On approximation theorems for the Euler characteristic with applications to the bootstrap

We study approximation theorems for the Euler characteristic of the Vietoris-Rips and Cech filtration. The filtration is obtained from a Poisson or binomial sampling scheme in the critical regime. We apply our results to the smooth bootstrap of the Euler characteristic and determine its rate of convergence in the Kantorovich-Wasserstein distance and in the Kolmogorov distance.


Introduction
In this manuscript we study approximation results and central limit theorems for the Euler characteristic (EC) χ of a simplicial complex K. The EC is a simple yet major functional in topological data analysis (TDA). Recent contributions concerning the EC in TDA include Adler [1], Turner et al. [41], Decreusefond et al. [14] and Crawford et al. [13]. See also [30,31,32]. Multivariate central limit theorems for the EC were proved in Hug et al. [22]. Ergodic theorems for the EC are given in Schneider and Weil [37]. Thomas and Owada [39] derive a functional strong law of large numbers and a functional central limit theorem (FCLT) for are needed in this paper. We refer to Boissonnat et al. [6] for a more thorough introduction to the subject.
Given a finite set P , an (abstract) simplicial complex K is a collection of non-empty subsets of P which satisfy (i) if x ∈ P , then {x} ∈ K and (ii) if σ ∈ K and τ ⊆ σ, then τ ∈ K. If σ ∈ K and #σ = k + 1, with k ∈ N 0 , then the simplex σ has dimension k, viz., dim σ = k.
The EC of a (finite) simplicial complex K is given by the alternating sum of its simplex counts S k (K) = #{σ ∈ K : dim σ = k}, viz., In this work, we consider theČech and Vietoris-Rips complex constructed from point clouds in R d . Given a finite subset P of the Euclidean space R d , thě Cech filtration C(P ) = (C t (P ) : t ≥ 0) and the Vietoris-Rips filtration R(P ) = (R t (P ) : t ≥ 0) are defined through the following simplicial complexes where B(x, t) = {y ∈ R d : x − y ≤ t} is a closed Euclidean ball, and diam(·) is the diameter of a set. We use a generic notation and write K t for either theČech or the Vietoris-Rips complex with parameter t obtained from a random point cloud in R d . Thě Cech filtration characterizes simplices in terms of the radius of their circumsphere, the smallest closed ball containing the given simplex. The Vietoris-Rips filtration relies on the pairwise distances between points in the simplex. This property not only makes theČech filtration analytically more complex than the Vietoris-Rips filtration but also more computationally intensive to work with in practice. The filtration time of a simplex σ, written as r(σ), is the smallest filtration parameter t such that σ is included in K t . r(σ) corresponds to the radius of the circumsphere of σ in theČech case, and to the maximum pairwise distance between the points of σ for the Vietoris-Rips filtration.
Given an ordering of the simplices in the simplicial complex K, we can separate each S k (K) in positive simplices S + k (K) and negative simplices S − k (K), so S k (K) = S + k (K) + S − k (K). A k-simplex is positive if it creates a k-dimensional feature. It is negative if it kills a (k − 1)-dimensional feature. The difference S + k (K) − S − k+1 (K) is the kth Betti number β k (K) of the simplicial complex K, see also [6]. Clearly, S 0 = S + 0 . This yields the well-known identity . We begin with an approximation result, which shows that the EC is locally Lipschitz-continuous in the underlying density function.
Note that the choice of ρ in the above result is arbitrary and in particular independent of the density κ. It is known that the EC tends to a Gaussian process for a Poisson sampling scheme and the Vietoris-Rips filtration, see [39]. We generalize this statement in the following to theČech filtration and the binomial sampling scheme and quantify the convergence. The given rate is asymptotically optimal when compared to the classical result of Berry-Esseen for the normalized empirical mean of iid data which is of order n −1/2 . The main reasons for this fast rate are the stabilizing properties of the EC, see Proposition 4.5, which correspond to m-dependent (and thus nearly iid) observations. Theorem 3.2 (Normal approximation). Consider theČech or the Vietoris-Rips filtration as well as the Poisson or binomial sampling scheme. Let κ be a bounded density on [0, 1] d and let t ∈ (0, T ]. There is a ρ > 0 and a corresponding C 1,κ ∈ R + , also depending on t, such that for all ν ∈ B ∞ (κ, ρ) d K χ ν,n (t) Var(χ ν,n (t)) 1/2 , N 0,1 ∨ d W χ ν,n (t) Var(χ ν,n (t)) 1/2 , N 0,1 ≤ C 1,κ n 1/2 .

(3.2)
Moreover, there are C 2,κ , C 2,κ ∈ R + , also depending on ρ, t, such that We detail the limiting covariance structure of the process (χ n (t)) t∈[0,T ] in Theorem 3.4 below. In particular, we show that lim n→∞ Var(χ κ,n (t)) ∈ R + for each t > 0. In order to obtain (3.3), we need to quantify the quotient Var(χ κ,n (t))/ Var(χ ν,n (t)) and its inverse. Both quotients are meaningful if the limiting variance is bounded away from zero and infinity. Note that the normal approximation given above does not immediately extend to a uniform result.
To obtain a FCLT and rates of convergence that consider the entire EC on an interval [0, T ], we need an understanding of the continuity properties of the filtration time as a function of the underlying simplex. These depend on the simplicial complex in use and we highlight this by writing r C (·), resp., r R (·), for the filtration time of theČech, resp., Vietoris-Rips complex. We also write r(·) to refer to either of them if no distinction is necessary. Assume that Z 0 , Z 1 , . . . , Z q are iid according to a density κ on [0, 1] d and let {Z 0 , Z 1 , . . . , Z q } denote the q-simplex spanned by Z 0 , . . . , Z q . If we use the Vietoris-Rips filtration, we can easily derive where α d is the d-dimensional Lebesgue measure of the d-dimensional ball B d (0, 1), see Lemma 4.3.
