- #1

Euge

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- #1

Euge

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- #2

Office_Shredder

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Converges in probability means ##P(|X_n-X|>\epsilon)\to 0## for all ##\epsilon>0##.

Converges in ##L_1## means ##E(|X_n-X|)\to 0##. One example where these aren't the same is: ##X## is identically zero, ##X_n## is ##n## with probability ##1/n## and 0 otherwise. ##P(|X_n-X|>\epsilon)\leq 1/n\to 0##, but ##E(|X_n-X|)=1## for all ##n##.

The expected value condition is interesting, I wonder if the ##1+## piece is necessary.

- #3

Euge

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Here is a hint: Convergence in probability implies convergence almost surely.

- #4

Euge

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