Euclidean setting

In this subsection, we show the strong law of large numbers for simplex counts for Euclidean spaces in the thermodynamic regime and give the proof of the statement (b) of Lemma3.4. The statement (a) is for the manifolds, so its proof is discussed in the next subsection. Note that the strong law forSj(·) in the thermodynamic regime may follow from the general theory of local functionals due to Penrose and Yukich [55, 57]. However, we present here an elementary proof by calculating the order of the fourth moments.

Our argument is based on the following two results, called Palm theorems, to calculate the expectation and higher moments of simplex counts. Let P(f(x)) be a Poisson point process on R^N with intensity function f(x), which is assumed to be integrable overR^N.

Theorem 3.7 (cf. [53, Theorem 1.6]). Suppose j ∈ N and h(Y) is a measurable function defined on all finite subsetsY ofR^N, satisfying h(Y) = 0 except whenY has j elements. Then

E

[ ∑

Y⊂P(f(x))

h(Y) ]

= 1 j!

∫

(R^N)^j

h(y)

∏j i=1

f(yi)dy,

where, fory= (y₁, y₂, . . . , y_j)∈(R^N)^j, we use the shorthandsh(y) :=h(y₁, y₂, . . . , y_j) anddy:=dy1· · ·dyj.

Proof. LetP^′ =P(f(x)). Sincef(x) is integrable overR^N,P^′(R^N) is finite almost

surely. To prove the theorem, we condition on the event{P^′(R^N) =n}. Therefore, E[ ∑

Y⊂P^′

h(Y) ]

=

∑∞ n=j

P(P^′(R^N) =n)E[ ∑

Y⊂P^′

h(Y)|P^′(R^N) =n ]

=

∑∞ n=j

P(P^′(R^N) =n) (n

j )

E[h(Y)]

= c^j

j!E[h(Y)], where c = ∫

R^Nf(x)dx and Y = {Y1, Y2, . . . , Yj} is a random vector of j random variables that are independently and identically distributed with density function c⁻¹f(x). Since

E[h(Y₁, Y₂, . . . , Y_j)] =c^−j

∫

(R^N)^j

h(y)

∏j i=1

f(y_i)dy, the proof is completed.

Theorem 3.8 (cf. [53, Theorem 1.7]). Suppose k ∈ N. Then under the same as-sumptions on the function h(Y) as in Theorem3.7,

E

[ ∑

Y1⊂P(f(x))

· · · ∑

Yk⊂P(f(x))

(∏^k

i=1

h(Yi) )

1_{{Yi∩Yj=∅for1≤i<j≤k}} ]

=

∏k i=1

E

[ ∑

Yi⊂P(f(x))

h(Yi) ]

,

where 1_{{Yi∩Yj=∅for1≤i<j≤k}} is the indicator function that is equal to 1 if and only if Yi∩ Yj =∅ for 1≤i < j≤k.

Proof. We prove this theorem fork= 2 and leave to the reader the straightforward generalization to higher values ofk. LetP^′ =P(f(x)) By the similar technique as in Theorem3.7, we have

E[ ∑

Y¹⊂P^′

∑

Y²⊂P^′

h(Y1)h(Y2)1_{{Y1∩Y2=∅}}

]

=

∑∞ n=2j

P(P^′(R^N) =n) (n

j

)(n−j j

)

E[h(Y1)]E[h(Y2)]

= c^2j

(j!)²E[h(Y1)]E[h(Y2)]

=

∏2 i=1

E[ ∑

Yi⊂P^′

h(Yi) ]

, wherec=∫

R^Nf(x)dxand the last equality is established in Theorem 3.7.

66 3.2. SLLN FOR SIMPLEX COUNTS Let (A, ρ) satisfy the assumptions in Theorem3.3. Lethj,rn,d(Y) be the indicator function which is equal to 1 if and only ifY ⊂R^N is aj-simplex of radiusrn, measured using the metric d. When r_n = 1 and dis the Euclidean metric, we write h_j(Y) for h_j,1,∥·∥(Y). Then the number of j-simplices in the ˇCech complex C(X, r, ρ) can be written as

S_j(X, r_n, ρ) = ∑

Y⊂X

h_j,rn,ρ(Y), (3.5)

whereX⊂ A is a finite set.

Forr ∈(0,∞) and x= (x1, x2, . . . , xj)∈(R^N)^j, define A^(N_j ⁾(r) = r^{N j}

(j+ 1)!

