Almost sure convergence

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Let $(X_n), X$ be random variables defined on the same probability space $(\Omega, A, P)$ with values in $\mathbb R$ (or $\mathbb R^d$).

Definition

Definition

We say that $X_n$ converges almost surely to $X$ if $\mathbb P(X_n \rightarrow X) = 1$

Results

Proposition

If $X_n$ converges to $X$ almost surely then $X_n$ converges in probability to $X$.