Suppose that I have a sequence of random variables $X=\langle X_1,…,X_n\rangle$ and I do not have any assumption about these continuous random variables, $X_i$'s, (they can be dependent/independent, identically distributed or not). Again suppose that $Y=\langle Y_1,…,Y_n\rangle$ is a stochastic process. Is there any difference between X and Y? can I say that X is also a stochastic process?
I would be grateful if you could help me.
Best Answer
According to the comments, there is a problem of comprehension at the heart of this question, which might be the following.
Consider a real valued random variable $Z$. This is a measurable function $Z:\Omega\to\mathbb R$ between a probability space $(\Omega,\mathcal F,P)$ and the measurable space $(\mathbb R,\mathcal B(\mathbb R))$. The distribution of $Z$ (aka the law of $Z$) is definitely not the probability measure $P$, but the image of $P$ by $Z$, often denoted $P_Z$, that is, the probability measure $\mu$ on $\mathcal B(\mathbb R)$ defined by the identity $\mu(B)=P(Z^{-1}(B))$ for every $B$ in $\mathcal B(\mathbb R)$.
Thus, to declare that two real valued random variables $X$ and $Y$ are identically distributed means that $X:\Omega\to\mathbb R$ and $Y:\Phi\to\mathbb R$ for some probability spaces $(\Omega,\mathcal F,P)$ and $(\Phi,\mathcal G,Q)$, and that the probability measures $P_X$ and $Q_Y$ on $\mathcal B(\mathbb R)$ coincide. The fact that $P=Q$ or that $P\ne Q$ (both can happen) is quite unrelated.
To declare that two real valued random variables $X$ and $Y$ are such that $X:\Omega\to\mathbb R$ and $Y:\Omega\to\mathbb R$ for some common probability space $(\Omega,\mathcal F,P)$, as you seem to be intent on saying, one usually simply mentions that $X$ and $Y$ are defined on the same probability space.