# Proving a characteristic function is infinitely differentiable

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

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 $$\int_{-\infty}^\infty (a_0+a_1x+\ldots+a_nx^n)^k \cdot e^{-(x-\mu)^2/(2\sigma^2)}\,dx$$
• 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. – NCh Dec 3 at 15:22