∫

(R^N)^j

hj(0,x)dx,

whereh_j(0,x) anddxstand for h_j(0, x₁, x₂, . . . , x_j) and dx₁· · ·dx_j respectively.

Recall thatPn is the Poisson point process onR^N with intensity functionnf(x), wheref(x) is supported onAand ∫

R^Nf(x)dx <∞. We now show the right order of the expectationsE[S_j(Pn, r_n, ρ)] whenr_ntends to zero. Then by calculating the order of the fourth central moments, we shall prove that in the thermodynamic regime, n⁻¹(Sj(Pn, rn, ρ) −E[Sj(Pn, rn, ρ)]) converges to zero almost surely as n tends to infinity. More explanation on the fourth moments will be given later.

Proposition 3.9. Assume that ∫

Af(x)^j+1dx <+∞ and lim_n→∞r_n= 0. Then

nlim→∞r_n^{−N j}n^−(j+1)E[S_j(Pn, r_n, ρ)] =A^(N)_j (1)

∫

A

f(x)^j+1 D(x)^j dx.

It follows from the equation (3.5) and Theorem 3.7that r_n^{−N j}n^−(j+1)E[Sj(Pn, rn, ρ)]

= r⁻n^{N j}

(j+ 1)!

∫

A^j+1hj,rn,ρ(x0,x)

∏j i=1

f(xi)f(x0)dxdx0

= r⁻n^{N j}

(j+ 1)!

∫

A

(∫

A^jh_j,rn,ρ(x₀,x)dx )

f(x₀)^j+1dx₀ + r_n^{−N j}

(j+ 1)!

∫

A^j+1h_j,rn,ρ(x₀,x) ( _j

∏

i=1

f(x_i)−f(x₀)^j )

f(x₀)dxdx₀

=I₁ⁿ+I₂ⁿ.

We claim thatI₁ⁿ is asymptotic toA^(N)_j (1)∫

Af(x)^j+1

D(x)^j dx, while I₂ⁿ tends to zero asn tends to infinity, from which Proposition 3.9 follows. Our claims are proved in the next two lemmas.

Lemma 3.10. As n→ ∞, I₁ⁿ= r⁻n^{N j}

(j+ 1)!

∫

A

(∫

A^jhj,rn,ρ(x0,x)dx )

f(x0)^j+1dx0 →A^(N)_j (1)

∫

A

f(x)^j+1 D(x)^j dx.

Proof. The idea is to use the Dominated Convergence Theorem (DCT) for the in-tegral with respect to dx0. We first show that for n large enough the integrand is dominated by an integrable function and then compute the point-wise limit. Let

F_n(x₀) =r⁻_n^{N j}

∫

A^jh_j,rn,ρ(x₀,x)dx.

Since the metric ρ satisfies the property (P2) and r_n → 0 as n → ∞, there exists n0 ∈Nsuch that for all n≥n0,hj,rn,ρ(x0,x) ≤hj,˜rn(x0,x), where ˜rn =rn/c with c being the constant in (P2). Therefore,

F_n(x₀)≤r⁻_n^{N j}

∫

A^jh_j,˜rn(x₀,x)dx.

Applying the change of variables x_i =x₀+ ˜r_ny_i for 1≤i≤j yields F_n(x₀)≤c^{−N j}

∫

B(0,2)^j

h_j(0,y)dy≤(c^−NLeb^N(B(0,2)))^j.

Thus for n large enough, the integrand is dominated by const ·f(x0)^j+1 which is integrable by our assumption.

Next we consider the point-wise limit of Fn. Let x0 ∈ A^◦. Since ρ satisfies (P1) and r_n→0, given ε >0, there exists n_x0 ∈Nsuch that for alln≥n_x0,

h_j,r⁺

n,dx0(x0,x)≤hj,rn,ρ(x0,x)≤h_j,r−

n,dx0(x0,x), (3.6) wherer⁺_n =r_n/(1 +ε) andr⁻_n =r_n/(1−ε). Let

I_n^+ε(x₀) =r⁻_n^{N j}

∫

A^jh_j,r+

n,dx0(x₀,x)dx, I_n^−ε(x₀) =r⁻_n^{N j}

∫

A^jh_j,r−

n,dx0(x₀,x)dx.

Then from the relation (3.6), we have that

εlim→0 lim

n→∞I_n^+ε(x0)≤lim inf

n→∞ Fn(x0)≤lim sup

n→∞ Fn(x0)≤lim

ε→0 lim

n→∞I_n^−ε(x0).

