Convergence of persistence diagram in the sparse regime

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