Why random walk is considered a stochastic process? Definition of stochastic processes assumes that each random variable is based on the same probability space.
If we consider random walk: $$X_{n}=V_{1}+V_{2}+...+V_{n}$$ where $V$ are bernoulli random variables then probability space of $X_{1}$ is different than probability space of $X_{10}$, $X_{10}$ can take different values than $X_{1}$
The random walk is considered a stochastic process because it is ;-)
Usually you use the "infinite coin toss" for modelling a random walk and then the problem doesn't occur so take
$\Omega = \left\{\omega = (\omega_1, \omega_2, \ldots), \omega_i \in \{0,1\}, i\in\Bbb N \right\} = \{0,1\}^\Bbb N$ and $V_i(\omega) = \omega_i$
with $V_i$s are bernoulli random variables.
Then $X_n(\omega) = V_1(\omega) + \ldots + V_n(\omega)$ and all r.v. are defined on the same probability space…