Wasserstein-2 bounds in normal approximation under local dependence

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1 Wasserstein- bounds in normal approximation under local dependence arxiv: v1 [math.pr] 16 Jul 018 Xiao Fang The Chinese University of Hong Kong Abstract: We obtain a general bound for the Wasserstein- distance in normal approximation for sums of locally dependent random variables. The proof is based on an asymptotic expansion for expectations of second-order differentiable functions of the sum. Applied to subgraph counts in the Erdős- Rényi random graph, our result shows that the Wasserstein-1 bound of Barbour, Karoński and Ruciński (1989) holds for the stronger Wasserstein- distance. AMS 010 subject classification: 60F05 Keywords and phrases: central limit theorem, local dependence, Erdős- Rényi random graph, Stein s method, Wasserstein- distance. 1 INTRODUCTION For two probability measures µ and ν on R d, the so-called Wasserstein-p distance, p 1, is defined as ( )1 W p (µ,ν) = x y p p dπ(x,y), inf π Γ(µ,ν) where Γ(µ,ν) is the space of all probability measures on R d R d with µ and ν as marginals and denotes the Euclidean norm. Note that W p (µ,ν) W q (µ,ν) if p q. For a random vector W whose distribution is close to ν, it is of interest to provide an explicit upper bound on their Wassersteinp distance. See, for example, Ledoux, Nourdin and Peccati (015), Bobkov (018), Zhai (018), Bonis (018) and Courtade, Fathi and Pananjady (018) for a recent wave of research in this direction. We consider the central limit theorem in dimension one where µ is the distribution of a random variable W of interest, ν = N(0,1) and d = 1 in the above setting. A large class of random variables that can be approximated 1

2 by a normal distribution exhibits a local dependence structure. Roughly speaking, with details deferred to Section.1, we assume that the random variable W is the sum of a large number of random variables {X i : i I} and that each X i is independent of {X j : j / A i } for a relatively small index set A i. Barbour, Karoński and Ruciński (1989) obtained a Wasserstein-1 bound in the central limit theorem for such W and Chen and Shao (004) obtained a bound for the Kolmogorov distance. We refer to these two papers for a number of interesting applications. Our main result, Theorem.1, provides a Wasserstein- bound in normal approximation under local dependence. We follow the argument of Rio (009) to prove our main result. We first obtain an asymptotic expansion for expectations of second-order differentiable functions of the sum of locally dependent random variables W. We then use this expansion and the upper bound for the Wasserstein- distance in terms of Zolotarev s ideal distance of order to control the Wasserstein- distance between the distributions of W and a sum of independent random variables. Finally, we use the triangle inequality and known Wasserstein- bounds in normal approximation for independent sums to prove our main result. Rio (009) used the asymptotic expansion of Barbour (1986) for independent sums and a Poisson-like approximation to obtain a Wasserstein- bound in normal approximation for independent sums. The approach, combined with the recent result on Wasserstein-p bounds of Bobkov (018) for independent sums, may yield a Wasserstein-p bound for general p > 1 under local dependence. We apply our main result to the central limit theorem for the number of copies of a fixed graph in the Erdős-Rényi random graph. Barbour, Karoński and Ruciński (1989) provided a Wasserstein-1 bound and we show that the same bound holds for the Wasserstein- distance. In the special case of triangle counts, Röllin (017) showed that the same bound holds for the Kolmogorov distance. The paper is organized as follows. In Section, we present a general Wasserstein- bound in normal approximation under local dependence and apply it to the central limit theorem for subgraph counts in the Erdős-Rényi random graph. Section 3 contains some related literature and the proofs of the results in Section. MAIN RESULTS In this section, we present a general Wasserstein- bound in normal approximation under local dependence and apply it to the central limit theorem for subgraph counts in the Erdős-Rényi random graph.

