in PROBABILITY
APPLICATIONS OF SIZE BIASED COUPLINGS FOR
CONCENTRA-TION OF MEASURES
SUBHANKAR GHOSH
Dept. of Mathematics, University of Southern California, 3620 South Vermont Avenue, Los Angeles,CA email: subhankar.ghosh@gmail.com
LARRY GOLDSTEIN
Dept. of Mathematics, University of Southern California, 3620 South Vermont Avenue, Los Angeles,CA email: larry@usc.edu
SubmittedSeptember 22, 2010, accepted in final formDecember 20,2010 AMS 2000 Subject classification: Primary 60E15; Secondary 60C05,60D05.
Keywords: Large deviations, size biased couplings, Stein’s method.
Abstract
Let Y be a nonnegative random variable with mean µ and finite positive variance σ2, and let Ys, defined on the same space asY, have theY size biased distribution, that is, the distribution characterized by
E[Y f(Y)] =µE f(Ys) for all functions f for which these expectations exist.
Under a variety of conditions on the coupling ofY andYs, including combinations of boundedness and monotonicity, concentration of measure inequalities such as
P
Y−µ σ ≥t
≤exp
− t
2
2(A+B t)
for allt≥0
are shown to hold for some explicit AandB in[8]. Such concentration of measure results are applied to a number of new examples: the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number ofm-runs in a sequence of coin tosses, the number of local maxima of a random function on a lattice, the number of urns containing exactly one ball in an urn allocation model, and the volume covered by the union ofnballs placed uniformly over a volumensubset ofRd.
1
Introduction
Theorem 1.1, from [8], demonstrates that the existence of a bounded size bias coupling to a nonnegative variableY implies bounds for the degree of concentration of the distribution ofY. In this work we explore a spectrum of new consequences of Theorem 1.1.
The couplings required here which yield concentration of measure results forY are to a random variable having the size biased distribution ofY, denotedYs. Size biasing of a random variable is
essentially sampling it proportional to its size, and is a well known phenomenon in the literature of both probability and statistics; see, for example, the waiting time paradox in Feller[7], Section I.4, and the method of constructing unbiased ratio estimators in[16]. Size biased couplings are used in Stein’s method for normal approximation (see, for instance,[11],[9]and[5]), and is a method which in some sense parallels the exchangeable pair technique. In fact, these two techniques are somewhat complementary, with size biasing useful for the approximation of distributions of nonnegative random variables such as counts, and the exchangeable pair for mean zero variates. Recently, the objects of Stein’s method have also proved successful in deriving concentration of measure inequalities, that is, deviation inequalities of the formP(|Y−E(Y)| ≥tpVar(Y)), where typically one seeks bounds that decay exponentially in t; for a guide to the literature on the concentration of measures, see [14] for a detailed overview. As far as the use of techniques related to Stein’s method is concerned, Raiˇc[22]obtained large deviation bounds for certain graph related statistics using the Stein equation (see[24]) along with the Cramér transform. Chatterjee [3] derived Gaussian and Poisson type tail bounds for Hoeffding’s combinatorial CLT and the net magnetization in the Curie-Weiss model in statistical physics in [3] using the exchangeable pair of Stein’s method (see[24]). Considering the complementary method, Ghosh and Goldstein [8] proved Theorem 1.1 which relies on the existence of bounded size bias couplings. Here we demonstrate the broad range of applicability of Theorem 1.1 by presenting a variety of examples. First recall that for a given nonnegative random variableY with finite nonzero meanµ, we say thatYshas theY-size biased distribution if
E[Y f(Y)] =µE[f(Ys)] for all functions f for which these expectations exist. (1)
Theorem 1.1. Let Y be a nonnegative random variable with mean and variance µandσ2 respec-tively, both finite and positive. Suppose there exists a coupling of Y to a variable Yshaving the Y -size bias distribution which satisfies|Ys−Y| ≤C for some C>0with probability one. Let A=Cµ/σ2.
