Let $F$ be a distribution on $\mathbb{Z}$. Let $(X_1,X_2,...)$ be an i.i.d. sequence of random variables with distribution $F$. Then $S_0=0, S_1=X_1, S_2=X_1+X_2,...$ is called the random walk with step size distribution $F$.
When I then have an expression like $$ P(S_1>0, S_2>0,...,S_n>0), $$ which probability measure P on which $\sigma$-algebra is meant?
Does this have something to do with a finite alphabet and the product-sigma-algebra generated by cylinder sets, i.e. with product measure
$P([s_1,...,s_n]=\prod_{i=1}^n p(s_i)$?