Reflected Simple Random Walk

Question

Suppose $W = (W_{n})_{n\geq0}$ is a symmetric random walk on $\mathbb{Z}$ with $SRW(\frac{1}{2})$. Define $\hat{W_{n}} = (\hat{W}_{n})_{n\geq0}$ by $\hat{W_{n}} := |W_{n}|$. Show that for $y \gt 0$:

$\Pr_{x}(\hat{W_{n}} = y) = \Pr_{x}(W_{n} = y) + \Pr_{x}(W_{n} = -y)$

I can understand this equation since $\hat{W_{n}}$ is the modulus and so would take into account the 'ups' and the 'downs' I'm not sure how I'd go about starting this question though?

Answer 1

I think it is just an application of the reflection principle as you already mentioned in the question's title.

Well, maybe if you start from the RHS you would see better what is going on.

$$P_{x}[W_n=y] = \frac{N_{n,y}}{T_n} $$

and

$$P_{x}[W_n=-y] = \frac{N_{n,-y}}{T_n} $$

Where $$N_{n,y} = \text{Number of paths that hit y in n steps}$$ and $$T_n = \text{Number of all paths of n steps}$$

Now, note that every path counted by $N_{n,y}$ or $N_{n,-y}$, if you appropriately reflect them to become positives, gives you a valid path in $\{\hat{W}(n) = y \}$. And this gives you all paths in $\{\hat{W}(n) = y \}$.

Hope this can help.

EDIT: I think it is simpler... Just intersect the event the modulus is equal y with each possible value Wn can assume.