- #1
Euge
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MHB
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Let ##\{X_n\}## be a sequence of integrable, real random variables on a probability space ##(\Omega, \mathscr{F}, \mathbb{P})## that converges in probability to an integrable random variable ##X## on ##\Omega##. Suppose ##\mathbb{E}(\sqrt{1 + X_n^2}) \to \mathbb{E}(\sqrt{1 + X^2})## as ##n\to \infty##. Show that ##X_n\xrightarrow{L^1} X##.