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

Is it true that $\displaystyle Pr(X_T\ge x) \le Pr(Y_T\ge x)$ ?

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