On limit theorems for persistent Betti numbers from dependent data
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We study persistent Betti numbers and persistence diagrams obtained from a time series and random fields. It is well known that the persistent Betti function is an efficient descriptor of the topology of a point cloud. So far, convergence results for the (r,s)-persistent Betti number of the qth homology group, β^r,s_q, were mainly considered for finite-dimensional point cloud data obtained from i.i.d. observations or a Poisson sampling scheme. In this article, we extend these considerations. We derive limit theorems for the pointwise convergence of persistent Betti numbers β^r,s_q in quite general dependence settings.
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