If we instead use theČech filtration, the situation is much more complex because it is no longer sufficient to study pairwise distances only. Instead, the filtration time is influenced by the geometry of the embedding space, R d , and is determined by the radius of the circumsphere. This radius can be calculated analytically with the result from Coxeter [12] using the Cayley-Menger matrix; we also refer to Le Caër [27] for more results on the circumsphere of q + 1 points in d-dimensional Euclidean space. We obtain a similar result in Lemma 4.4, for a certain continuous real-valued function g * d depending on d only. With these preparations, we are now able to give the approximation property in the functional Kantorovich-Wasserstein distance d D W . Here it is of course necessary that sup t∈[0,T ] |χ κ,n (t) − χ ν,n (t)| be measurable, which is true because the EC functional t → χ(K t,n ) is càdlàg. Theorem 3.3 (Functional approximation). Let κ be a bounded density on [0, 1] d and let ρ ∈ R + . Let ν ∈ B ∞ (κ, ρ). Consider theČech or the Vietoris-Rips filtration. Let [0, T ] be partitioned into J equidistant intervals of length T /J.
There are coupled Poisson processes (P n , Q n ) with intensities (nκ, nν), coupled binomial processes (X n , Y n ) of length n with densities (κ, ν), respectively, and there are constants C 3,κ , C 4,κ ∈ R + depending on κ, T > 0 and ρ but neither on ν ∈ B ∞ (κ, ρ) , nor on n nor on J, such that the following holds: In particular, there are constants C 5,κ , C 6,κ ∈ R + depending on κ, T > 0 and ρ, but neither on ν ∈ B ∞ (κ, ρ), nor on n nor on J, such that Obviously, the result in (3.5) is also valid for general (uncoupled) Poisson processes P n , Q n with intensity functions nκ, nν, and general n-binomial processes X n , Y n with density functions κ, ν, respectively.
Moreover, using the continuity properties of theČech filtration, we now extend the findings of [39] who provide a functional central limit theorem for the Vietoris-Rips complex and a Poisson sampling scheme. We remark that a functional central limit theorem for the binomial sampling scheme has not been established yet for either filtration type and follows from a Poissonization argument covered in the technical details of Section 4.
Using the strong stabilizing property of the EC from Proposition 4.5, the following limits exist for each t ∈ R + and z ∈ Z d and can be expressed in terms of a finite and deterministic radius of stabilization We assume the following technical condition for the FCLT. We call a density function blocked if it has the form For example, if κ can be approximated uniformly by continuous density functions, then it can also be approximated uniformly by blocked density functions. We present the FCLT, which enables us to capture the dynamic topological evolution of Vietoris-Rips andČech complex as the filtration time runs through a given interval [0, T ].
Theorem 3.4 (Functional central limit theorem). Let T ∈ R + . Let the filtration be obtained either from the Vietoris-Rips or theČech complex. Let κ satisfy (3.8). There is a Gaussian process G = (G(t) : t ∈ [0, T ]) such that, as n → ∞, The covariance structure of G depends on the sampling scheme. In the Poisson sampling scheme, where the random variable Z has density κ and for s, t ∈ [0, ∞) and for F 0 being the σ-field generated by {P ∩ Q(z) : z 0}. Then sup 0≤s,t≤T γ(s, t) < ∞ by the representation in (3.7).
In the binomial sampling scheme For both the Poisson and the binomial sampling scheme, the process G has a continuous modification which is β-Hölder continuous for each β ∈ (0, 1/2).
An immediate consequence of the functional central limit theorem is the weak convergence of continuous functionals applied to the EC curve. Let (S, d S ) be a metric space and let J : D([0, T ]) → (S, d S ) be continuous (w.r.t. to d S and the J 1 -topology). Then, under the assumptions of the above theorem, (J(χ n (t)) : t ∈ [0, T ]) converges weakly to J(G) as n → ∞.
As an example, consider the smooth EC-transform, which is the image of χ n under the continuous integration mapping Crawford et al. [13] consider a similar transform of the EC curve with practical applications in functional data analysis. Further potential applications of the smooth EC-transform I(χ n ) are goodness-of-fit tests as an exploratory tool in topological data analysis. We refer to [5] and [24] for similar applications in the context of persistent Betti numbers.

The bootstrap
Our bootstrap procedure merely requires an estimate for the true density function κ of the random variables X i underlying the Poisson or binomial process. Denote this estimate byκ n , where the index n refers to the sample P n , resp. X n . So when considering P n , we assume implicitely the knowledge of the Poisson parameter of N n , which is n. For instance,κ n can be obtained from a kernel density estimate, see [33] and [20].
The bootstrap procedure works as follows: Conditional on the sample P n or X n and the density estimateκ n , we resample a Poisson process P * n = {X * 1 , . . . , X * or a binomial process X * n = {X * 1 , . . . , X * n }, where the X * i are iid with densitŷ κ n and the random variable N * n is independent (of all other random variables) and Poisson distributed with mean n.
Using the sample P * n or X * n , we compute the EC of the correspondingČech or Vietoris-Rips complex K * t , which is either equal to K t (n 1/d P * n ) or to K t (n 1/d X * n ), t ∈ [0, T ]. The related empirical process is where E * denotes the expectation conditional on the sample P n or X n , respectively. In practice we use a kernel estimateκ n ; this smooth bootstrap is proposed in [36]. In that contribution we also address in detail possible problems with the "standard" bootstrap from the empirical distribution, which we sketch in the following. Hence, the present approach is an alternative, even though estimation of the true underlying density κ can be challenging, especially in high dimensions. When compared to the direct bootstrap from the empirical distribution, our smooth bootstrap procedure has certain advantages. As the empirical distribution is discrete, the number of unique values in a given bootstrap sample is random and strictly smaller than n, with an expected number of points approximately 0.632n. This can be problematic because in the critical regime, we rescale according to sample size by a factor of n 1/d . Moreover, since the support of the empirical distribution is discrete, the developed asymptotic theory does not apply, requiring at least an underlying distribution with a density. As such, there is a need for a smooth bootstrap procedure; we refer to [36] for a more thorough discussion with examples. Our first result applies to the EC evaluated at a specific point t. Furthermore, for each t ∈ [0, T ] Consider the case of n iid data points Z i , where the density κ has a continuous p th derivative on [0, 1] d and where the kernel density estimateκ n is obtained from a p th order kernel for an integer p ≥ 1 (see [40] for the definition of the order of a kernel). In this case, a.s., (3.11) see e.g. [20]. Hence, for the Kantorovich-Wasserstein distance A similar result is true for the Kolmogorov distance for fixed t (and not uniformly in t ∈ [0, T ]), viz., for each t ∈ [0, T ]. Moreover, we have the following functional result.

