Random filtered complexes built over marked point processes on Euclidean spaces are considered. Examples of these filtered complexes include a filtration of

Much attention has been paid to topological data analysis (TDA) over the last few decades and persistent homology has been playing a central role as one of the most important tools in TDA. Persistent homology measures persistence of topological feature, in particular, appearance and disapperance of homology generators in each dimension and enables us to view data sets in multi-resolutional way. There are several aspects to be discussed in the theory of persistent homology, among which we focus on the random aspect. Data sets to be analyzed are often represented as binomial processes if each data point is regarded as a sample from a certain probability distribution and as stationary point processes if data points are considered as part of a huge object. There have been many works on the topology of binomial processes from the viewpoint of manifold learning [

Given data as a finite point configuration Ξ in

When we look at an atomic configuration, it is natural to consider the influence of atomic radii. In the usual setting, as explained above, we start from a finite set of points in Ξ and attach balls of radius

Now we introduce some notations to state our main theorems. We say that a nonempty finite subset Ξ of

For a

New feature of this theorem is two-fold: marks and averaging nets. The same limit theorem as above is first established in [

The paper is organized as follows. We give the statement of our results after introducing some notation and fundamental facts in Section

For a topological space

There exists an increasing function

Given a simple marked point set Ξ, we construct a filtration

For a fixed

Let

Let

In what follows, we fix a function

Now we consider marked point processes. Let

Let

Let

Let

Incidentally, for

Other examples and basic facts for marked point processes are available in [

In order to state the main results we introduce the notion of convex averaging nets in

Now we are in a position to state the main theorem.

Theorem

The proofs of Theorem

The aim of this section is to prove Theorem

Let

Proposition

Next we need a version of the multi-dimensional ergodic theorem for stationary ergodic marked point processes.

Take any

Let

The proof is similar to that of [

Now we state a basic estimate on the persistent Betti numbers for nested filtered complexes

Now we give the proof of Theorem

In order to prove the second assertion, we assume that Φ is ergodic. By virtue of the multi-dimensional ergodic theorem mentioned in Proposition

In this section we prove Theorem

An example of convergence-determining classes satisfying the conditions in Lemma

We finish with the proof of Theorem

Suppose that