Convergence in probability

Definition

Let $X, X_1, X_2, \dots$ be real random variables on one probability space $(\Omega, \mathcal{F}, P)$. The sequence $(X_n)$ converges in probability to $X$ if for each $\varepsilon > 0$, $$\lim_{n \to \infty} P\left(|X_n - X| > \varepsilon\right) = 0.$$ One writes $X_n \xrightarrow{P} X$.

Remarks

The limit is unique up to equality almost surely. If $X_n \to X$ in probability, then a subsequence of $(X_n)$ converges to $X$ almost surely. The weak law of large numbers is a result about convergence in probability. Example: let $X_n$ be $1$ with probability $1/n$ and $0$ otherwise. Then $X_n \to 0$ in probability.

Explanations

L1 human

Convergence in probability means that, for large $n$, $X_n$ is very likely to be close to $X$. It does not say that one fixed run of the experiment gives a convergent sequence.