What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should be motivated by the sort of intuition one gains from a study of probability, e.g. games, information, behavior of random walks and other processes. This is very vague, but hopefully some of you will know what I mean (and perhaps have a better description for what this intuition is).
I'll give one example that comes to mind, which I found quite inspiring when I worked through the details. Every Lipschitz function (in this case, $[0,1] \to \mathbb{R}$) is absolutely continuous, and thus is differentiable almost everywhere. We can use a probabilistic argument to actually construct a version of its derivative. One begins by considering the standard dyadic decompositions of [0,1), which gives us for each natural n a partition of [0,1) into $2^{n-1}$ half-open intervals of width $1/{2^{n-1}}$. We define a filtration by letting $\mathcal{F}_n$ be the sigma-algebra generated by the disjoint sets in our nth dyadic decomposition. So e.g. $\mathcal{F}_2$ is generated by $\{[0,1/2), [1/2,1)\}$. We can then define a sequence of random variables $Y_n(x) = 2^n (f(r_n(x)) – f(l_n(x))$ where $l_n(x)$ and $r_n(x)$ are defined to be the left and right endpoints of whatever interval contains x in our nth dyadic decomposition (for $x \in [0,1)$). So basically we are approximating the derivative. The sequence $Y_n$ is in fact a martingale with respect to $\mathcal{F}_n$, and the Lipschitz condition on $f$ makes this a bounded martingale. So the martingale convergence theorem applies and we have that $Y_n$ converges almost everywhere to some $Y$. Straightforward computations yield that we indeed have $f(b) – f(a) = \int_a^b Y$.
What I really like about this is that once you get the idea, the rest sort of works itself out. When I came across the result it was the first time I had thought of dyadic decompositions as generating a filtration, but it seems like a really natural idea. It seems much more structured than just the vague idea of "approximation", since e.g. the martingale condition controls the sort of refinement the next approximating term must yield over its predecessor. And although we could have achieved the same result easily by a traditional argument, I find it interesting to see from multiple points of view. So that's really my goal here.
Best Answer
One nice example is Bernstein's proof of the Weierstrass theorem. This proof analyses a simple game: Let $f$ be a continuous function on $[0,1]$, and run $n$ independent yes/no experiments in which the “yes” probability is $x$. Pay the gambler $f(m/n)$ if the answer “yes” comes up $m$ times. The gambler's expected gain from this is, of course, $$p_n(x)=\sum_{k=0}^n f(k/n)\binom{n}{k}x^k(1-x)^{n-k}$$ (known as the Bernstein polynomial). The analysis shows that $p_n(x)\to f(x)$ uniformly.
S. N. Bernstein, A demonstration of the Weierstrass theorem based on the theory of probability, first published (in French) in 1912. It has been reprinted in Math. Scientist 29 (2004) 127–128 (MR2102260).