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Two stochastically continuous processes on $[0,T]$ with the same finite dimensional distribution on a dense subset of $[0,T]$ have the same finite dimensional everywhere? The processes live on different spaces.

I suspect that this is true since Skorokhod seems to be using this in his book "Studies in the Theory of Random Processes" which I am self studying . Can somebody point me to some references where I could find a proof or give me a hint.

Thank you

user3503589
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1 Answers1

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If $X$ and $Y$ are processes agreeing on a dense set $S$, then

$$ \mathbf{P}(X_t < A) = \mathbf{P}( \limsup_{s < t} X_s < A) = \limsup_{s < t} \mathbf{P}(X_s < A) = \limsup_{s< t} \mathbf{P}(Y_s < A) = \mathbf{P}(Y_t < A) $$

The biggest jump in the calculation is moving the limsip out of the probability, but this follows by continuity, because $X_t$ is less than $A$ only when the $X_s$ to the left are eventually less than $A$. This idea can easily be generalized to finite sequences of random variables, so all f.d.d. agree.

Jacob Denson
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  • Can you please elaborate on the first and the second equality? How would this reasoning work if $X_t$ takes values in an arbitrary borel set? – user3503589 Dec 07 '17 at 10:55
  • Does it also work if Y is not necessarily stochastic continuous? – user3503589 Dec 07 '17 at 11:09
  • @user3503589 This distribution of a random variable is uniquely determine by it's cumulative distribution function, i.e. the values $\mathbf{P}(X_t < A)$, or the values $\mathbf{P}(X_t \leq A)$, and so on and so forth (you can see this from $\pi$ $\lambda$ theory, if you're interested. – Jacob Denson Dec 07 '17 at 22:34
  • @user3503589 No, then the last equality fails, where we move the limsup back into the calculation. Both processes need to be continuous. – Jacob Denson Dec 07 '17 at 22:35
  • In my particular case only one process is stochastically continuous from the right, the other is defined only on the a dense subset of $[t_0,T]$ i.e it not a continuum of random variables but r.vs only only dense subset – user3503589 Dec 07 '17 at 22:37
  • But you said they were both continuous? – Jacob Denson Dec 12 '17 at 04:34
  • I am sorry Jacob , I didnt think that the solution presented would use the stochastic continuity of both the processes. I know that one process is stochastic continuous and i get the second process as a result of the Skorokhod construction of rvs only in the dense subset of $[t_0,T]$. I need to somehow show that i can extend the newly constructed discrete process on the dense subset of $[t_0,T]$ to every where on $[t_0,T]$ such that the two processes have the same finite dimensional distribution. Do you get what I am saying? – user3503589 Dec 12 '17 at 15:55