Definition: A sequence of random variables $\x_n$ converges almost surely to the random variable $\x$, denoted $\x_n \inAS \x$, if
\[\Pr \{ \lim_{n \to \infty} \x_n = \x \} = 1.\]An equivalent definition is given below.
Definition: A sequence of random variables $\x_n$ converges almost surely to the random variable $\x$, denoted $\x_n \inAS \x$, if for every $\eps > 0$,
\[\Pr \{ \norm{\x_k - \x} < \eps \textrm{ for all } k \ge n \} \to 1\]as $n \to \infty$
Comparing this definition to that of convergence in probability, it is clear that almost sure convergence implies convergence in probability.
Theorem: $\x_n \inAS \x \implies \x_n \inP \x$.
If $\bth \inAS \bt$, then $\bth$ is said to be a strongly consistent estimator of $\bt$. The sample mean, for example, is a strongly consistent estimator of $\Ex(X)$; this result is known as the strong law of large numbers.
Theorem (Strong law of large numbers): Let $\x, \x_1, \x_2, \ldots$ be independently and identically distributed random vectors such that $\Ex\norm{\x} < \infty$. Then $\bar{\x}_n \inAS \bmu$, where $\bmu = \Ex(\x)$.