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I have a sequence $(\mu_n)_{n\in \mathbb N}$ of probability measures on the measurable space induced by the Borel $\sigma$-field over $\mathbb R^{\mathbb N}$ equipped with its canonical topology (product topology).

I know that, in general, $(\mu_n)_{n\in \mathbb N}$ converges to a tight measure $\mu$ iff $(\mu_n)_{n\in \mathbb N}$ is tight and the finite-dimensional marginal distributions converge to those of $\mu$.

My question is: Do I really need tightness of $(\mu_n)_{n\in \mathbb N}$ when working on $\mathbb R^{\mathbb N}$? I saw a proof (Example 2.6 in Convergence of Probability Measures, 2nd Edition by Billingsley) that shows tightness of the sequence is not required. But I lack the intuition why it should be possible to say something about the joint distribution of an infinite dimensional object from only finitely many dimensions without any compactness assumption like tightness.

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  • $\begingroup$ It's not clear to me that what you are referring to, for example what kind of proof and proof of what? It seems too general... $\endgroup$
    – LNT
    Jan 22 at 10:42
  • $\begingroup$ I added a reference $\endgroup$
    – lmaosome
    Jan 22 at 12:13

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