# Convergence in probability

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Convergence of a sequence of random variables $X_1,X_2,\ldots$ defined on a probability space $(\Omega,\mathcal{F},\mathbb{P})$, to a random variable $X$, defined in the following way: $X_n \stackrel{\mathrm{P}}{\rightarrow} X$ if for any $\epsilon > 0$, $$\mathbb{P}\{ |X_n-X| > \epsilon \} \rightarrow 0 \ \ \text{as}\ \ n \rightarrow \infty \ .$$