The local limit theorem describes how the density of a sum of random variables follows the normal curve. However the local limit theorem is often seen as a curiosity of no particular importance when compared with the central limit theorem. Nevertheless the local limit theorem came first and is in fact associated with the foundation of probability theory by Blaise Pascal and Pierre de Fer...

Consider the heat kernel ℘(t,x,y) on universal cover M˜ of a closed Riemannian manifold negative sectional curvature. We show local limit theorem for ℘: limt→∞t3∕2eλ0t℘(t,x,y)=C(x,y), where λ0 is bottom spectrum geometric Laplacian and C(x,y) positive λ0-harmonic function which depends x,y∈M˜. also that λ0-Martin boundary equal to its topological boundary. The Martin decomposition gives family ...

On montre que dans tout système dynamique probabilisé ergodique et apériodique, il existe une fonction de carré intégrable à valeurs entières dont les sommes Birkhoff satisfont le théorème limite local.