Pointwise Limit Inferior of Stopping Times is Stopping Time

Theorem

Let $\struct {\Omega, \Sigma, \sequence {\FF_n}_{n \ge 0}, \Pr}$ be a filtered probability space.

Let $\sequence {T_n}_{n \in \N}$ be a sequence of stopping times with respect to $\sequence {\FF_n}_{n \ge 0}$.

Let:

$\ds T = \liminf_{n \mathop \to \infty} T_n$

be the pointwise limit inferior of the $\sequence {T_n}_{n \in \N}$.

Then $T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

Proof

From the definition of the limit inferior we have:

$\ds \liminf_{n \mathop \to \infty} T_n = \sup_{n \in \N} \paren {\inf_{m \ge n} T_m}$

Writing:

$\ds \inf_{m \ge n} T_m = \inf_{m \in \N} T_{m + n - 1}$

From Pointwise Infimum of Stopping Times is Stopping Time, we have:

$\ds \inf_{m \ge n} T_m$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

Then, from Pointwise Supremum of Stopping Times is Stopping Time:

$\ds \sup_{n \in \N} \paren {\inf_{m \ge n} T_m}$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

That is:

$T$ is a stopping time with respect to $\sequence {\FF_n}_{n \ge 0}$.

$\blacksquare$