Probability and Mathematical Physics Seminar

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Limit theorems for signatures and applications

Speaker: Yuri Kifer, Hebrew University

Location: Warren Weaver Hall 1302

Date: Friday, October 3, 2025, 11:10 a.m.

Synopsis:

I'll talk about various limit theorems for iterated sums and integrals of the form S^{(ν)}_N(t) = N^{-ν/2} ∑_{0≤k1<...<kν≤Nt} ξ(k1) ⊗ ··· ⊗ ξ(kν), t ∈ [0, T] and S^{(ν)}_N(t) = N^{-ν/2} ∫_{0≤s1≤...≤sν≤Nt} ξ(s1)⊗ ··· ⊗ ξ(sν) ds1 ··· dsν, where {ξ(k)}_{−∞<k<∞} and {ξ(s)}_{−∞<s<∞} are centered stationary vector processes with some weak dependence properties. Collections of such iterated sums and integrals were called signatures in papers on the rough paths theory and their applications to data science, machine learning and neural networks were discussed recently. I'll speak on the law of large numbers, strong limit theorems, law of iterated logarithm and large deviations for such objects. An application to averaging will be mentioned, as well. All proofs are direct and they do not rely on the rough paths theory.