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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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  • $\begingroup$ The integral on RHS does not exist when $T=\infty$. $\endgroup$ – Kavi Rama Murthy Aug 14 at 7:30
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For finite $T$ use the fact that $|e^{ix} -1| \leq |x|$ to see that $|\frac {e^{-it(b-a)}-1} {it} e^{-it(X-b)}| \leq |b-a|$. DCT can be applied now. To be explicit write down the Riemann sums for the integral on the left, use the fact that expectation is linear and then take the limit. The constant function $|b-a| $ is integrable on $\Omega \times [-T,T]$ and this is the dominating function.

For $T=\infty$ the integral on RHS does not exist.

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  • $\begingroup$ Sir , I am aware of the inequality . Can you explicitly state the $\{ X_n\}$ that we can take here for applying the DCT. $\endgroup$ – John Aug 14 at 9:19
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    $\begingroup$ @John I have edited my answer. $\endgroup$ – Kavi Rama Murthy Aug 14 at 9:26
  • $\begingroup$ Here , the do we have riemann sums or lebeque sums . The rv may not have a density ? I am sorry for my stupid questions , but it would be extremely helpful if you could write down the ${X_n}$'s explicitly . Thank you . $\endgroup$ – John Aug 14 at 9:53
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    $\begingroup$ We are only using Riemann sums for the integral w.r.t $t$ for fixed $\omega$. Riemann sums are not for the integral over the product space. $\endgroup$ – Kavi Rama Murthy Aug 14 at 9:55
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For finite $T$, you should be able to use Fubini's theorem (since you're just interchanging integrals)

For $T=\infty,$ I can't even see whether it's true or not. You might use DCT in this case, but all the majorizing random variables I see don't seem to work.

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