Simulation study
In this section, we provide the results for a series of simulations using the smoothed bootstrap procedure described in Section 3.2, establishing its efficacy in producing valid uniform confidence bands for the mean Euler characteristic curve of theČech complex. Due to computational constraints, data generating distributions were chosen in dimensions 2 and 3 only. A description of the distributions considered is given in Table 1. Visual illustrations are given in Figure 1 for F 1 to F 4 in dimension 2. For a given distribution and sample size, the true mean curve of the Euler characteristic (E χ(K t (n 1/d X n )) : t ∈ [0, T ]) was estimated using the average over a large (n µ = 50000) number of iid replicates from the true distribution. Betti number calculations for theČech complex were done using the GUDHI library via alphaComplexDiag from the TDA R package. Evaluation was done at a dense (n t = 1000) grid within [0, T ], with the exact value of T changing depending on sample size and distribution. T was chosen large enough as to not influence the analysis. The estimation error included in these steps is considered negligible.
Next, for a given sample size, we generate an original sample X n , and B = 1000 bootstrap replications, using the smoothed bootstrap procedure. Bandwidth selection was done using Hpi.diag from the ks R package.  The mean curve (E * χ(K t (n 1/d X * n )) : t ∈ [0, T ]) was estimated using the average Euler curve over the B bootstrap replicates, again evaluated at a dense (n t = 1000) grid within [0, T ]. For each bootstrap sample X * n,i , we calculate To establish coverage, the 0.95 quantile of the e 1 , ..., e B gives the width of the corresponding uniform confidence band, and is compared to using the established estimate of the true mean curve. The entire data generation and bootstrap procedure was repeated n p = 500 times to estimate the coverage proportion. Coverage proportions and details are provided in Table 2.
We see that the bootstrap procedure is generally conservative, yielding higher than the nominal 95% coverage proportion in the majority of cases, moving towards the nominal level with larger sample size. In the case of F 5 , the uniform distribution on [0, 1] 3 , the poor coverage for n = 30 and n = 50 is likely due to boundary effects not present in the other continuous cases. The stand-out case, however, is F 2 , which seems to diverge from the stated level for large samples.
In this case, the density approaches ∞ towards the origin, in such a way that no L p norm is bounded. This is likely the driving factor behind the poor coverage in this case.
For the results in this work, we consider only the case of a bounded density on [0, 1]. As shown by the provided coverage proportions, it is likely that these conditions can be greatly weakened, while still providing for bootstrap consistency.

Technical results
Throughout all our proofs, we will use the same terminology and notation. In the following lines, we introduce more definitions which are exclusively needed in this section and in the appendix.
Convention about the connectivity. Since we are studying simplicial complexes built from theČech and the Vietoris-Rips filtration for filtration parameters in the range [0, T ], an upper bound on the diameter of the simplex is 2T , resp. T . We abbreviate this upper bound by δ, e.g., we only need to know the points in a δ-neighborhood of a given point x in order to determine the simplices containing x.
Convention about the densities. Throughout this section and the appendix κ is an arbitrary but fixed bounded density on [0, 1] d . Moreover, for a given ρ ∈ R + , we study density functions ν ∈ B ∞ (κ, ρ). The choice of the neighborhood parameter ρ can depend on κ, however, this will then be mentioned. As already pointed out in Section 3, the constants depend then on κ and ρ but not on Convention about constants. To ease notation, most constants in this paper will be denoted by c, c , C, etc. and their values may change from line to line. These constants may depend on parameters like the dimension and often we will not point out this dependence explicitly; however, none of these constants will depend on the index n, used to index infinite sequences, or on the index i, used to index martingale differences. Furthermore, these constants will not depend on ν as long as ν satisfies ν − κ ∞ ≤ ρ. If we point out this property explicitly, we say "C is independent* of ν". Specific constants carry a subscript C 1 , c 1 etc.
Notation in the Poisson sampling scheme. Let P, P be independent Poisson processes on R d × [0, ∞) with unit intensity. We assume the following couplings Note that as in Section 3 the density κ is related to the Poisson processes P(n), P (n) whereas the density ν belongs to Q(n), Q (n). The Poisson processes P(n), P (n) and Q(n), Q (n) are supported on the cube B n = [−n 1/d /2, n 1/d /2] d and have intensity functions κ(·/n 1/d + e d /2) and ν(·/n 1/d + e d /2), respectively, where here e d is the all-one vector (1, . . . , 1) ∈ R d .
Recall that P n (resp. Q n ) is a non-homogenous Poisson process with intensity function nκ (resp. nν). Obviously, the distribution of n 1/d P n and P(n) are equal modulo the shift; the same holds for n 1/d Q n and Q(n). We write B n = {z ∈ Z d : Q(z) ∩ B n = ∅} and denote the cardinality of B n by b n . We will use an enumeration of B n given by In slight abuse of notation we write P i (n) rather than P {i} (n), and we write P z (n) when replacing the points in P(n) ∩ Q(z) by points in P (n). We also use a similar notation with P replaced by Q.
The following filtrations of simplicial complexes will be used to construct martingale differences: K t,n := K t (P(n)), K t,n := K t (Q(n)), and n ∈ N. The next two filtrations are needed for approximation arguments: and n ∈ N. Notice that filtrations without a "tilde" in their notation are based on P(n) (and P (n)), while those with a "tilde" are based on Q(n) (and Q (n)). The notation "tilde-star" indicates filtrations based on Q(n) with small parts replaced by points in P(n) and P (n), respectively.
