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I was reading about random walks and i encountered one step of a proof which i dont know how to derive in a mathematically rigorous way.

the problem is in the attached file and S is a random walk with X_i as increments, X_i =

{-1,+1}

I know that intuitively we can switch the indices to obtain the second equation from the first but how do we prove it rigorously?

EDIT: btw, im just looking for hints, not the entire solution. i think one of the possible hints is that the X_i's are i.i.d. but i cant think of a way to use this

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# Random walk probability

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