 Convergence in probability

Random variables $X_n$ converge in probability to $X$ when $\mathbb{P}(|X_n-X|>\epsilon )\to 0$ for every fixed $\epsilon >0$. It makes a fixed-size error unlikely at a late deterministic time; it does not assert that every path eventually avoids such errors. For uniform $U$ on $[0,1]$, $1_{\{U\le 1/n\}}$ converges to zero in probability since its probability of being one is $1/n$. Almost sure convergence and convergence in Lp each imply convergence in probability. No shared rate is part of the definition, and convergence of expected values requires further conditions.