Question

4. Let Xi, X2,... be uncorrelated random variables, such that Xn has a uniform distribution over -1/n, 1/n]. Does the sequence converge in probability? 5. Let Xi,X2 be independent random variables, such that P(X) PX--) Does the sequence X1 +X2+...+X satisfy the WLLN? Converge in probability to 0?

Answer 1

$E(X_i)=\frac{i}{2}\int_{-\frac{1}{i}}^{\frac{1}{i}}xdx=0\\ Var(X_i)=E(X_i^2)=\frac{i}{2}\int_{-\frac{1}{i}}^{\frac{1}{i}}x^2dx=i\int_0^{\frac{1}{i}}x^2dx=i\times \frac{1}{3i^3}=\frac{1}{3i^2}\\ \bar{X}_n=\frac{1}{n}\sum_{i=1}^nX_i\\ E(\bar{X}_n)=0\\ Var(\bar{X}_n)=\frac{1}{n^2}\sum_{i=1}^nVar(X_i),~since~X_i's~are~uncorrelated.\\ =\frac{1}{n^2}\sum_{i=1}^n\frac{1}{3i^2}\rightarrow 0~as~n\rightarrow \infty~since~\sum_{i=1}^n\frac{1}{i^2}~is~a~converging~series.\\ Hence~X_n\overset{p}{\rightarrow}0\\ 5.~E(X_n)=0,~Var(X_n)=E(X_n^2)=n^2\\ \bar{X}_n=\frac{X_1+X_2+...+X_n}{n}\\ E(\bar{X}_n)=0\\ Var(\bar{X}_n)=\frac{1}{n^2}\sum_{i=1}^nVar(X_i),~since~X_i's~are~independent\\ =\frac{1}{n^2}\sum_{i=1}^ni^2=\frac{1}{n^2}\times \frac{n(n+1)(2n+1)}{6}=\frac{\left(1+\frac{1}{n} \right )(2n+1)}{6}\rightarrow \infty~as~n\rightarrow \infty\\ Hence~WLLN~does~not~hold.$