Stochastic Convergence

In order to understand what ”stochastic converge“ is, we have to remember first what ”deterministic convergence“ means.

Deterministic Convergence

Definition: A deterministic sequence $\{ x_n \} = x_1, x_2, \ldots$ is said to converge to the limit $C$ if for every $\varepsilon>0$ we can find an $n_0$ such that for $n>n_0$ we have
\begin{eqnarray}
|x_n-C|<\varepsilon, \quad n>n_0
\end{eqnarray}
If this is the case, we write
\begin{eqnarray}
\lim_{n \rightarrow \infty} x_n = C
\end{eqnarray}
$\Box$