# Random walk and definition of Stochastic Processes

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 $X_{1}$

• You consider all the 'values' that $X_n$ takes for some fixed $n$ normall (eg positiom after $10$ steps). So you are right. Apr 26 '17 at 13:16
• So what is definition of random walk? Sequence of $X_{n}$ defined as above? And such thing is not a stochastic process? Apr 26 '17 at 13:20
• Whose definition of stochastic processes assumes that each random variable is based on the same probability space? See for example math.stackexchange.com/a/17617/6460 In any case, all your $X_n$ take values in the integers (though some with probability $0$) Apr 26 '17 at 13:22
• Yes, the sum of the $V_i$'s. Apr 26 '17 at 13:23
• I am only talking about a random walk which is a very basic type of stochastic process. Apr 26 '17 at 13:25

$\Omega = \left\{\omega = (\omega_1, \omega_2, \ldots), \omega_i \in \{0,1\}, i\in\Bbb N \right\} = \{0,1\}^\Bbb N$ 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…