# Math Help - Comparison of stochastic processes at a random time 2

1. ## Comparison of stochastic processes at a random time 2

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 $Pr(X_T\ge x) \le Pr(Y_T\ge x)$ ?

Thanks

2. Good morning. Please can you help me to solve this exercise?. I must to use limiting distribution and change of variable. Thank you my friend.
Let Yn denote the nth order statistic, with Yn=qXn:n – ln(n), of a random sample X1,X2,…………..Xn from a exponential distribution Xi~exp(q). Find the limiting distribution of Yn.

3. I think that still conditioning over T it should be ok, right?

4. yes i would think so.