# Thread: Comparison of stochastic processes at a random time

1. ## Comparison of stochastic processes at a random time

Let $X_t$ and $Y_t$, $t \in \mathbb{N}$, be two discrete-time Markov chains such that $Pr(X_t\ge x) \le Pr(Y_t\ge x)$, for all $x$. Let $T$ be a non-negative random variable in $\mathbb{N}$ (T may depend on X and Y).

Is it true that $Pr(X_t\ge T) \le Pr(Y_t\ge T)$ ?

Thanks

2. Hint: Condition on the value of T

3. Thanks, you're right, after conditioning we found that that is true.

My only concern was that with martingales, for a stopping time T, it is not true that its expectation is zero at T (which is random) even though is zero for all fixed t>0... This does not apply in my case maybe. I have to study more theory.