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3 months ago in Probability Theory By Shraddha

Can you have a local central limit theorem in infinite dimensions?

We have local limit theorems for sums of random vectors in finite dimensions. What about when the random variables live in Wiener space—like paths of Brownian motion?

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By Meera Answered 2 months ago

Yes, these exist, and they're mathematically beautiful. They extend the classical CLT to infinite-dimensional Gaussian spaces. The tools come from Malliavin calculus, which gives you a way to prove that the laws of normalized sums have smooth densities and converge locally. Key results involve Fourth Moment Theorems and non-degeneracy conditions. It's deep, technical, and very much alive in contemporary stochastic analysis. Nualart is your gateway.

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