[[Convergence concepts in probability MOC]]
# Convergence in probability

A sequence $(X_i)_{i=1}^\infty$ of [[real random variable|real random variables]] is said to **converge in probability** to $X$ iff either of the equivalent conditions are met for all $\epsilon > 0$: #m/def/prob 
$$
\begin{align*}
\lim_{ i \to \infty } \mathbb{P}(|X_{i}-X| >\epsilon) &= 0 \\
\lim_{ i \to \infty } \mathbb{P}(|X_{i}-X | < \epsilon) &= 1
\end{align*}
$$
This is precisely [[convergence]] under the [[Ky Fan metric]].

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