Convergence of persistence diagram in the sparse regime

The objective of this paper is to examine the asymptotic behavior persistence diagrams associated with Čech filtration. A diagram a graphical descriptor topological and algebraic structure geometric objects. We consider filtration over scaled random sample rn−1Xn={rn−1X1,…,rn−1Xn}, such that rn→0 as n→∞. treat point process establish their limit theorems in sparse regime: nrnd→0, In setting, we show asymptotics kth depends on value sequence nk+2rnd(k+1). If nk+2rnd(k+1)→∞, converges deterministic Radon measure almost surely vague metric. rn decays faster so nk+2rnd(k+1)→c∈(0,∞), weakly limiting without normalization. Finally, if nk+2rnd(k+1)→0, probability distributions should be normalized, resulting convergence will treated terms M0-topology.