my problem is that I have a two Bernoulli Random walks \begin{align} X(n) = X_0 + \sum_{k \in \lbrace 1,...,n \rbrace} X_k \end{align} with $X_0= (y_0,z_0) \in \mathbb{R}^2$ and $X_k = (Y_k, Z_k)$ whereby $Y_k, Z_k$ are both independently Bernoulli distributed.
In addition, for both random walks should hold: $Y_k > Z_k \forall k \in \lbrace 1,...,n \rbrace$.
I am looking for the transition probability:
\begin{align} p(X_n, (i,j)) := \mathbb{P}(X_{n+1} = (i,j)| \sigma(X_0,...,X_n), Y_k > Z_k \forall k \in \lbrace 1,...,n +1 \rbrace ) \end{align}
for $i,j \in \lbrace -1,1 \rbrace$. It would as well help me to fix the notation for the line above, since mixing an σ-algebra and a set seems to be inappropriate for me.
Thanks for taking your time.