Conditional distribution for the random walk on $\mathbb{Z}$

Question

Define a random walk on the integers $\mathbb{Z}$ with step distribution $F$ and initial state is zero which is a sequence $S_n$ of random variables and its increments are iid random variables $X_i$ with common distribution $F$, that is, $$S_n=\sum_{i=1}^n X_i$$

Can we find a distribution $F$ such that for some $0<i<n$, when conditioned to go from 0 to 0 versus from 0 to 1 does not stochastically order, that is, $S_i$ condition on $S_n=1$ versus $S_i$ condition on $S_n=0$ does not stochastically order.

In fact, this is an example of a discrete random walk which when conditioned to form a bridge violates monotone coupling.

Answer 1

Suppose the possible increments are $+3$, $+2$, $-1$, and $-3$. The specific probabilities don't matter, as long as they're positive.

Then conditional on $S_2=1$, we have $S_1\leq 2$ with probability $1$ (the first two steps must be $2$ and $-1$ in some order).

But conditional on $S_2=0$, we have $S_1=3$ with probability $1/2$ (the first two steps must be $3$ and $-3$ in some order).