limite presque sure

Question

I just want to know why for a continue process X such $X_{t} \rightarrow Z$ p .s when $t \rightarrow \infty$ then lim inf $X_{s}^{2}$=Z when p .s $t \rightarrow \infty$ inf is on $\frac{t}{2}\leq s \leq t$. Thanks

Answer 1

This is a deterministic result: assume that $x_t\to z$ when $t\to\infty$ and that $A_t\subseteq\mathbb R$ is such that for every $t$ there exists some $s$ such that $A_u\subseteq[t,+\infty)$ for every $u\geqslant s$, then $\inf\{x_s\mid s\in A_t\}\to z$ when $t\to\infty$. (Likewise, $\sup\{x_s\mid s\in A_t\}\to z$ when $t\to\infty$.)

Note: Obviously, $\inf\{x_s^2\mid s\in A_t\}\to z^2$ when $t\to\infty$.