If Ys≥Y with probability one, then P
Y−µ σ ≤ −t
≤exp
−t
2
2A
for all t>0. (2)
If the moment generating function m(θ) =E(eθY)is finite atθ=2/C, then
P
Y−µ σ ≥t
≤exp
− t
2
2(A+B t)
for all t>0, where B=C/2σ. (3)
In typical examples the variable Y is indexed byn, and the ones we consider have the property that the ratioµ/σ2remains bounded asn→ ∞, andCdoes not depend onn. In such cases the bound in (2) decreases at rate exp(−c t2)for somec>0, and ifσ→ ∞asn→ ∞, the bound in (3) is of similar order, asymptotically.
In[8], the number of lightbulbs switched on at the terminal time in the lightbulb process was shown to obey the hypothesis of Theorem 1.1 and concentration of measure inequalities were obtained. In Section 3 we apply Theorem 1.1 to the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number ofm-runs in a sequence of coin tosses, the number of local maxima of a random function on the lattice, the number of urns containing exactly one ball in the uniform urn allocation model, and the volume covered by the union ofnballs placed uniformly over a volumensubset ofRd. In Section 2, we review the
2
Construction of size bias couplings
In this section we will review the discussion in[11]which gives a procedure for a construction of size bias couplings whenY is a sum; the method has its roots in the work of Baldi et al.[1]. The construction depends on being able to size bias a collection of nonnegative random variables in a given coordinate, as described Definition 2.1. LettingFbe the distribution ofY, first note that the characterization (1) of the size bias distribution Fs is equivalent to the specification ofFs by its
Radon Nikodym derivative
d Fs(x) = x
µd F(x). (4)
Definition 2.1. LetA be an arbitrary index set and let{Xα:α∈ A }be a collection of nonnegative
random variables with finite, nonzero expectations EXα = µα and joint distribution d F(x). For
β ∈ A, we say that Xβ ={Xαβ :α∈ A }has theXsize bias distribution in coordinateβ ifXβ has joint distribution
d Fβ(x) =xβd F(x)/µβ.
Just as (4) is related to (1), the random vectorXβ has theXsize bias distribution in coordinateβ
if and only if
E[Xβf(X)] =µβE[f(Xβ)] for all functions f for which these expectations exist.
Letting f(X) = g(Xβ)for some function g one recovers (1), showing that theβthcoordinate of
Xβ, that is,Xββ, has theXβ size bias distribution.
The factorization
P(X∈dx) =P(X∈dx|Xβ =x)P(Xβ∈d x)
of the joint distribution of Xsuggests a way to construct X. First generate Xβ, a variable with
distribution P(Xβ ∈ d x). If Xβ = x, then generate the remaining variates {Xαβ,α 6= β} with
distributionP(X∈dx|Xβ=x). Now, by the factorization ofd F(x), we have
d Fβ(x) = xβd F(x)/µβ (5)
= P(X∈dx|Xβ=x)xβP(Xβ∈d x)/µβ=P(X∈dx|Xβ=x)P(X β β ∈d x).
Hence, to generateXβ with distributiond Fβ, first generate a variableXββ with theXβ size bias
distribution, then, when Xββ = x, generate the remaining variables according to their original conditional distribution given that theβthcoordinate takes on the value x.
Definition 2.1 and the following special case of a proposition from Section 2 of[11]will be applied in the subsequent constructions; the reader is referred there for the simple proof.
Proposition 2.1. Let A be an arbitrary index set, and let X = {Xα,α ∈ A }be a collection of
nonnegative random variables with finite means. Let Y =P
β∈AXβ and assumeµA=EY is finite and
positive. LetXβ have theX-size biased distribution in coordinate β as in Definition 2.1. Let I be a random index taking values in A with distribution
P(I=β) =µβ/µA, β∈A.
