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, 
$$S\_n=\sum\_{i=1}^n X\_i$$



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



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