ConsiderI_n^+ε(x₀). By applying the change of variables x_i =x₀+r⁺_ny_i for 1≤i≤j, the indicator function h_j,r⁺

n,dx0(x0, x0+r⁺_ny1, . . . , x0+r_n⁺yj) is equal to hj,1,dx0(0,y) for all large enoughnsincex₀∈ A^◦ and r_n→0. Thus, forx₀ ∈ A^◦,

nlim→∞I_n^+ε(x₀) = (1 +ε)^{−N j}

∫

(R^N)^j

h_j,1,dx

0(0,y)dy=A^(N)_j (1)(1 +ε)^{−N j} D(x₀)^j . Similarly, for x₀ ∈ A^◦,

nlim→∞I_n^−ε(x₀) =A^(N)_j (1)(1−ε)^{−N j} D(x₀)^j .

Therefore, Fn(x0) converges to A^(N)_j (1)/D(x0)^j almost everywhere because of the assumptionLeb^N(∂A) = 0, from which the desired result follows.

68 3.2. SLLN FOR SIMPLEX COUNTS Remark 3.11. (i) Ifρis homogeneous and translation invariant, the change of

vari-ables can be applied directly inFn(x0).

(ii) IfA=R^N then forx₀∈R^N, the functionh_j,r⁺

n,dx0(x₀, x₀+r⁺_ny₁, . . . , x₀+r_n⁺y_j) is equal to hj,1,dx0(0,y) for all n∈N.

Lemma 3.12. As n→ ∞, I₂ⁿ= rn^{−N j}

(j+ 1)!

∫

A^j+1h_j,rn,ρ(x₀,x) ( _j

∏

i=1

f(x_i)−f(x₀)^j )

f(x₀)dxdx₀→0.

Proof. The absolute value ofI₂ⁿ is bounded by

∫

R^N

(∫

Bρ(x0,2rn)^j

r_n^{−N j}

∏j i=1

f(xi)−f(x0)^j dx

)

f(x0)dx0.

We claim that the above expression tends to zero asntends to infinity. To show this, DCT is used again. Let

F_n(x₀) = (∫

Bρ(x0,2rn)^j

r⁻_n^{N j}

∏j i=1

f(x_i)−f(x₀)^j dx

) f(x₀).

Let us first show thatFnis uniformly bounded innby an integrable function. Clearly, F_n(x₀)≤r_n^{−N j}f(x₀)

∫

Bρ(x0,2rn)^j

∏j i=1

f(x_i)dx+r⁻_n^{N j}f(x₀)^j+1(

Leb^N(B_ρ(x₀,2r_n)))j

. Since ρ satisfies (P2), for all large enoughn,Bρ(x0,2rn) ⊂B(x0,2rn/c). Therefore, withC=(

Leb^N(B(0,2/c)))j

, we have F_n(x₀)≤Cf(x₀)

(

1

Leb^N(B(x₀,2r_n/c))

∫

B(x0,2rn/c)

f(x)dx )j

+Cf(x₀)^j+1. Let (M f)(x) be the centered Hardy–Littlewood maximal function, i.e.,

(M f)(x0) = sup 1

Leb^N(B(x0, r))

∫

B(x0,r)

f(x)dx,

where the supremum is taken over all the balls inR^N whose center is x0. Then Fn(x0)≤Cf(x0)(M f)(x0)^j+Cf(x0)^j+1.

Thus, we only need to show that the functionf(x0)(M f)(x0)^j is integrable to conclude thatF_nis dominated by an integrable function fornlarge enough. Sincef ∈L^p(R^N) for 1 < p ≤ ∞ implies M f ∈ L^p(R^N), by H¨older’s inequality with p = j+ 1 and q= (j+ 1)/j,

∫

R^Nf(x0)(M f)(x0)^jdx0≤ (∫

R^Nf(x0)^j+1dx0

) ¹

j+1(∫

R^N(M f)(x0)^j+1dx0

) ^j

j+1

<∞.

It remains to show that Fn(x0) converges to zero almost everywhere. Let x0 be a Lebesgue point off such thatf(x0)<∞. The convergence is proved by the induction on j. The inductive step is to bound F_n(x₀) by

( ∫

B(x0,2rn/c)^j

r_n^{−N j}|f(xj)−f(x0)|

j−1

∏

i=1

f(xi)dx )

f(x0)

+ (∫

B(x0,2rn/c)^j

r⁻_n^{N j}

j∏−1 i=1

f(xi)−f(x0)^j−1

f(x0)dx )

f(x0).

