# Proofs of main probability results from other fields

Making connections between different areas is very exciting and probability has already made connections with other fields (BM used in proving complex analysis and PDE results).

To keep it short, I will greatly appreciate any references to multidisciplinary proofs for any of the following results:

1)Central limit theorem (simple version, Lindeberg's or Donsker's theorem)

2)Law of large numbers

3)Recurrence in d=1,2 and transience in $d>2$ of random walk.

4)Conformal and time change: if f is conformal then $f(B_{t})\stackrel{d}{=}\widetilde{B}_{\int_{0}^{t}|f'(B_{s})|^{2}ds}$ where B,$\widetilde{B}$ are planar Brownian motions.

5)Feynman-Kac formula of Brownian motion

The more far-fetched the proofs the better. For example, I will be surprised (and glad) to see proofs using mainly topology, number theory or algebra.

For example CLT has a very interesting approach from information theory:

"An information-theoretic proof of the central limit theorem with the Lindeberg condition", Theory of Probability and its applications. 1959, Vol IV, n o 3, 288-299.

As for (3) (recurrence in $d\leq 2$ and transience in $d\geq 3$ of simple random walk), there are "electric networks"-proofs of these facts. See the classical book of Doyle and Snell "Random walks and electric networks", http://arxiv.org/abs/math/0001057