Consider two one-dimensional discrete time simple, symmetric random walks on:
a) a finite lattice of $L$ sites
b) a finite lattice of $2L-1$ sites
In both cases, once the walker starts from $1$ and when it reaches the right boundary, it reflects back with probability $1$. Is the distribution of first return time in a), the same as distribution of $\min\{$first return to $1$, first passage to $2L-1$ in b)$\}$
Problem is a). I feel that b) is just Gambler's ruin. Is this reasoning correct? Any help would be appreciated.
Yes. The equivalence classes of the states in b) under the reflection symmetry are in bijection with the states in a), and the transition probabilities between these equivalence classes are those of the states in a); thus the return times on these equivalence classes are also the same, and the return time of the equivalence class formed by the two ends is your minimum.