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I wonder if the following statement is true: Suppose you have a probability space $(\Omega,\mathcal{F}, \mathbb{P})$ and random variables $(X_n)_{n \in \mathbb{N}},X,(Y_n)_{n \in \mathbb{N}},Y$. If for all $\omega \in \Omega$ it holds [$X_n(\omega) \to X(\omega)$ $\Rightarrow Y_n(\omega) \to Y(\omega)$], can we then conclude that the same implication holds for stochastic convergence, so [$X_n \overset{\mathbb{P}}{\to} X \Rightarrow Y_n \overset{\mathbb{P}}{\to} Y$] ? My intuition is that this sounds useful, but i couldn't make up a sufficient proof yet.

I'm not actually asking for a proof, maybe just an answer by someone experienced, because i guess it is either trivial or not true at all (which perhaps can be seen by a counter example). Thanks in advance.

lower_rhine_stoc
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    I don't think you have formulated the question in a way that captures what you wanted. Consider a sequence $X_n$ such that $X_n \to X$ in probability but there does not exist an $\omega$ such that $X_n(\omega)$ is convergent e.g. the typewriter sequence. Then your implication is vacuously true for any sequence $Y_n$ so just pick a $Y_n$ that doesn't converge in probability. – Rhys Steele Sep 06 '22 at 09:38
  • i see your point and changed the formulation, but I'm not sure if this is what i actually meant. I think the assumption is too weak now. – lower_rhine_stoc Sep 06 '22 at 09:50
  • so the statement was too strong actually, now i think i finally have found the correct formulation – lower_rhine_stoc Sep 06 '22 at 10:06
  • okay I'm really sorry but I changed to the initial version again. Now the thing is that i wanted to generalize my actual problem because it would be too confusing. I think in my problem, there are $\omega$ that fullfill the convergence. – lower_rhine_stoc Sep 06 '22 at 10:20
  • I don't know how you expect people to answer this since we can't guess what conditions are true in your problem and my first comment shows some condition is needed to get anything sensible. Note that assuming some $\omega$ exists such that convergence holds does nothing at all, since I can always modify $X_n, X, Y_n, Y$ on a null set without changing anything to do with convergence in probability.

    If you want a sensible answer you have to do the work of formulating a sensible question.

     – Rhys Steele Sep 06 '22 at 13:51

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