Almost sure convergence ($X_n\stackrel{a.s.}{\to }X$)

A sequence of random variables converges almost surely when the set of outcomes on which pointwise convergence fails has probability zero.

Random variables $X$ and $Y$ are almost surely equal when $\mathbb{P}(X=Y)=1$. Probability theory conventionally identifies random variables that differ only on a null event.

If $(X_n)$ are independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$, thenalmost surely.

If ${\sup }_n\mathbb{E}|X_n{|}^2<\infty$ and the $X_n$ are pairwise uncorrelated random variables, then $n^{-1}{\sum }_{k=1}^n(X_k-\mathbb{E}{X}_k)\to 0$ almost surely. Independence is unnecessary for this version of the strong law of large numbers. A uniformly bounded martingale difference sequence meets its hypotheses, since the differences have mean zero and are orthogonal in $L^2$.

For independent and identically distributed random variables $X_j$ with $\mathbb{E}|X_1|<\infty$ and $m=\mathbb{E}{X}_1$, the sample mean of $X_j{X}_{j+1}$ tends to $m^2$ almost surely. Split into odd and even indexed products; each subsequence consists of independent and identically distributed random variables that are integrable random variables. Apply the strong law of large numbers separately.

If independent centered random variables $(Y_n)$ satisfy ${\sum }_n\mathrm{Var}(Y_n)<\infty$, then ${\sum }_n{Y}_n$ converges almost surely. It follows from Kolmogorov's maximal inequality and a Cauchy-tail argument.

For independent identically distributed centered variables of variance one under standard moment assumptions, the law of the iterated logarithm describes the almost-sure envelope of their partial sums on the scale $\sqrt{2n\log \log n}$.

For standard Brownian motion, the Brownian reflection principle bounds the probability that its maximum up to $a^n$ exceeds $(1+\epsilon )\sqrt{2a^n\log \log (a^n)}$ by $2(n\log a{)}^{-(1+\epsilon {)}^2}$. This is summable for every $a>1$ and $\epsilon >0$. The Borel-Cantelli first lemma controls these geometric times, monotonicity of the normalizing function controls the intervening times, and countable choices $a\downarrow 1$, $\epsilon \downarrow 0$ give the bound. Equality requires a separate lower-bound proof.

For a simple symmetric random walk, geometric blocking and the Borel-Cantelli first lemma yield the upper estimate ${\mathrm{limsup}}_n{S}_n/\sqrt{2n\log \log n}\le 1$ almost surely. Equality requires a separate lower-bound argument.

For a simple symmetric random walk, almost surely