The first term with V_n=Leb^N(B(x₀,2r_n/c)) can be written as

C (

1 Vn

∫

B(x0,2rn/c)

f(x)dx

)_j−1( 1 Vn

∫

B(x0,2rn/c)

|f(xj)−f(x0)|dxj

) f(x0), which converges to zero since x0 is a Lebesgue point of f with f(x0) < ∞. The second term also tends to zero by the inductive hypothesis. Hence, by the Lebesgue differentiation theorem and the almost everywhere finiteness of f, the function F_n tends to zero almost everywhere. This completes the proof.

In the thermodynamic regime, Proposition 3.9is restated as follows.

Corollary 3.13. Assume that ∫

Af(x)^j+1dx < +∞ and limn→∞n^1/Nrn = r ∈ (0,∞). Then

nlim→∞

E[S_j(Pn, r_n, ρ)]

n =A^(N_j ⁾(r)

∫

A

f(x)^j+1 D(x)^j dx.

By defining ˆS_j^(N)(λ, r) := A^(N)_j (r)λ^j+1, the limiting constant in the above corol-lary can also be written as ∫

ASˆ_j^(N)(f(x)/D(x), r)D(x)dx. Note that ˆS_j^(N)(λ, r) is nothing but the limiting constant of L⁻¹E[Sj(PL(λ), r)] when L → ∞ (for the def-inition of PL(λ), see Subsection 3.1.1). The following corollary is also a result on homogeneous Poisson point processes but with the metricdx0, which is needed in the proof of Proposition3.20.

Corollary 3.14. Let x₀ ∈ A^◦. Let E ⊂R^N be a Borel set with Leb^N(E) >0, and let E_n=n^1/NE. Then for any λ≥0 and r∈(0,∞),

nlim→∞

E[S_j(P(λ)|En, r, d_x0)]

Leb^N(En) = ˆS_j^(N) ( λ

D(x0), r )

D(x0), where P(λ)|En is the restriction of P(λ) onEn.

Proof. Take f(x) =λ,A=E and rn=r/n^1/N in Corollary3.13, we get

nlim→∞

E[Sj(Pn, rn, dx0)]

n =A^(N)_j (r)Leb^N(E) λ^j+1

D(x₀)^j. (3.7)

70 3.2. SLLN FOR SIMPLEX COUNTS Consider the mapx→n^1/Nx withn^1/Nrn=r∈(0,∞) on the setE. By the scaling property of Poisson point processes, we have

E[Sj(Pn, rn, dx0)] =E[Sj(P(λ)|En, r, dx0)]. (3.8) Substituting the equation (3.8) into (3.7) yields

nlim→∞

E[S_j(P(λ)|En, r, d_x0)]

Leb^N(E_n) =A^(N_j ⁾(r) λ^j+1 D(x₀)^j, which completes the proof.

LetY1,Y2, . . . ,Yk⊂ Pnwith|Yi|=j+ 1,(1≤i≤k). Note that Theorem3.8can be applied only when the sets {Yi}^ki=1 are disjoint. In order to calculate the fourth moment of S_j(Pn, r_n, ρ), we need to deal with the non-disjoint case. Let us consider the case when k = 2, and leave to the reader the straightforward generalization to higher values ofk.

Lemma 3.15. Assume that∫

Af(x)^2j+1dx <+∞ and limn→∞n^1/Nrn=r∈(0,∞).

Then for any 1≤l≤j+ 1,

nlim→∞

1

nE[ ∑

Y1⊂Pⁿ

∑

Y2⊂Pⁿ

hj,rn,ρ(Y1)hj,rn,ρ(Y2)1_{{|Y1∩Y2|=l}} ]

=const.

Proof. For each finite subsetZ ⊂R^N, let h^′_j,rn,ρ(Z) =1_{{|Z|=2j+2−l}} ∑

Y⊂Z

∑

Y^′⊂Z

hj,rn,ρ(Y)hj,rn,ρ(Y^′)1_{Y∪Y′=Z}. Then

E[ ∑

Y1⊂Pn

∑

Y2⊂Pn

hj,rn,ρ(Y1)hj,rn,ρ(Y2)1_{{|Y1∩Y2|=l}} ]

=E[ ∑

Z⊂Pn

h^′_j,rn,ρ(Z) ]

, from which all the arguments in the proof of Proposition3.9 can work here to show the desired result.

