How to explain this using DCT

by John   Last Updated August 14, 2019 07:20 AM

Is it possible to use Dominated Convergence Theorem to explain the following calculation :

$\displaystyle\int_{-T}^T \mathbb{E}\left[\left( \frac{e^{-it(b-a)}-1}{it} \right) e^{-it(X-b)}\right]dt = \mathbb{E}\left[ \displaystyle\int_{-T}^T \left( \frac{e^{-it(b-a)}-1}{it} \right) e^{-it(X-b)}dt \right] $

Here , X is any random variable , and $T \in \mathbb{R} $ or $T = \infty$ . If not , then please explain when this step is valid .

I am aware of the following formulation of DCT :

If $\{X_n\}_{n \in \mathbb{N}} $ is a sequence of random variables such that $X_n(\omega) \rightarrow X(\omega)$ for all $\omega \in \Omega$ , such that $|X_n| \leq Y$ where $\mathbb{E}[Y] < \infty$ , then $\mathbb{E}[X_n] \rightarrow \mathbb{E}[X]$



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