For each n define a filtration of σ-fields by . Also set G n,0 = {∅, Ω}. The following notation is convenient for statements regarding the asymptotic normality. We write for first order differences in a specific point z ∈ Z d . Moreover, we use the following notation for first order differences tied to specific indices for A ⊆ [b n ] and j ∈ [b n ]: Notation in the binomial sampling scheme. In order to ease the notation, we set b n :≡ n, so that we can treat the binomial and the Poisson sampling scheme with the same notation. We use coupled binomial processes X = (X i : These have the property that (X, Y) and (X , Y ) are independent and the components of X, X and Y, Y have a density κ and ν, respectively, such that see for instance [15], Theorem 2.12. (Later, we will apply this coupling to the case where κ − ν ∞ is small.) In what follows, we will use the fact that the binomial processes are defined as sequences and not as point clouds. Define the filtration of σ-fields G n,i = σ{X j , Y j : j ∈ [i]} for i ∈ [n] and G n,0 = {∅, Ω}. Also, write X n for the elements X i of X with i ∈ [n], and similarly define Y n , X n and Y n . Furthermore, let We write X i n for X {i} n . A similar notation is used with X n replaced by Y n . The following definitions of filtrations of simplicial complexes parallel those of the Poisson case: and n ∈ N, as well as and n ∈ N. Compared to the Poisson case, we replace P(n), P (n) by n 1/d X n , n 1/d X n and Q n , Again, if A = ∅, we omit A in the superscript on the left-hand side.
Recall that in the Poisson sampling scheme the processes χ n were defined in Section 2 from the Poisson processes n 1/d P n (resp. n 1/d Q n ) and not P(n) (resp. Q(n)). However, it is not difficult to see that we can define P n and Q n on the same probability space such that the joint distributions of (n 1/d P n , n 1/d Q n ) and (P(n), Q(n)) are equal (modulo the shift by e d /2). To see this define The joint distribution of (P(n), Q(n)) is determined by the random variables (P(n)(A), Q(n)(B)), where A, B are Borel sets of R d . The same holds for (n 1/d P n , n 1/d Q n ). Using the independence property of the Poisson process, it is sufficient to consider the distributions of the type For the rest of the manuscript we will use the following definitions that apply to both the Poisson and the binomial case: Proof. The result relies on the Stirling formula √ 2πn n+1/2 e −n ≤ n! < en n+1/2 e −n for n ∈ N.
It is well-known that the binomial coefficient is maximal at m/2 if m is even and at (m A similar result is valid if m is odd. The claim regarding the moment of the Poisson random variable follows immediately because E[e δX ] = exp(λ(e δ − 1)) is finite for all δ < ∞. This completes the proof.
in the Poisson and in the binomial sampling scheme and for both theČech and the Vietoris-Rips complex.
Proof of Lemma 4.2. Let Y, Z be two point clouds. Then Consequently, it suffices to study the expression in the binomial case, where i is a generic index. (Exchanging the roles of Y and Z, the following arguments stay the same.) In the Poisson case, the result is an immediate consequence of Lemma 4.1. Indeed, let δ > 0 be defined as in the beginning of Section 4. Let U be a Poisson random variable with mean |Q(0) (δ) | (sup κ + ρ). Then there is a constant such that (4.3) is at most where we use that conditional on m there are at most m k+1 possible k-simplices, so the last result follows from Lemma 4.1. Clearly, the constant C p is independent of t ∈ [0, T ], z ∈ Z d , n and ν as long as ν − κ ∞ ≤ ρ.
In the binomial case, the reasoning is quite similar and we can use that the number of points is deterministic. Conditional on the realization where the constant C(d, T ) depends on d, T but not on the location x. For the last sum we have

Continuity properties of the filtration time
The following properties are crucial for the desired tightness results of the EC which we will prove below. We begin with the Vietoris-Rips complex, see also [39] for a similar result for this filtration.
The next lemma gives the corresponding continuity properties in theČech filtration.
Lemma 4.4 (Continuity in theČech filtration). Let q ≥ 1 and Z 0 , Z 1 , . . . , Z q be independent and identically distributed on Proof. To ease the notation we write r for the filtration time and begin as follows. Let z = (z 0 , z 1 , . . . , z q ) ∈ B(0, 1) q+1 be (q + 1) points in general position on B(0, 1). The circumsphere is the smallest d-dimensional ball containing all elements of z; its center is the circumcenter. Let . . , i m } is smaller). Indeed, the circumsphere and the circumcenter of q + 1 points in general position in R d is determined by at most d + 1 points. Thus, depending on z, we find such a subset with cardinality at most d + 1. In particular, given d and q the number of these minimal index sets is bounded above by In the following, let {i j,0 , . . . , i j,mj }, j ∈ [L], be an enumeration of these subsets. The above insights allow us to construct the following upper bound given an arbitrary density function κ on B(0, 1).

Approximation properties
Proof of Theorem 3.1. Let t ∈ [0, T ]. Using martingale differences and the definition of the σ-fields from (4.1), we obtain Var(χ κ,n (t) − χ ν,n (t)) For the summands in (4.6) we have by using the definition of the EC that (4.8) With ε := κ − ν ∞ , it remains to show that each of the last three expectations is of order ε uniformly in i ∈ [b n ], n and t ∈ [0, T ]. For this, we consider the two sampling schemes separately. The Poisson case. We define W (n) as the union of P(n), Q(n), P (n), Q (n). Also define symmetric differences of Poisson processes: W n,ε = P(n) Q(n) and W n,ε = P (n) Q (n).
When restricted to a cube Q = Q(z n,i ), W n,ε and W n,ε are not empty with a probability of order at most ε. Clearly, given a point Z in W n,ε , this point can only be involved in simplices which lie inside the δ-neighborhood Q (δ) of Q, where δ is the upper bound on the diameter of the simplices (defined at the beginning of Section 4), which is only depending on T and d but neither on n nor on i ∈ [b n ].
