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
Best Answer
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 $$ \int_{-\infty}^\infty (a_0+a_1x+\ldots+a_nx^n)^k \cdot e^{-(x-\mu)^2/(2\sigma^2)}\,dx $$