Deducing results about continuous time random walks from the corresponding discrete time result

Question

Is there any standard way to prove results about continuous time random walks from the corresponding results for discrete time random walks?

Specifically, my problem is that I was reading Lawler and Limic's Random Walk: A Modern Introduction and there is a result that bounds the transition probability which feels like is should be true still for continuous time random walks.

The result states $\exists c>0$ such that $\forall x,n$

$$p_n(x)\leq\frac{c}{n^{d/2}}. $$

So my question is, can this be used to deduce the analogous result with a continuous time parameter $t$ in place of $n$?

Answer 1

You can discretize a continuous 1- dimensional random walk $X_t$: let $p=\mathbb P(X_t \mbox {hits 1 before hitting -1}|X_0=0)$ and $q=1-p$. Define a simple discrete random walk $Y_n$ with transition probabilities p and q. This gives you $t_1<t_2<..$ such that $X_{t_n}=Y_n$ and some version of the mentioned result must hold for $X_t$.

I think I read this in Durrett's "Probability theory and applications".