I am looking for an example of a sequence of random variables $X_n$ such that $\lim \inf E[X_n] \le E[\lim \inf X_n]$ . Please help . Thanks in advance
2026-03-31 05:41:37.1774935697
Looking for example of a sequence of random variables $X_n$ such that $\lim \inf E[X_n] \le E[\lim \inf X_n]$
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Let $X_n=-n^21_{[0,\frac{1}{n^2}]}$ independent then $EX_n=-1$ for every $n$. By Borel Cantelli, $X_n\to0$ a.e. Thus $E(\liminf X_n)=E(0)=0$ so here $-1=\liminf E(X_n)\leq E(\liminf X_n)=0$.
This shows why Fatou's Lemma needs the non-negativity criterion. The usual Fatou's lemma is for $X_n\geq0$, $E(\liminf X_n)\leq \liminf E(X_n)$.