3 .1 A Wasserstein- bound under local dependence Let W = i I X i for an index set I with X i = 0, W = 1 and satisfies the following local dependence structure: (LD1): For each i I, there exists A i I such that X i is independent of {X j : j / A i }. (LD): For each i I and j A i, there exists A ij A i and we can construct {(X k ) k/ Aij,( X k ) k Aij } which is independent of {X i,x j } and has the same distribution as {(X k ) k/ Aij,(X k ) k Aij }. (LD3): For each i I, j A i and k A ij, there exists A ijk A ij and we can construct{(x l ) l/ Aijk,( X l ) l Aijk }whichisindependentof{x i,x j,x k, X k } and has the same distribution as {(X l ) l/ Aijk,(X l ) l Aijk }. Theorem.1. Under the above setting, we have W (L(W),N(0,1)) C [ β +(γ 1 +γ +γ 3 +γ 4 +γ 5 ) 1 ], (.1) where C is a universal constant and β = X i X j X k + X i X j X k, i I j,k A i i I k A ij \A i j A i γ 1 = X i X j X k X l, i I j A i k A ij l A ijk γ = X i X j X k X l, i I j A i k A ij l A ijk γ 3 = X i X j X k X l, i I j A i k A ij l A ijk γ 4 = X i X j X k X l, i I j A i k A ij l A ijk γ 5 = X i X j X k Xl. i I k,l A ij j A i Remark.1. The conditions (LD1) (LD3) and the bound (.1) is a natural extension of (.1) (.5) and (.7) of Barbour, Karoński and Ruciński (1989). The sizes of neighborhoods A ij and A ijk are typically smaller than those used in Chen and Shao (004). It would be interesting to prove a bound for the Kolmogorov distance under the above setting. 3

4 . Subgraph counts in the Erdős-Rényi random graph Let K(n,p) be the Erdős-Rényi random graph with n vertices. Each pair of vertices is connected with probability p and remain disconnected with probability 1 p, independent of all else. Let G be a given fixed graph. For any graph H, let v(h) and e(h) denote the number of its vertices and edges, respectively. Theorem.1 leads to the following result. Theorem.. Let S be the number of copies (not necessarily induced) of G in K(n,p), and let W = (S S)/ Var(S) be the standardized version. Then { ψ 1 if 0 < p 1 W (L(W),N(0,1)) C(G) n 1 (1 p) 1 if 1 < p < 1, (.) where C(G) is a constant only depending on G and ψ = min {n v(h) p e(h) }. H G,e(H)>0 Remark.. Barbour, Karoński and Ruciński (1989) proved the same bound as in (.) for the weaker Wasserstein-1 distance. It seems natural to conjecture that the same bound holds for the Wasserstein-p distance for general p > 1. Remark.3. In the special case where G is a triangle, the bound in (.) reduces to n 3 p 3 if 0 < p n 1 C n p 1 n 1 (1 p) 1 if n 1 < p 1 if 1 < p < 1, where C is a universal constant. Röllin (017) proved the same bound for the Kolmogorov distance in this special case. 3.1 Preliminaries 3 PROOFS To prepare for the proof of Theorem.1, we need the following lemmas. The first lemma relates Wasserstein-p distances to Zolotarev s ideal metrics. Definition 3.1. Forp > 1,letl = p 1bethelargestintegerthatissmaller than p and Λ p be the class of l-times continuously differentiable functions f : R R such that f (l) (x) f (l) (y) x y p l for any (x,y) R. The 4

5 ideal distance Z p of Zolotarev between two probability distributions µ and ν is defined by } Z p (µ,ν) = sup fdµ fdν. f Λ p { R R Lemma 3.1 (Theorem 3.1 of Rio (009)). For any p > 1 there exists a positive constant C p, such that for any pair (µ,ν) of laws on the real line with finite absolute moments of order p, W p (µ,ν) C p [ Zp (µ,ν) ]1 p. We use Stein s method to obtain the asymptotic expansion in the proof of Theorem.1. Stein s method was discovered by Stein (197) to prove central limit theorems. The method has been generalized to other limit theorems and drawn considerable interest recently. We refer to the book by Chen, Goldstein and Shao (011) for an introduction to Stein s method. Barbour (1986) used Stein s method to obtain an asymptotic expansion for expectations of smooth functions of sums of independent random variables. Rinott and Rotar (003) generalized it for dependency-neighborhoods chain structures. Our expansion can be regarded as a refined version of the expansion of Rinott and Rotar (003) for the local dependence structure (LD1) (LD3) in Section.1. For a suitable function h, define Nh = h(z), where Z N(0,1). Let f = f h be the bounded solution to Stein s equation f h can be written as f (w) wf(w) = h(w) Nh. (3.1) f h (w) = w = We will use the following lemma. w e 1 (w t ) { h(t) Nh } dt e 1 (w t ) { h(t) Nh } dt. (3.) Lemma 3. (special case of Lemma 5 of Barbour (1986)). For any positive integer p > 1, let h Λ p where Λ p is defined in Definition 3.1, with h (s) (0) = 0,0 s p 1. Then f h in (3.) is p times differentiable, and satisfies for any x,y R. f (p) h (x) f(p) h (y) C p x y 5