Then if XI has the mixture distributionPβ∈AP(I =β)L(Xβ), the variable Ys=P α∈AX
I
α has the
In our examples we use Proposition 2.1 and the random index I, and (5), to obtain Ys by first generatingXII with the size bias distribution ofXI, then, if I =αandXαα= x, generating{X
α β :
β∈A\ {α}}according to the (original) conditional distributionP(Xβ,β6=α|Xα=x).
3
Applications
We now consider the application of Theorem 1.1 to derive concentration of measure results for the number of relatively ordered subsequences of a random permutation, the number ofm-runs in a sequence of coin tosses, the number of local extrema on a graph, the number of nonisolated balls in an urn allocation model, and the covered volume in a binomial coverage process. Without further mention we will use that when (2) and (3) hold for someAandBthen they also hold when these values are replaced by larger ones, also denoted byAandB, and that moment generating functions of bounded random variables are everywhere finite.
3.1
Relatively ordered sub-sequences of a random permutation
Forn≥m≥3, letπandτbe permutations ofV ={1, . . . ,n}and{1, . . . ,m}, respectively, and let
Vα={α,α+1, . . . ,α+m−1} forα∈ V, (6)
where addition of elements ofV is modulon. We say the patternτappears at locationα∈ V if the values{π(v)}v∈Vα and{τ(v)}v∈V1 are in the same relative order. Equivalently, the patternτ appears atαif and only ifπ(τ−1(v) +α−1),v∈ V
1is an increasing sequence. Whenτ=ιm, the
identity permutation of lengthm, we say thatπhas a rising sequence of length mat positionα. Rising sequences are studied in[2]in connection with card tricks and card shuffling.
Lettingπ be chosen uniformly from all permutations of{1, . . . ,n}, and Xαthe indicator that τ
appears atα,
Xα(π(v),v∈ Vα) =1(π(τ−1(1) +α−1)<· · ·< π(τ−1(m) +α−1)),
the sumY =P
α∈V Xαcounts the number ofm-element-long segments ofπ that have the same
relative order asτ.
Forα∈ V we may generateXα={Xβα,β∈ V }with theX={Xβ,β ∈ V }distribution size biased
in directionα, following[9]. Letσαbe the permutation of{1, . . . ,m}for which
π(σα(1) +α−1)<· · ·< π(σα(m) +α−1), and set
πα(v) =
π(σα(τ(v−α+1)) +α−1), v∈ Vα
π(v) v6∈ Vα.
In other wordsπαis the permutationπwith the valuesπ(v),v∈ V
αreordered so thatπα(γ)for
γ∈ Vαare in the same relative order asτ. Now letXβα=Xβ(πα(v),v∈ Vβ), the indicator thatτ
appears at positionβin the reordered permutationπα. Asπαandπagree except perhaps for the mvalues inVα, we have
Hence, as
joint event in this case demands two different relative orders on the segment ofπof lengthm−k of which both Xα andXα+k are a function. If Ik = 1 then a given, common, relative order is
demanded for this same length ofπ, and relative orders also for the two segments of lengthkon which exactly one ofXαandXβ depend, and so, in total a relative order onm−k+2k=m+k values ofπ, and therefore
EXαXα+k=Ik/(m+k)! and Cov(Xα,Xα+k) =Ik/(m+k)!−1/(m!)2.
As the relative orders of non-overlapping segments ofπare independent, now takingn≥2m, the varianceσ2ofY is given by all 1 ≤ m≤ m−1, and as mentioned, this case corresponds to counting the number of rising sequences. In contrast, the variance lower bound given whenIk=0 for all 1≤k≤m−1 is
, attained at the permutation τ(j) =
3.2
Local Dependence
The following lemma shows how to construct a collection of variablesXαhaving theXdistribution biased in directionαwhenXαis some function of a subset of a collection of independent random
variables.