We are now in the position to state and prove the strong law of large numbers for simplex counts.

Proposition 3.16. Assume that ∫

Af(x)^4j+1dx < +∞ and lim_n→∞n^1/Nr_n = r ∈ (0,∞). Then as n→ ∞,

S_j(Pn, r_n, ρ)

n →A^(N_j ⁾(r)

∫

A

f(x)^j+1 D(x)^j dx a.s.

This section is concluded with the proofs of Proposition 3.16 and the statement (b) of Lemma 3.4. In both the proofs, we show that the fourth moments of the appropriate quantities are of O(n^δ), where 0≤ δ <3. This is a sufficient condition for the strong laws. For instance, let ξ_n = (S_j(Pn, r_n, ρ)−E[S_j(Pn, r_n, ρ)]). We

shall show that E[ξ⁴_n] ≤ Kn², for some constant K. Then by Markov’s inequality, P(|ξn| ≥ nε) ≤ Kn⁻²ε⁻⁴. Since ∑

n⁻² < ∞, by the first Borel–Cantelli lemma, P(lim sup_n|n⁻¹ξ_n| ≥ε) = 0. This means n⁻¹ξ_n converges to zero almost surely. The strong law for n⁻¹Sj(Pn, rn, ρ) in the thermodynamic regime then follows from the convergence of expectation, which has been established in Corollary3.13. Now let us get into more detail on the proof of Proposition3.16.

Proof of Proposition 3.16. The presentation of this proof is similar to some parts of the proof of Theorem 3.9 in [53]. To ease notation, we write S_j for S_j(Pn, r_n, ρ) and h^′_j forhj,rn,ρ. Using the binomial expansion, we have

E[(S_j−E[S_j])⁴] =

∑4 k=0

(4 k )

(−E[S_j])^4−kE[S_j^k], (3.9) where

E[S_j^k] =E



 ∑

Y¹⊂Pⁿ

∑

Y²⊂Pⁿ

· · · ∑

Yk⊂Pⁿ

h^′_j(Y1)h^′_j(Y2)· · ·h^′_j(Yk)



. (3.10) Note that E[S_j^k] =O(n^k). So at the first look it seems that the equation (3.9) is of O(n⁴). However, this order is not exact. To calculate the exact order, we express E[S_j^k] in (3.10) as a sum of finitely many terms according to the intersection of{Yi}^k_i=1 and analyze the order of each term. For instance, when k = 2, there are (j + 2) terms corresponding to the conditions |Y1∩ Y2| = l,(l = 0, . . . , j + 1). The term corresponding to l = 0 coincides with E[Sj]² by Theorem 3.8, and hence, has order n², while the other terms have ordernby Lemma3.15. In general, each term in the expression (3.10) has order n^k′, where k^′ ≤ k is the number of disjoint components in{Yi}^k_i=1.

The only term in (3.10) that gives an order n^k comes from the case where all Y1,Y2, . . . ,Yk are disjoint. By Theorem 3.8, this term is equal to E[Sj]^k. Putting back this contribution for each k in (3.9), we see that the coefficient of the leading order term, the term of ordern⁴, is zero.

Now consider terms of order n^k−1 in (3.10). They should come from ordered k-subsets Y1,Y2, . . . ,Yk when two of them have 1 ≤l ≤ j+ 1 points in common and the remaining subsets have neither any point in common with each other nor with the two subsets. Clearly, in this case there is no contribution whenk = 0 ork = 1.

For fixed land 2≤k≤4, this contribution, denoted by T^k,l, is T^k,l=

(k 2 )

E[ ∑

Y1⊂Pn

∑

Y2⊂Pn

h^′_j(Y1)h^′_j(Y2)1_{{|Y1∩Y2|=l}} ]∏k

s=3

E[ ∑

Ys⊂Pn

h^′_j(Ys) ]

= (k

2 )

E[ ∑

Y1⊂Pn

∑

Y2⊂Pn

h^′_j(Y1)h^′_j(Y2)1_{{|Y1∩Y2|=l}} ]

E[S_j]^k−2.

Putting back the contributionT^k,l fork ≥2 in (3.9) again makes all terms of order n³ disappear. Thus, we deduce that E[(Sj−E[Sj])⁴] ≤Kn², for some constant K.

The proof is complete.

72 3.2. SLLN FOR SIMPLEX COUNTS Remark 3.17. Similarly, under the setting of Corollary3.14, asn→ ∞,

Sj(P(γ)|En, r, dx0)

Leb^N(E_n) →Sˆ_j^(N) ( γ

D(x₀), r )

D(x₀) a.s.