First we consider (4.8), here we give the details for the first term only; the second term can be treated very similarly. Write W (n) = W n,ε ∪ W n,ε ∪ W (n), where W (n) is the Poisson process that collects all remaining points from W (n) \ (W n,ε ∪ W n,ε ), so it has a finite intensity. Then |S k ( K t,n,i ) − S k ( K * t,n,i )| is stochastically dominated by the random variable where the last inequality follows from Lemma 4.1, and by bounding W n,ε (Q) by W (n)(Q (δ) ). We can compute moments of this last expression by exploiting the independence between (P, Q) and (P , Q ). Indeed, the components W n,ε , W n,ε and W (n) are independent, and so it is sufficient to consider (for C ∈ R + ) for constants C 1 , C 2 , C 3 < ∞; the last inequality follows by the mean-value theorem. This completes the considerations for (4.8).
Second, we study the term in (4.7). This second order difference can only be non zero if W n,ε (Q (δ) ) > 0. The conclusion follows now in a similar fashion as before (see (4.9)). Reasoning as we did after (4.2), we deduce (3.1).
The binomial case. The structure of the proof works in the same fashion. Since after applying the decomposition in martingale differences both (4.7) and (4.8) do not depend on the σ-field G n,i , we consider w.l.o.g. the case i = 1 (this simplifies the notation). We begin with the first term in (4.8): where the second sum is taken over all combinations (i 1 , . . . , i k ) in {2, . . . , n} with pairwise different indices.
Also, for k 1 , k 2 ≤ n−1 and for two sets {i 1 , . . . , i k1 }, {j 1 , . . . , j k2 }, which have common elements, the probabilities where the constant C d,T only depends on d and T . Using this last insight, elementary combinatorial calculations unveil for a constant C < ∞.
The bound for the second order difference in (4.7) follows in a similar fashion; we omit most of the details here, but refer to the proof of Theorem 3.3, where second order differences are studied in great detail. We have, where the sums are taken over all n−1 k k-element subsets (i 1 , . . . , i k ) of {2, . . . , n}. Clearly, a term only contributes to the sum if there is at least one index u ∈ {i 1 , . . . , i k } for which X u = Y u . By using similar arguments as in the proof of Theorem 3.3), we arrive at: This yields (3.1). The rate of convergence in the Kantorovich-Wasserstein distance is an immediate consequence.
Proof of Theorem 3.3. The proof has the same structure as the proof of Theorem 3.1. Let ν be arbitrary but fixed with ε := κ − ν ∞ ≤ ρ. Let [0, T ] be partitioned into J equidistant intervals of length T /J marked by the points t 0 , t 1 , . . . , t J . We apply the fundamental decomposition (4.10) The first term in (4.10) can be treated with the result in (3.1). In order to obtain a bound on the second term, we use the monotonicity of the EC. It is enough to study a generic index i ∈ [J], so we set a = t i−1 and b = t i . Also, we write t * for the time in [a, b], where the supremum is attained. So t * is random and measurable. First, we decompose the EC in two terms which contain the simplices of even, resp. odd, dimension. So, the first term is indexed by I 1 = {k ∈ N 0 : k is even}, the second by I 2 = N 0 \ I 1 . We only consider the index set I 1 , I 2 works in a similar fashion. The part of the second term in (4.10), which is related to I 1 , is then

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Clearly, it is enough to study the first term (4.12). If it is positive, then the double sum in the first term in (4.12) is at most Otherwise, if it is negative, then the double sum in the first term in (4.12) is at most and this term is already contained in the estimate in (4.13). Hence, it is enough to derive upper bounds for 1{r(σ) ∈ (a, b]} 2 (4.14) and separately. We begin with (4.14), which can be treated with an MDS approach similar as in the proof of Lemma 4.2 and Theorem 3.1. By using this MDS approach we obtain that the expression in (4.14) is at most We continue by estimating (4.16) and (4.15) in the Poisson and the binomial case separately.
The Poisson case. Using the fact that the simplices σ with σ ∩ Q(z n,i ) = 0 appear in both of the double sums inside the expectation in (4.16), we can bound (4.16) by (4.17) Clearly, it is enough to study the first term in (4.17). We show that there is a constant, which is uniform in i and n, such that each expectation is at most C|b − a|ε, where ε is an upper bound for the supremum distance between the densities κ and ν. Let i ∈ [b n ] be arbitrary but fixed and set Q = Q(z n,i ). Moreover, we let W n = P(n) ∩ Q(n), P n,ε = P(n) \ Q(n), Q n,ε = Q(n) \ P(n).
These processes are independent. First, we compute the expectation on the cube Q given that W n (Q (δ) ) = m and P n,ε (Q (δ) ) = m, so that we can write note that this expression is 0 if m = 0 because each simplex necessarily contains at least one Poisson point of P n,ε . In order to compute the expectation, we need to control for arbitrary tuples (i 1 , . . . , i k ), (j 1 , . . . , j k ) and indices k, k , u, u . For this, we simply omit the simplex with the higher dimension. We obtain for the Vietoris-Rips filtration the uniform upper bound C d,T, κ ∞,ρ (k ∧ k ) 2 |b − a|, as in Lemma 4.3. The constant C d,T, κ ∞,ρ only depends on d, T , κ ∞ , ρ and is independent* of ν. Then (4.18) is at most for some p ∈ R + , and a constant C independent of i ∈ [b n ], n and independent* of ν.
If theČech filtration is used instead, we bound above the probability in (4.19) by C d,T, κ ∞,ρ (k ∧ k ) d+2 |b − a| for a constant C d,T, κ ∞,ρ , which only depends on d, T , κ ∞ , ρ and which is independent* of ν, see Lemma 4.4. So the upper bound in (4.21) changes only in terms of the constants but not in its structure.
Finally, we have to weigh this last upper bound according to the distribution of P n,ε (Q (δ) ) and W n (Q (δ) ). Note that the Poisson parameter of P n,ε (Q (δ) ) is at most ε |Q (δ) |. It is now straightforward to show that (for each C < ∞) m∈N0 m∈N0 for a constant C that is also independent of i ∈ [κ] and n and independent* of ν. This shows the claim for (4.14) because |b − a| = T /J. The claim regarding (4.15) follows in a similar fashion by partitioning again the simplices according to their position: This last upper bound is of order CT 2 ε 2 nJ −2 .
