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Let $\mathcal{F} = \{ X_n, n = 1,2, \ldots\}$ be a family of exponential distributions with parameters $\lambda_n, n=1,2, \ldots$. I am looking for necessary and sufficient conditions for $\mathcal{F}$ to be tight.

Would be grateful, if you could give some ideas or insights. Thanks.

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    yes, I know Prokhorov's theorem.2012-12-02

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If $R>0$, then for each $n$, $P(|X_n|>R)=\int_R^{+\infty}\lambda_ne^{-\lambda_nx}dx=\int_{\lambda_n R}^{+\infty}e^{-t}dt=e^{-\lambda_nR}.$ Let $\lambda:=\inf_{n\geqslant 1}\lambda_n$. What can we say if $\lambda>0$?

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    The condition is actually necessary. If $\inf \lambda_n = 0$ there exist a sequence of variables with parameter tending to $0$, and then by prokhorov's theorem, $\mathcal{F}$ is not tight because the dirac in $0$ is not in $\mathcal{F}$.2012-12-03