Let be a sequence such that as . Let be a trajectory of a simple symmetric random walk on . How do you find the limit of the following conditional probabilities:

Where and are fixed numbers?

Using independence of the increments of the random walk and Moivre-Laplace Theorem (e.g. Central Limit Theorem in discrete case), I find an expression of the form:

Which I am far from certain about. I would appreciate if anyone could come out with a proper proof.

Thanks for your help.