I would like to get some feedback on the answer I wrote below:
The problem is from a textbook I'm reading: problem 3.2.8 of Probability Theory Lecture Notes by Panchenko. I posted about 3.2.7 here which is in relation to 3.2.8. Here's the question:
3.2.8. Let $f$ be a continuous function on $\mathbb{R}$ with $f(0) = 1$, $f(t) = f(-t)$ for all $t$, and such that $f$ restricted to $[0,\infty)$ is convex, with $f(t)\to 0$ as $t \to \infty$. Show that $f$ is a characteristic function. (Hint: Approximate the general case by piecewise linear. Then use that $\max(1-|t|,0)$ is a characteristic function and that characteristic functions are closed under convex combinations and scaling $f(t)\to f(at)$ for $a\in\mathbb R$.)
Answer: I looked this up and found an answer in Feller's intro to probability and its applications pages 505 and 509 (link to pdf). Here it goes:
As it is given in the hint, the idea is to approximate $f$ with piecewise linear functions.
- Piecewise linear functions can be written as a convex combination of rescalings of $\phi(t):= \max(1-|t|,0)$. Proof: Think of a piecewise linear function that is convex and is even as a convex polygon that is symmetric w.r.t. to the $y$-axis. Then, we can write $$f(t)= \sum_{i=1}^n p_i\phi(t/a_i)$$ where $\sum_i^np_i=1$ so that $f(0)=1$, and $a_i>0$, and $n$ is the number of sides on the positive side. This is clearly a convex combination of rescalings of $\phi(t)$.
- By 3.2.7 (linked post), we know that $\max(1-|t|,0)$ is a characteristic function.
- Therefore, since rescalings of characteristic functions are characteristic functions, and since convex combination of them is also a characteristic function, any piecewise linear function is a characteristic function.
- Now, let $f_n$ be a piecewise linear function that approximates $f$. By Levy's continuity theorem (theorem 3.5 of the same book), since $f_n\to f$ pointwise and each $f_n$ is a characteristic function, $f$ is also a characteristic function.