The binomial case. Again, we begin with (4.16). Here the martingale difference sequence is constructed by replacing points X i and Y i with independent points X i and Y i , respectively. For a fixed i, we have the following relation. A k-simplex σ in K b,n not containing n 1/d X i , also lies in K b,n,i , and thus cancels in (4.16). The same holds for a k-simplex σ not containing n 1/d X i : if σ ∈ K b,n,i , then σ ∈ K b,n . Again, these simplices cancel in (4.16). Clearly, this relation similarly holds for the simplicial complexes K b,n and K b,n,i . Hence, (4.16) is at most It suffices to consider the expectation in (4.22) for an arbitrary but fixed index i. Let N be the number of observations X j in the δ-neighborhood of X i , so, Conditional on N and X i , we can then compute the expectation. To this end, consider a generic simplex from X n with a strictly positive and bounded density function. In the same fashion, we write Z i for the corresponding elements from Y n . Then, for the Vietoris-Rips complex, by using Lemma 4.3, for certain constants q, c, C < ∞. The conclusion now follows from a Poissonization argument as the probability of . This shows that (4.16) (and thus also (4.14)), is at most C T J −1 ε for a certain C < ∞ which is independent of i ∈ [n], n and independent* of ν.
It is now straightforward to see that the term in (4.15) is bounded above in a similar fashion by Cn(T J −1 ε) 2 . We omit the details.

Asymptotic normality
In order to verify the asymptotic normality, we first show the strong stabilizing property of the EC. Write ∆(t, P ) = χ(K t ({0}∪P ))−χ(K t (P )). Define the radius of stabilization S := 2t. There is a random variable ∆ ∞ (t, P ) ∈ R such that In particular, the equalities in (3.6) and (3.7) are true.
Proof. Consider the difference that is, the limit variances of (χ n (t)) n are positive for each t > 0.
Proof. We begin with the case of a uniform distribution κ ≡ 1. Then the limit variance in the binomial sampling scheme equals for each t > 0, the positivity follows from the fact that the distribution of ∆ ∞ (t), t > 0, is non degenerate and Penrose and Yukich [35,Theorem 2.1]. (Observe that the second term in the last formula does not occur in the Poisson sampling scheme). Let now κ be a general density. We infer from Proposition A.1 that the limit variance in the binomial sampling scheme takes the form which is positive for t > 0 by (4.25).
Proof of Theorem 3.2. Let t ∈ (0, T ] be arbitrary but fixed. The proof is divided in two parts. First we derive (3.2), then (3.3).
Derivation of (3.2). Recall the definitions of D n (t, i) and D A n (t, i) given early in Section 4. Let µ n,t denote the distribution of χ ν,n (t)/ Var(χ ν,n (t)) 1/2 , and define .  where σ 2 = Var(χ ν,n (K t )). It thus remains to bound the terms on the right-hand sides.
Using Fatou's lemma, lim inf n→∞ Var(χ ν,n (t)) ≥ c * for some positive c * ∈ R for all ν ∈ B ∞ (κ, ρ). Here ρ must be sufficiently small, see (4.28) below. Moreover, we will see that c * depends on t. Indeed, lim inf n→∞ Var(χ κ,n (t)) 1/2 ≥ c 1 > 0 by Proposition 4.6, where c 1 depends on t. Furthermore, using the result of Theorem 3.1, there is a constant c 2 such that E[(χ κ,n (t) − χ ν,n (t)) 2 ] 1/2 ≤ which is positive if ρ is sufficiently small. This implies, σ 2 ≥ c * n for all but finitely many n and uniformly in ν ∈ B ∞ (κ, ρ) for some c * > 0, which depends on t. Furthermore, Lemma 4.2 says that, for each p ∈ N, E[| D n (t, i)| p ] is bounded above uniformly over all n, i ∈ [b n ] and ν ∈ B ∞ (κ, ρ). Hence given t, the second term in (4.26) and the third and fourth term in (4.27) are of order b n /n 3/2 which is of order n −1/2 . It remains to obtain bounds for Var(S) and Var(S ). We only study Var(S) in detail. The calculations will show that Var(S ) admits a very similar upper bound.
where all covariances are uniformly bounded by Lemma 4.2.
First consider the Poisson case. Clearly, D n (t, i), D A n (t, i) both only involve Poisson points in a δ-neighborhood of z n,i and δ does not depend on t, A, z n,i , see also Proposition 4.5. Consequently, exploiting the independence property of the Poisson process, for a given i, the number of indices i such that the covariances in (4.29) are non zero does not depend on n and is bounded above by some constant. Moreover, a combinatorial argument shows that Hence, Var(S) is of order b n . This completes the calculations in the Poisson case. Now we bound (4.29) in the binomial case. If i = i , then the remaining double sum is bounded above by a constant. To see this we use Cauchy-Schwarz and the boundedness of the variances together with the previous displayed formula. If i = i , then we need to consider the covariances between simplices in a δneighborhood around X i , X i and X i , X i . The covariance is only non zero if the distance between {X i , X i } and {X i , X i } is at most δ. This happens with a probability proportional to n −1 . This shows once more that Var(S) is of order n and establishes the claim in the case of a binomial sampling scheme. This demonstrates (3.2).
for some constant C ∈ R + not depending on n. To see this, let 0 ≤ s ≤ t ≤ T and s, s, t, t ∈ Γ n with s ≤ s ≤ s, t ≤ t ≤ t and |t − t| ≤ T /n, |s − s| ≤ T /n. Then, due to monotonicity of s → Σ i,n (s) , This gives together with a similar lower bound It remains to show that the last summand can be bounded by a constant C > 0. To this end, let 0 ≤ s ≤ t ≤ T with a distance of at most T /n. Then for a constant C q which is uniform in n, i ∈ [b n ] and s, t, moreover C q is independent* of ν and satisfies C q ≤ ce cq . In the Poisson sampling scheme, one uses that M has a Poisson distribution with a parameter λ ∈ R + that is uniformly bounded above in n and i ∈ [b n ]. Then using Lemma 4.1, we find that the right-hand side of (4.35) is of order Since |s − t| ≤ T /n and b n /n → 1, we see that, in the Poisson sampling scheme, E [|Σ i,n (s) − Σ i,n (t)|] is bounded above by a constant uniformly in s, t with |s − t| ≤ T /n and n ∈ N.
