Convergence in Probability vs Distribution

Random sequences can converge in several distinct senses, and knowing which mode you have determines what you can actually conclude. From strongest to weakest:

Almost sure: $P(X_n \to X) = 1$. Each realization of the sequence converges.In probability: $P(|X_n - X| > \epsilon) \to 0$ for every $\epsilon > 0$. The sequence is eventually within $\epsilon$ with high probability.In distribution: $F_n(x) \to F(x)$ at every continuity point of $F$. Only the distribution converges, not the values themselves.In $L^p$: $E[|X_n - X|^p] \to 0$. Convergence in $p$-th moment.

Almost sure implies in probability implies in distribution; the reverse implications fail in general. The Strong Law of Large Numbers is a statement about almost-sure convergence; the Weak Law is in probability. The Central Limit Theorem is about convergence in distribution — the sample mean's distribution approaches Normal, but individual realizations don't converge to a Normal.

Distinguishing these matters. Convergence in distribution alone is too weak to swap inside an expectation safely; you usually need uniform integrability or stronger.