  The objective of this paper is to examine the asymptotic behavior of
persistence diagrams associated with \v{C}ech filtration. A persistence diagram
is a graphical descriptor of a topological and algebraic structure of geometric
objects. We consider \v{C}ech filtration over a scaled random sample
$r_n^{-1}\mathcal X_n = \{ r_n^{-1}X_1,\dots, r_n^{-1}X_n \}$, such that
$r_n\to 0$ as $n\to\infty$. We treat persistence diagrams as a point process
and establish their limit theorems in the subcritical regime: $nr_n^d\to0$,
$n\to\infty$. In this setting, we show that the asymptotics of the $k$th
persistence diagram depends on the limit value of the sequence
$n^{k+2}r_n^{d(k+1)}$. If $n^{k+2}r_n^{d(k+1)} \to \infty$, the scaled
persistence diagram converges to a deterministic Radon measure almost surely in
the vague metric. If $r_n$ decays faster so that $n^{k+2}r_n^{d(k+1)} \to c\in
(0,\infty)$, the persistence diagram weakly converges to a limiting point
process without normalization. Finally, if $n^{k+2}r_n^{d(k+1)} \to 0$, the
sequence of probability distributions of a persistence diagram should be
normalized, and the resulting convergence will be treated in terms of the
$\mathcal M_0$-topology.
