# Stochastic process and sequence of random variable

The question is actually very simple:

Could I define the stochastic process $$X=(X_t)_{t\in J}$$ from $$(\Omega,\mathcal F, \mathbb P)$$ into $$(E, \mathcal E)$$, as a sequence of random variable?

What I mean is, there is some difference between a stochastic process and a sequence of random variable?

I see that the question is quite silly, and in my opinion they are the same thing, but I would like to be sure. Thank you for the patience.

Actually this viewpoint is very adequate because we often see e.g. Random Walks with $$J = \mathbb{N}$$ under this exact idea with the sequence interpreted as a timeline. Notice however that you are limiting yourself to certain subsets $$J$$ as we are losing important aspects of a sequence otherwise. If you ment sequence rather in a more describing way we still get (and use) this perspective for example with the Brownian Motion which is a stochastic process with $$J = \mathbb{R}_+$$ as a stochastic process in continuous time.
• in practice a sequence of random variable it correspond to a stochastic process in discrete time? while if it is in continuous time, the concept are still similar but with the difference that in the stochastic process $J \in \mathbb R_+$ while in the sequence $J \in \mathbb N$ Nov 29, 2019 at 15:55