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I'm trying to follow a proof of Lévy's Continuity Theorem. Let $X_n$ be a sequence of random variables with characteristic functions $\phi_n$, such that $\phi_n\to\phi$ pointwise and such that $\phi$ is continuous in a neighbourhood of $0$. Given

$$\mathbb{P}\left[|X_n|\geq \frac{1}{h}\right]\leq \frac{(1-\sin 1)^{-1}}{2h}\int_{-h}^h(1-\phi_n(t))dt$$

for each $n$, the author obtains, by the Dominated Convergence Theorem,

$$\limsup_n \mathbb{P}\left[|X_n|\geq \frac{1}{h}\right]\leq \frac{(1-\sin 1)^{-1}}{2h}\int_{-h}^h(1-\phi(t))dt$$

I'm not sure what function to use for the Dominated Convergence Theorem here. It seems strange that we can keep the same constant on the RHS in the process.

It's Theorem 29 of these notes.

Thank you!

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up vote 4 down vote accepted

You are actually using the fact that the $\phi_n$ are bounded by 1 (they are characteristic functions). Since the interval in question, $[-h, h]$, is bounded, all is copacetic.

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