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Let $X$ be a Gaussian random variable, and let $a_0, a_1, \ldots$ be constants. Prove that the characteristic function of the random variable

$$Y = a_0 + a_1X + a_2X^2 + \cdots + a_n X^n$$

is infinitely differentiable.

I am really stuck on this problem and I do not know how to show infinite differentiability either. I tried to start off by calculating the characteristic function of different moments of a normal random variable, and multiplying them together. But I am not able to get a closed form, so I don't even know how to approach it. I would greatly appreciate anyone's help

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For every $k\geq 1$, if $\mathbb E[Y^k]$ exists, then characteristic function of $Y$ has $k$ continuous derivatives. So all you need is to show the existence of moments of $Y$ of all orders. This easily follows from absolute convergence of integral $$ \int_{-\infty}^\infty (a_0+a_1x+\ldots+a_nx^n)^k \cdot e^{-(x-\mu)^2/(2\sigma^2)}\,dx $$

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  • $\begingroup$ How do we know that this integral is absolutely convergent? I have been trying to show it with no luck. $\endgroup$ – user666614 Dec 3 at 13:35
  • $\begingroup$ How did you tried? Show your efforts. $\endgroup$ – NCh Dec 3 at 15:13
  • $\begingroup$ I tried to do the entire problem with induction. Let me update my post with my attempt. $\endgroup$ – user666614 Dec 3 at 15:19
  • $\begingroup$ Can you prove that normal distribution has finite moments of any order $\mathbb E[|X|^m] <\infty$? Or can you prove that $\mathbb E[X^2] <\infty$? There is no difference. $\endgroup$ – NCh Dec 3 at 15:22

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