It is a standard routine to verify the same statement for the binomial sampling scheme, using the fact that in this case M n,i tends to a Poisson distribution. This concludes the proof of (4.34).
Part 2. We verify the moment condition from Bickel and Wichura (1971) for the maps where I ∈ {I 1 , I 2 }. We write ∆ q,n (s, t) for S q (K t,n ) − S q (K s,n ). Let 0 ≤ r ≤ s ≤ t ≤ T be elements in Γ n . We show for a constant C not depending on n and r, s, t ∈ Γ n , r ≤ s ≤ t.
First, for any s ≤ t, we rewrite ∆ q,n (s, t) − E [∆ q,n (s, t)] as a martingale difference sequence where the filtration (G n,i : i = 1, . . . , b n ) is introduced at the beginning of Section 4 and where ∆ q,n,i (s, t) = S q (K t,n,i )−S q (K s,n,i ). With these abrevations the left-hand side of (4.36) equals i,j,k, ∈DM q1,...,q4∈I n −2 E E ∆ q1,n (s, t) − ∆ q1,n,i (s, t) | G n,i E ∆ q2,n (s, t) − ∆ q2,n,j (s, t)) | G n,j E ∆ q3,n (r, s) − ∆ q3,n,k (r, s) | G n,k E ∆ q4,n (r, s) − ∆ q4,n, (r, s) | G n, , (4.37) where the outer summation is extended over DM, the set of quadruples with "(at least a) double maximum" meaning the set of elements in [b n ] 4 for which the greatest index appears at least twice. All other index combinations have expectation 0 and thus do not enter the sum. We divide the remainder of the proof in two steps.
Step 1. We show that we can reduce the sum over DM to a few smaller sums, each of which is extended over only O((b n ) 2 ) indices, and one sum over three indices (of the order O(b n ) 3 indices), but with an extra factor of n −1 . This then enables us to derive the desired bounds, which is indicated in Step 2 below.
Due to the symmetry of the situation, it is enough to study the subcases of (4.37) with i < j < = k and i < k < = j. In these cases the summands in (4.37) have the structure E[F (W, V )G(W, V, R)], where W are the observations with index less than i, V are the observations with index i and R are the observations larger than i. To be more precise, the first conditional expectation in (4.37) is F (W, V ) with W = P(n) ∩ (Q(z n,1 ∪ · · · ∪ Q(z n,i−1 ), or W = (X 1 , . . . , X i−1 ), respectively, and V = P(n) ∩ Q(z n,i ), or V = X i , respectively. The last three conditional expectations in (4.37) are not only functions of V and W , but also of R = P(n) ∩ (Q(z n,i+1 ) ∪ · · · ∪ Q(z n, ), or R = (X i+1 , . . . , X ), respectively.
Clearly, W, V, R are independent. Hence, if we omit the variable V as input in the second factor, we see by using an independence argument that E[F (W, V )G(W, R)] = 0 because E[F (w, V )] = 0 for almost every realization w = W . It thus suffices to study the difference F (W, V )(G(W, V, R) − G(W, R)).
To make this more clear, let imsart-ejs ver. 2020/08/06 file: NormalApproxEuler-EJS.tex date: September 21, 2021 Each of the last three summands involves a factor of the form for some q and j = i (note that (r, s) can also take the role of (s, t) here). We now study this difference inside this conditional expectation. In the Poisson case, we write i j for Q(z n,i ) (δ) ∩ Q(z n,j ) (δ) = ∅, and i j otherwise. In the binomial sampling scheme, the notation i j (resp. ) simply means i = j (resp. =).
Using this notation, one finds that in the Poisson sampling scheme (4.38) is only non zero if i j because both ∆ q,n (s, t) − ∆ q,n,j (s, t)) and ∆ q,n,i (s, t) − ∆ q,n,j,i (s, t)) only involve q-simplices with filtration times in (s, t] which intersect with Q(z n,j ); and these are identical if i j. So the sum over three indices in (4.36) reduces to a sum over essentially two indices only; more precisely, the number of summands is of the order O (b n ) 2 .
If i j in the binomial sampling, then we obviously have a sum over two indices only, and (4.38) consists of the q-simplices containing an element of {X j , X j }. If i j, (4.38) consists of q-simplices containing and element of both {X j , X j } and {X i , X i }, and crucially, this event is of order n −1 . So the latter sum, which is a sum over three indices, has an additional correction factor n −1 . With these insights and the techniques presented in the next step, it is straightforward to verify the claim for these subcases.
Similar arguments hold in the case i < k < j = . We omit further details in this step and continue with Step 2.
Step 2. After having reduced the sums in Step 1, we now go back to (4.37) and study this sum in the reduced settings. We verify the claim for the index combinations containing two pairs or one triple, so that the relevant index set has order (b n ) 2 . Due to the symmetry of the situation, it is sufficient to study (a) i = j, k = , (b) i = k, j = and (c) i < j = k = . So, we have (at most) two indices only in each subcase (a) to (c); we write i and for these.
The difference ∆ q,n (s, t)− ∆ q,n,i (s, t) consists only of simplices in a δ-neighborhood of Q(z n,i ), or in a δ-neighborhood of X i or X i , respectively, with a filtration time in (s, t], i.e.,  Furthermore, we can apply the following super-positioning principle of point processes. Clearly, we can compute the conditional expectation in (4.37) using five independent processes P (0) (n), . . . , P (4) (n), resp. X n . We use the process indexed by 0 for the outer expectation and the other four for each In the Poisson sampling scheme, we have to consider the quantity  In the binomial sampling scheme, we replace the "cube intersection conditions" in (4.43) with the second line in (4.41) containing the "point intersection conditions".