6 In the final step of the proof of Theorem.1, we will invoke the known Wasserstein- bounds in the central limit theorem for sums of independent random variables. The following result was recently proved by Bobkov (018). Lemma 3.3 (Theorem 1.1 of Bobkov (018)). Let V n = n i=1 ξ i where {ξ 1,...,ξ n } are independent, with ξ i = 0 and Vn = 1. Then for any real p 1, [ n W p (L(V n ),N(0,1)) C p ξ i p+]1 p, (3.3) where C p continuously depends on p. The result for p (1,] was first proved by Rio (009), who also showed thattheratein(3.3)isoptimal. Thecaseforgeneral p > 1where{ξ 1,...,ξ n } are independent and identically distributed was first proved by Bonis (018). 3. Proof of Theorem.1 In this subsection, we use C to denote positive constants independent of all other parameters, possibly different from line to line. As noted in the Introduction, the proof consists of three steps. We first obtain an asymptotic expansion for h(w) for h Λ. We then use the expansion and Lemma 3.1 to control the Wasserstein- distance between the distributions of W and a sum of independent random variables. Finally, we use the triangle inequality and known Wasserstein- bounds in Lemma 3.3 for independent sums to prove our main result Asymptotic expansionfor h(w). Inthisstep, weprovethefollowing proposition. Proposition 3.1. Let W be as in Theorem.1, let h Λ with h(0) = h (0) = 1 and let f = f h be as in (3.). We have h(w) Nh+ β Nf ] C [ β W (L(W),N(0,1))+γ 1 +γ +γ 3 +γ 4 +γ 5, where β, γ 1 γ 5 are as in Theorem.1. i=1 Proof of Proposition 3.1. From Lemma 3., we have (3.4) f (x) f (y) C x y (3.5) 6

7 for any x,y R. As f solves (3.1), we have h(w) Nh = f (W) Wf(W). (3.6) For each index i I, let W (i) = W j A i X j. By (LD1), X i is independent of W (i). From X i = 0, Taylor s expansion and (3.5), we have Wf(W) = i I X i f(w) = i I X i [f(w) f(w (i) )] where = i I j A i X i X j f (W (i) )+ 1 R 1 C i I i I j,k A i X i X j X k f (W (i) )+R 1, j,k,l A i X i X j X k X l Cγ 1. (3.7) We begin by dealing with the first term on the right-hand side of (3.7). The second term can be dealt with similarly. In (LD), let W (ij) = X k + Xk. k/ A ij k A ij By construction, W (ij) is independent of {X i,x j } and has the same distribution as W. we have { X i X j f (W (i) ) = X i X j f ( W (ij) )+ [ f (W (i) ) f ( W (ij) ) ]} = X i X j f (W)+ X i X j (W (i) W (ij) )f ( W (ij) )+R,ij, where by (3.5), Note that R,ij C X i X j (W (i) W (ij) ). W (i) W (ij) = By independence, we have [ ( R,ij C X i X j C k A ij \A i X k k A ij \A i X k k,l A ij \A i X i X j X k X l +C k A ij Xk. ) + ( ) ] Xk k A ij k,l A ij X i X j X k X l, and hence i I j A i R,ij C(γ 1 +γ ). 7

8 By the assumption that W = i I j A i X i X j = 1, we have X i X j f (W) = f (W). i I j A i To deal with X i X j (W (i) W (ij) )f ( W (ij) ) = k A ij X i X j (X k I(k / A i ) X k )f ( W (ij) ), let Ŵijk = l/ A ijk X l + l A ijk Xl in (LD3). By construction, Ŵ(ijk) is independent of {X i,x j,x k, X k } and has the same distribution as W. Moreover, We have, W ij Ŵijk = l A ijk \A ij X l + l A ij Xl l A ijk X. (3.8) X i X j (X k I(k / A i ) X k )f ( W (ij) ) = X i X j (X k I(k / A i ) X { k ) f (Ŵijk )+ [ f ( W (ij) ) f (Ŵijk ) ]}. (3.9) By independence in (LD3) and (LD), we have, X i X j (X k I(k / A i ) X k )f (Ŵijk ) = X i X j X k I(k / A i ) f (W). (3.10) By (3.5) and (3.8), we have X i X j (X k I(k / A i ) X k ) [ f ( W (ij) ) f (Ŵijk ) ] C Xi X j (X k I(k / A i ) X k ) X l + Xl X l A ijk \A ij l A ij l A ijk C X i X j X k I(k / A i )X l +C X i X j X k I(k / A i ) X l l A ijk \A ij l A ij +C X i X j X k I(k / A i ) X l +C X i X j Xk X l l A ijk l A ijk \A ij +C X i X j Xk Xl +C X i X j Xk Xl l A ij l A ijk 8

