Definition: A sequence of random variables $\x_n$ converges in probability to the random variable $\x$, denoted $\x_n \inP \x$, if for all $\delta > 0$,
\[\Pr\{ \norm{\x_n -\x} > \delta \} \to 0.\]Equivalently, we can write this condition as
\[\Pr\{ \norm{\x_n -\x} < \delta \} \to 1\]or (see neighborhood)
\[\Pr\{ x_n \in N_\delta(\x) \} \to 1.\]