$lim E[f^2(X_n)]neq E[f^2(X)]$ even if $X_n rightarrow ^d X$
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Let $X_nsim N(0,1/n)$ . Is there a continuous function $f$ such that $E[f^2(X_n)]<infty$ $lim_{nrightarrow infty} E[f^2(X_n)] neq f^2(0)$ ? Also, what would happen if I add the condition $E[|f(X)f(Y)|]<infty$ for all jointly normal $X,Y$ such that $EX=EY=0$ I know that there is no such $f$ if we additionally require $f$ to be bounded since $X_n stackrel{d}{rightarrow}delta_0$ . However, I am totally clueless when it comes to proving (or disproving) the existence of such $f$ if we drop out boundedness condition. I appreciate every hint!
real-analysis functional-analysis probability-theory convergence normal-distribution
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