Var(X_i)=\sigma^2~\forall i=1,2,3,.....\\ Then~from~(2)~Var(\bar{X}_n)\rightarrow 0~as~n\rightarrow \infty.....(3)\\$

$Now~from~(1)~and~(3),\\ \bar{X}_n\overset{P}{\rightarrow}\mu.\\ Hence ~WLLN~holds~for~sequence~\{ X_n\}$.