9 which can be simplified by the independence assumptions in(ld) and(ld3) to C X i X j X k X l +C X i X j X k Xl l A ijk l A ij +C X i X j X k X l +C X i X j X k X l (3.11) l A ijk l A ijk \A ij +C X i X j X k X l +C X i X j X k X l. l A ij l A ijk From (3.9), (3.10) and (3.11), we have where X i X j (X k I(k / A i ) X k )f ( W (ij) ) i I j A i k A ij = X i X j X k f (W)+R 3, i I k A ij \A i j A i R 3 C(γ 1 +γ +γ 3 +γ 4 +γ 5 ). In summary, for the first term on the right-hand side of (3.7), we have X i X j f (W (i) ) i I j A i = f (W)+ (3.1) X i X j X k f (W)+R, i I k A ij \A i where j A i R C(γ 1 +γ +γ 3 +γ 4 +γ 5 ). For the second term on the right-hand side of (3.7), using the similar argument as for the first term, we have 1 X i X j X k f (W (i) ) i I j,k A i = 1 X i X j X k f (W)+R, i I j,k A i (3.13) where R = 1 i I j,k A i X i X j X k [ f (W (i) ) f (Ŵijk ) ] 9

10 and R C(γ 1 +γ 3 ). From (3.6), (3.7), (3.1) and (3.13), we have h(w) Nh = f (W) Wf(W) = X i X j X k f (W) 1 X i X j X k f (W)+R i I k A ij \A i i I j,k A i j A i = β f (W)+R, where R C(γ 1 +γ +γ 3 +γ 4 +γ 5 ). From (3.5) and the equivalent definition of the Wasserstein-1 distance W 1 (µ,ν) = sup gdµ gdν, g Lip 1 (R) wherelip 1 (R)denotestheclassofLipschitzfunctionswithLipschitzconstant 1, we have f (W) Nf CW1 (L(W),N(0,1)) CW (L(W),N(0,1)). This proves (3.4) W bound for approximating L(W) by the distribution of an independent sum. Note that in proving Theorem.1, we can assume that β is sufficiently small, say, β 1. Let n = β so that nβ 1. Let {ξ i : i = 1,...,n} be independent and identically distributed such that ξ i = 0, ξ i = 1, ξ 3 i = nβ, ξ 4 i C. Let V n = 1 n n i=1 ξ i. Note that κ 3 (V n ) = β, where κ r denotes the rth cumulant, and n ξi 4 i=1 C n n Cβ. The expansion in Theorem 1 of Barbour (1986) implies h(v n ) Nh+ β Nf Cβ. (3.14) From Lemma 3.1 and the expansions (3.4) and (3.14), we have W (L(W),L(V n )) { [ C sup h(w) h(vn ) ]}1 h Λ { [ =C sup h(w) h(vn ) ]}1 h Λ : h(0)=h (0)=0 { C β + [ β W (L(W),N(0,1)) ]1 +(γ 1 +γ +γ 3 +γ 4 +γ 5 ) 1 10 }, (3.15)

11 where the equality is obtained by considering h(x) = h(x) h(0) x h (0). Note that Rio (009) used a Poisson-like approximation for L(W). The approximation in(3.15) seems more likely to be generalized to obtain Wassersteinp bounds under local dependence for general p > Triangle inequality and the final bound. By Lemma 3.3, { n W (L(V n ),N(0,1)) C i=1 ξi 4 }1 n Using the triangle inequality, (3.15) and (3.16), we obtain W (L(W),N(0,1)) C β. (3.16) W (L(W),L(V n ))+W (L(V n ),N(0,1)) { C β + [ } β W (L(W),N(0,1)) ]1 +(γ 1 +γ +γ 3 +γ 4 +γ 5 ) 1. Finally, we use the inequality ab 1a + ǫ b with a = β and b = ǫ W (L(W),N(0,1)), choose a sufficiently small ǫ and solve the recursive inequality for W (L(W),N(0,1)) to obtain the bound (.1). 3.3 Proof of Theorem. In this subsection, the constants C are allowed to depend on the given fixed graph G. Let the potential edges of K(n,p) be denoted by (e 1,...,e ( n ). Let ) v = v(g),e = e(g). In applying Theorem.1, let W = i I X i, where the index set is { ( ) n } I = i = (i 1,...,i e ) : 1 i 1 < < i e,g i := (e i1,...,e ie )is a copy of G, X i = σ 1( Y i p e), σ := Var(S), Y i = Π e l=1e il, and E il is the indicator of the event that the edge e il is connected in K(n,p). It is known that (cf. (3.7) of Barbour, Karoński and Ruciński (1989)) For each i I, let For each i I and j A i, let σ C(1 p)n v p e ψ 1. A i = {j I : e(g j G i ) 1}. A ij = {k I : e(g k (G i G j )) 1}, 11

