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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.

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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.

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  • $\begingroup$ 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 $J \in \mathbb N$ $\endgroup$
    – Buddy_
    Nov 29, 2019 at 15:55
  • $\begingroup$ As far as I understand you I would say yes, as long as they are defined on the same probability space $\endgroup$
    – claimes
    Nov 29, 2019 at 17:00

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