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

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 $$0, when conditioned to go from 0 to 0 versus from 0 to 1 does not stochastically order, that is, $$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.

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).