12 and construct {(X k ) k/ Aij,( X k ) k Aij } by deleting all the edges in G i G j and connect each of them with probability p, independent of all else. For each i I, j A i and k A ij, let A ijk = {l I : e(g l (G i G j G k )) 1}, and construct {(X l ) l/ Aijk,( X l ) l Aijk } by deleting all the edges in G i G j G k and connect each of them with probability p, independent of all else. Then these constructions satisfy (LD1) (LD3) of Section.1. Note that the Y s are all increasing functions of the E s. By the arguments leading to (3.8) of Barbour, Karoński and Ruciński (1989), we have γ := γ 1 +γ +γ 3 +γ 4 +γ 5 { C } { C } (Y σ 4 i Y j Y k Y l ) (1 Y σ 4 i ). i I l A ijk i I l A ijk j A i k A ij j A i k A ij For 1 < p < 1, the latter term directly yields the estimate γ Cσ 4 n v n 3(v ) (1 p) Cn 4v 6 (1 p)[n v (1 p)] Cn (1 p) 1. Let = denote graph homomorphism. For 0 < p 1, the former term gives γ Cσ 4 H G e(h) 1 Cσ 4 H G e(h) 1 i,j I G i G j =H { L (G i G j G k ) e(l) 1 i,j I G i G j =H { Cσ 4 ψ 1 n v p e H G e(h) 1 Cσ (ψ 1 n v p e ), K (G i G j ) e(k) 1 k I G k (G i G j )=K l I G l (G i G j G k )=L K (G i G j ) e(k) 1 L (G i G j G k ) L Gm for some m,e(l) 1 i,j I G i G j =H k I G k (G i G j )=K p 4e e(h) e(k) e(l) } n v v(l) p 4e e(h) e(k) e(l) } K (G i G j ) e(k) 1 k I G k (G i G j )=K p 3e e(h) e(k) where in the last step, we used (3.10) of Barbour, Karoński and Ruciński (1989). This gives γ Cψ 1. 1

13 In summary, we have proved that γ 1/ is bounded by the right-hand side of (.). By a similar and simpler argument which is essentially the same as (3.10) of Barbour, Karoński and Ruciński (1989), we also have that β is bounded by the right-hand side of (.). Theorem. is now proved by invoking Theorem.1. ACKNOWLEDGEMENTS The author was partially supported by Hong Kong RGC ECS , a CUHK direct grant and a CUHK start-up grant. REFERENCES Barbour, A. D. (1986). Asymptotic expansions based on smooth functions in the central limit theorem. Probab. Theory Relat. Fields 7, no., Barbour, A. D., Karoński, M. and Ruciński, A. (1989). A central limit theorem for decomposable random variables with applications to random graphs. J. Combin. Theory Ser. B 47, no., Bobkov, S. G. (018). Berry-Esseen bounds and Edgeworth expansions in the central limit theorem for transport distances. Probab. Theory Related Fields 170, no. 1-, 9 6. Bonis, T. (018). Rate in the central limit theorem and diffusion approximation via Stein s method. Preprint. Available at Chen, L.H.Y., Goldstein, L. and Shao, Q.M. (011). Normal approximation by Stein s method. Probability and its Applications (New York). Springer, Heidelberg, 011. xii+405 pp. Chen, L.H.Y. and Shao, Q.M. (004). Normal approximation under local dependence. Ann. Probab. 3, no. 3A, Courtade, T.A., Fathi, M. and Pananjady, A. (018). Existence of Stein kernels under a spectral gap, and discrepancy bound. Preprint. Available at Ledoux, M., Nourdin, I. and Peccati, G. (015). Stein s method, logarithmic Sobolev and transport inequalities. Geom. Funct. Anal. 5, no. 1, Rio, E. (009). Upper bounds for minimal distances in the central limit theorem. Ann. Inst. Henri Poincar Probab. Stat. 45, no. 3,

14 Rinott, Y. and Rotar V. (003). On Edgeworth expansions for dependencyneighborhoods chain structures and Stein s method. Probab. Theory Related Fields 16, no. 4, Röllin, A. (017). Kolmogorov bounds for the normal approximation of the number of triangles in the Erdös-Rényi random graph. Preprint. Available at Stein, C. (197). A bound for the error in the normal approximation to the distribution of a sum of dependent random variables. Proc. Sixth Berkeley Symp. Math. Stat. Prob., Univ. California Press. Berkeley, Calif., Zhai, A. (018). A high-dimensional CLT in W distance with near optimal convergence rate. Probab. Theory Related Fields 170, no. 3-4,