# 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 $$\mathbb{E}[X_n] \rightarrow \mathbb{E}[X]$$

Tags :

## Related Questions

Updated November 24, 2018 05:20 AM

Updated March 01, 2017 04:20 AM

Updated February 24, 2018 12:20 PM

Updated December 31, 2018 21:20 PM

Updated April 30, 2019 06:20 AM