We now present the proof of the statement (b) of Lemma3.4.

Proof of Lemma 3.4 (b). For anym, letSj(m, n) =|Sj(Xm, rn, ρ)−Sj(Xn, rn, ρ)|. We first bound the probability of the event{X₁∈B_ρ(x, r_n)∩ A}. Clearly,

P(X1 ∈Bρ(x, rn)∩ A) =

∫

Bρ(x,rn)∩Af(x)dx. (3.11) Applying H¨older’s inequality, we obtain that

P(X1 ∈Bρ(x, rn)∩ A)≤ (∫

R^Nf(x)^pdx )1/p(

Leb^N(Bρ(x, rn)∩ A))1/q

, where p = 4j+ 1 and q = (4j+ 1)/4j. Since ∫

R^Nf(x)^4j+1dx < ∞, ρ satisfies (P2) and r_n → 0, it follows that in the thermodynamic regime P(X₁ ∈ B_ρ(x, r_n)∩ A) is bounded byc1n^−1/q for some constantc1.

Form > n≥j, since eachj-simplex inC(Xm, rn, ρ)\ C(Xn, rn, ρ) must contain at least one vertex in{X_n+1, . . . , X_m}, we have

S_j(m, n)≤

∑m i=n+1

ξ(X_i,X_m),

whereξ(Xi,Xm) counts the number ofj-simplices with one vertexXi inC(Xm, rn, ρ).

It follows that

E[S_j(m, n)⁴]≤(m−n)³

∑m i=n+1

E[ξ(X_i,X_m)⁴] = (m−n)⁴E[ξ(X₁,X_m)⁴]. (3.12) Note thatξ(X₁,Xm) can be written as follows

ξ(X1,Xm) = ∑

{i1,...,ij}

hj,rn,ρ(X1, Xii, . . . , Xij) =: ∑

i={i1,...,ij}

η_i,

where {i1, . . . , ij} is a subset of {2,3, . . . , m}. We estimate the fourth moment of ξ(X₁,X_m),

E[ξ(X1,Xm)⁴] = ∑

i,j,k,l

E[ηiηjηkηl]. (3.13) Letwt(i,j,k,l) =|{i∪j∪k∪l}|. Then

E[η_iη_jη_kη_l]≤P(X_i ∈B_ρ(X₁, r_n)∩ A:i∈i∪j∪k∪l)≤( c₁ n^1/q

)wt(i,j,k,l)

. Note that j ≤ wt(i,j,k,l) ≤ 4j and given wt(i,j,k,l) = w, the number of terms in the sum (3.13) is bounded by

(m−1 w

)(w j

)4

≤c(w, j)m^w,

where c(w, j) is a constant depending on w and j. Therefore, the sum (3.13) is bounded by

E[ξ(X1,Xm)⁴]≤c ( m

n^1/q )_4j

,

wherecis a constant which does not depend onm andn. Combining the formula for the fourth moment (3.12) and the above estimate, we have

E[S_j(m, n)⁴]≤c(m−n)⁴ ( m

n^1/q )4j

. When j≤m < n, we change the role ofm and nto get

E[S_j(m, n)⁴]≤c(n−m)⁴n^4j/p=c(n−m)⁴n^δ, whereδ = 4j/(4j+ 1)∈(0,1). Combining two estimates, we have

E[S_j(m, n)⁴]≤c(m−n)⁴ [

n^δ+ ( m

n^1/q )4j]

. Therefore,

E[

|S_j(Pn, r_n, ρ)−S_j(X_n, r_n, ρ)|⁴]

≤cE [

(N_n−n)⁴ (

n^δ+(N_n)^4j n^4j/q

)]

≤cE[(Nn−n)⁸]^1/2E [ (

n^δ+(Nn)^4j n^4j/q

)2]_1/2 . Here in the last inequality, we have used H¨older’s inequality. Note thatE[(Nn)^j] is a polynomial innof degreej. Thus the second factor in the above estimate is ofO(n^δ).

It is easy to check that E[(N_n−n)⁸] is a polynomial of degree 4 inn. Therefore, E[(Sj(Pn, rn, ρ)−Sj(Xn, rn, ρ))⁴] =O(n^2+δ).

This completes the proof.

ドキュメント内 Topology of Random Geometric Complexes inThermodynamic Regime (ページ 79-88)