Reflection principle for walk with possible steps right, left and stay

Question

I need to use reflection principle for one dimensional walk with equally possible steps right, left and stay.

I would like to know if hold a similar identity to that of question Is there an intuitive way to see this property of random walks?

If I go down through the proof in an accepted answer, I got

\begin{align*} P(M_n = r) &= P(M_n \geq r) - P(M_n \geq r+1) \\ &= P(X_n \geq r) + P(X_n \geq r+1) - P(X_n \geq r+1) - P(X_n \geq r+2) \\ &= P(X_n = r) + P(X_n = r+1). \end{align*}

But since $n$ does not have to be even, I cannot simplify this further. Is that right? If you go through accepted answer, I think that this is the only place where it deviates for my type of walk. Am I right?

Thank a lot for any help!

Answer 1

Yes, this is true. Only bottleneck is indeed nonparity of $n$.