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
Central limit theorem (simple version, Lindeberg's or Donsker's theorem)
Law of large numbers
Recurrence in $d=1,2$ and transience in $d>2$ of random walk.
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.
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"An information-theoretic proof of the central limit theorem with the Lindeberg condition". Ju. V. Linnik (Transl. by R.A. Silverman). Theory of Probability & Its Applications. 1959,. Vol IV, n oNo 3,. 288-299.
