Convergence in Probability vs Distribution — Section 7: Limit Theorems

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.

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.

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