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Tight

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Definition
Probability

A sequence of probability measures {μn}n\{ \mu_{n} \}_{n} on (X,B)(\mathbb{X},\mathcal{B}) is said to be tight such that ϵ>0, compact KϵX:μn(Kϵ)>1ϵ,n1\forall\epsilon>0,\exists \text{ compact }K_{\epsilon}\subset \mathbb{X}:\mu_{n}(K_{\epsilon}) > 1 — \epsilon,\forall n\ge 1

That is, all of the measures give most of their mass to the same Compact set (or finite interval); mass does not “escape to” \infty.

The set of probabilities AMA ⊂ M is Precompact if and only if it is tight.

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