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.