TITLE: Examples of analysis results using probability theory
QUESTION [19 upvotes]: Sometimes, nice results from analysis appear unexpectedly in probability theory.
Here are a couple of examples:
$1.$ If $Z \sim \mathcal{N}(0,1)$, then $Z^2 \sim \Gamma(1/2,2)$
When we want to prove this, we find that $Z^2$ has density function $x \mapsto \sqrt{2\pi}^{-1} x^{-1/2} e^{-x/2}$ for $x \geq 0$
and comparing this to the density function of the gamma $(1/2,2)$ distribution, and using the fact that $\int_{-\infty}^{+ \infty} f(x)dx = 1$ for a density function $f$, it follows that $\boxed{\Gamma(1/2) = \sqrt{\pi}}$
$2.$ If $X \sim \Gamma(\alpha_1, \beta), Y \sim \Gamma( \alpha_2, \beta)$ and $X,Y$ are independent, then $X+Y \sim\Gamma(\alpha_1 + \alpha_2, \beta)$
While proving this, one can find the identity
$$\boxed{\int_0^1 u^{\alpha_1 -1}(1-u)^{\alpha_2 -1}du = \frac{\Gamma(\alpha_1)\Gamma(\alpha_2)}{\Gamma(\alpha_1 + \alpha_2)}}$$
So my question is: what are other examples where we can find interesting results from analysis (or other other branches of mathematics) using probability theory?

REPLY [5 votes]: 1) A classic problem first investigated by Erdos is:
Let $a_{0}=1$ and
$$a_{n}=a_{\left\lfloor n/2\right\rfloor}+a_{\left\lfloor n/3 \right\rfloor}+a_{\left\lfloor n/6\right\rfloor}.$$  Show that
$$\lim_{n\to\infty}\dfrac{a_{n}}{n}=\dfrac{12}{\log{432}},$$
where $\lfloor x \rfloor$ is the largest integer not greater than $x$. There's a wonderful solution involving optional stopping and Markov processes: How prove this nice limit $\lim\limits_{n\to\infty}\frac{a_{n}}{n}=\frac{12}{\log{432}}$