Suppose that X N2 and that Xn Nn2 n n 1 Prove that Xn d X as

Suppose that X N(,2), and that Xn N(n,2 n), n 1. Prove that Xn d X as n n and n as n

Solution

Here X_nX with probability and X_nX with distribution are equivalent , since X is a degenerate random variable. By Khinchins weak law of large numbers , let phi(t) be the charateristic function of the common distribution of X_n, then the charateristic function of X_n bar is [phi(t/n)]ⁿ. And we have.   [phi(t/n)]ⁿ=[1+(it)/n+o(t/n)]ⁿexp(it), as n tends to infinity. But exp(it) is the charateristic function of a degenerate random variable, degenerate at . Similarily for sigma also.