Almost Sure Convergence of Random Variables

Definition and Relationship to Other Modes of Convergence

Almost sure convergence is one of the most fundamental concepts of convergence in probability and statistics. A sequence of random variables (X _n ) _{n ≥ 1}, defined on a common probability space (Ω, $\mathcal{F}$, P), is said to converge almost surely to the random variable X, if

$$P(\{\omega \, :\,\lim \limits_{n\rightarrow \infty }{X}_{n}(\omega ) = X(\omega )\}) = 1.$$

Commonly used notations are $X_n \mathop \to \limits^{a.s.} X$ or lim _{n → ∞} X _n = X (a. s. ). Conceptually, almost sure convergence is a very natural and easily understood mode of convergence; we simply require that the sequence of numbers (X _n (ω)) _{n ≥ 1} converges to X(ω) for almost all ω ∈ Ω. At the same time, proofs of almost sure convergence are usually quite subtle.

There are rich connections of almost sure convergence with other classical modes of convergence, such as convergence in probability, defined by lim _{n → ∞} P( | X _n − X | ≥ ε) = 0 for all ε>...