I'm working with a discrete-time Markov Chain $\{Y_j, j \geq 0 \}$ that evolves untill a stoppingtime $\tau$ is reached. $X$ is een stochastic variable which depends on the state of the Markov Chain. I want to estimate $\mathbb{E}[X|Y_0]$ and to do that I use Importance Sampling.
It is assumed that $\mathbb E[\tau] < \infty$, but I don't understand why we need this assumption. Please help
Sincerely MKTEL
The reason one asks that $\tau$ is integrable is probably to ensure that $X$ is integrable, which is needed if one wants to consider the conditional expectation $\mathbb E[X\mid Y_0]$. But the lack of details in the question makes it difficult to say more.