We begin with the Poisson sampling scheme. Denote by p n,i (m) (resp. p n, (m)) the probability that Q(z n,i ) (δ) (resp. Q(z n, ) (δ) ) contains m Poisson points of P * (n). We write p n,0 (m) for the probability that Q(z n,i ) (δ) ∪ Q(z n, ) (δ) contain m Poisson points of P * (n). p n,i , p n, and p n,0 follow a Poisson distribution with a Poisson parameter which can depend on the indices but which is uniformly bounded above.
We begin with the subcase (a), where i = j and k = . Then (4.43) amounts to where the last inequality follows because the interval length is bounded below by n −1 . The subcases (b) and (c) follow similarly, the only difference is that the binomial coefficients change somewhat. The conclusion is the same. So we find in all three cases that (4.43) is at most C|s − r||t − s|.
In the binomial sampling scheme, we replace the Poisson distributions p n,i , p n, and p n,0 by their binomial approximations conditional on the sets {X factor n −1 by |t − s| or |s − r| because the interval length is bounded below by n −1 . With these preparations, it is a routine to verify that (4.43) is bounded above by C|s − r||t − s| for a C ∈ R + independent of n.
Proof of Proposition 4.8. First we prove the claim for the Poisson sampling scheme. The claim for the binomial sampling scheme is then an immediate consequence as shown below.
Let 0 ≤ s ≤ t ≤ T . We write G κ for the Gaussian limit of the EC, when the underlying density is κ. We use G for the Gaussian limit if κ is the uniform distribution on [0, 1] d .
Since G κ (t) − G κ (s) follows a normal distribution, we have for each k ∈ N Using the representation of the covariance function, we obtain in the Poisson case Consider the expectation in (4.45). Using the definition of γ, we have for s, t ≥ 0, Given a simplicial complex K and the set Q = Q 0 , we write S k (K; Q) for the number of k-simplices in K with one vertex in Q. Given the upper bound T , there is an R ≥ 0 such that for all 0 ≤ t ≤ T , the limit D ∞ (t, 0) admits the representation We can use the representation in (4.47) to obtain the following upper bound for (4.46) (up to a universal multiplicative constant) (4.48) We follow the calculations as in (A.1) to see that the expectation in (4.48) is at most C|s − t| for a universal constant C, which only depends on T . Combining the estimates from (4.44) to (4.48) yields that E (G κ (t) − G κ (s)) 2k ≤ C k |t − s| k for all 0 ≤ s, t ≤ T for a universal constant C k ∈ R + for all k ∈ N. Hence, by the Kolmogorov-Chentsov continuity theorem there is a continuous modification of G, which is β-Hölder continuous with exponent β ∈ ∪ k≥1 (0, (k − 1)/(2k)).
If the binomial sampling scheme is used, we have Consequently, the claim follows from the previous arguments.

Results on the bootstrap
Proof of Theorem 3.5. The estimateκ n is uniformly consistent, i.e. there is a random integer N 0 such that κ n − κ ∞ ≤ ρ for all n ≥ N 0 . So, we can apply (3.1) from Theorem 3.1 and (3.3) from Theorem 3.2 to obtain the desired result.
Proof of Theorem 3.6. The assertion of Theorem 3.6 is an immediate application of Theorem 3.3.
The proof is quite similar to the proof of Theorem 2.1 and 3.1 in [35]. However, we cannot immediately apply their theorem because it formally only applies to the density function κ = 1. Moreover, the EC is not necessarily polynomially bounded. Straightforward calculations show, however, that it is exponentially bounded. Indeed, let P be a finite point cloud, then Furthermore, as we treat the multivariate case, we want to obtain an analytic expression of the covariance structure of the Gaussian process appearing in the limit. For this we have to carry out the entire proof. However, since the EC is closely related to persistent Betti numbers, we can use the ideas laid out in [25] as a blueprint and so we will only sketch the main points here.
The positivity of σ follows from Proposition 4.6. Conditions (1) and (2) follow from Lemma 4.2. Indeed, consider (1), which is less than sup n∈N max 1≤i≤b n E D 2 n,i . Now E[D 2 n,i ] is bounded above in terms of the single differences E[|χ(K tu,n ) − χ(K tu,n,i )| 2 ], 1 ≤ u ≤ m, and these expressions are bounded above uniformly in n and i by Lemma 4.2. Regarding the property (2), we use that where D n,i (t) = E χ(K t (P(n))) − χ(K t (P i (n))) | G n,i . We show that for each pair (s, t) the random variable b n −1 b n i=1 D n,i (s)D n,i (t) attains the limit stated in the theorem. This is done in three steps. (i) We derive the covariance structure in the case where κ ≡ 1. (ii) We verify the claim if κ is a blocked density of the form m d i=1 b i 1 Ai . (iii) Finally, using the approximation result Theorem 3.1, we show the result for general density functions which satisfy (3.8).
Step (i). In this step, let κ ≡ 1 and let s, t ∈ R + . By construction P(n) is the restriction of a homogeneous Poisson process to the cube [−n 1/d /2, n 1/d /2] d . It is an immediate consequence of the strong stabilizing property outlined in Proposition 4.5 that there is an N 0 depending on z and T such that for all n ≥ N 0 and t ≤ T χ(K t (P(n))) − χ(K t (P z (n))) = D n (t, z) =: D ∞ (t, z) a.s.
Step (ii). Let κ be a blocked density of the form for a random variable Z, which is distributed according to the blocked density κ. The calculations are similar to those in the proof of Proposition 5.7 in [25], we omit the details. This completes step (ii).
The binomial sampling scheme. The result follows as in [25] using Poissonization arguments and the ideas of [35]; we only give a sketch and omit the technical details. (The arguments are very straightforward for the EC because the radius of stabilization is bounded, see also [23] for approximation results in the binomial sampling scheme.) Using the just cited sources, it is not difficult to show that for a general density κ on [0, 1] d Cov(χ κ,n (s), χ κ,n (t))