If $X_nto X$ in $L^p$, is $E(X_n^p)to E(X^p)$ true?
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Let $p$ denote a positive integer. If $X_nto X$ in $L^p$ , is $E(X_n^p)to E(X^p)$ true? If each $X_n$ and $X$ are nonnegative, then it follows directly from Minkowski's inequality. Then how about the general case? I separated each $X_n$ into its positive and negative parts as $X_n=X_n^+-X_n^-$ , but it might not be the case that $X_n^+to X^+$ and $X_n^-to X^-$ in $L^p$ .
probability-theory lp-spaces
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edited Dec 2 at 12:42
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asked Dec 2 at 11:37