Almost Sure Convergence

Definition

Let

• $X_n$ : Sequence of Random variables
• $X$ : Random variable

If $P\left[{\lim }_{n\to \infty }{X}_n=X\right]=1$

Then

• We say $X_n$ converges almost surely to $X$
• We write $X_n\stackrel{a.s.}{\to }X$

Remark

Almost sure convergence is stronger than convergence in probability, i.e., if $X_n\stackrel{a.s.}{\to }X$, then $X_n\stackrel{P}{\to }X$, but not necessarly the other way around.

Like convergence in probability, sometimes $X_n$ converges almost surely to a constant $c$. In that case, we may write $X_n\stackrel{\text{a.s.}}{\to }c$.

Related theorems

• Theorem Strong law of large numbers : ${\bar{X}}_n\stackrel{a.s.}{\to }\mu$