Lemma 3.1. Let{Cg,g∈ V }be a collection of independent random variables, and for eachα∈ V let Vα ⊂ V and Xα = Xα(Cg,g ∈ Vα) be a nonnegative random variable with a nonzero, finite
expectation. Then if{Cgα,g∈ Vα}has distribution
d Fα(cg,g∈ Vα) =
Xα(cg,g∈ Vα)
EXα(Cg,g∈ Vα)
d F(cg,g∈ Vα)
and is independent of{Cg,g∈ V }, letting
Xβα=Xβ(Cαg,g∈ Vβ∩ Vα,Ch,h∈ Vβ∩ Vαc),
the collectionXα={Xβα,β∈ V }has theXdistribution biased in directionα.
Furthermore, with I chosen proportional to EXα, independent of the remaining variables, the sum
Ys=X
β∈V XβI
has the Y size biased distribution, and when there exists M such that Xα≤M for allα,
|Ys−Y| ≤bM where b=max
α |{β:Vβ∩ Vα6=;}|. (10)
Proof. By independence, the random variables
{Cαg,g∈ Vα} ∪ {Cg,g6∈ Vα} have distribution d Fα(cg,g∈ Vα)d F(cg,g6∈ Vα).
Thus, withXαas given, we find
EXαf(X) =
Z
xαf(x)d F(cg,g∈ V)
= EXα Z
f(x)xαd F(cg,g∈ Vα) EXα(Cg,g∈ Vα)
d F(cg,g6∈ Vα)
= EXα Z
f(x)d Fα(cg,g∈ Vα)d F(cg,g6∈ Vα)
= EXαE f(Xα).
That is,Xαhas theXdistribution biased in directionα, as in Definition 2.1. The claim onYsfollows from Proposition 2.1, and finally, sinceX
β =XβαwheneverVβ∩ Vα=;,
|Ys−Y| ≤ X
β:Vβ∩VI6=;
|XβI −Xβ| ≤bM.
3.2.1 Slidingmwindow statistics with the periodic conventionξn+k = ξk. In[23], the authors develop smooth function bounds for normal approximation for Y. Note that the construction given in Lemma 3.1 for this case is monotone, as for anyi, size biasing the Bernoulli variablesξjfor j∈ {i, . . . ,i+m−1}by setting non-overlapping segments of the sequence are independent,
σ2 = Hence (2) and (3) of Theorem 1.1 hold with
A= 2m−1
3.2.2 Local extrema on a lattice
Let G = {V,E } be a given graph, and for every v ∈ V letVv ⊂ V be a collection of vertices depending on v; we think ofVv as some ‘neighborhood’ of the vertex v. Let{Cg,g ∈ V } be a collection of independent and identically distributed random variables, and letXvbe the indicator that vertexvcorresponds to a local maximum value with respect to the neighborhoodVv, that is
The sumY =P
v∈V Xvcounts the number of local maxima. Size biased couplings to Y, for the
purpose of normal approximation, were studied in[1]and[9].
In general one may define the neighbor distancedbetween two verticesv,w∈ V by
d(v,w) =min{n: there∃v0, . . . ,vninV so thatv0=v,vn=wand(vk,vk+1)∈ E fork=0, . . . ,n}, and forr∈N, therneighborhood ofv∈ V consisting of vertices at distance at mostrfromv,
Vv(r) ={w∈ V :d(w,v)≤r}.
We consider the case where there is some r such that the graphsGv = {Vv,Ev},v∈ V, where
Vv=Vv(r)andEv={{w,u} ⊂ Vv(r),{w,v} ∈ E }, are isomorphic, and the isomorphism fromGv1
toGv2 mapsv1tov2. Then ifd(v1,v2)>2r, and(w1,w2)∈ Vv1× Vv2, rearranging 2r<d(v1,v2)≤d(v1,w1) +d(w1,w2) +d(w2,v2)
and usingd(vi,wi)≤r,i=1, 2, yieldsd(w1,w2)>0. Hence, d(v1,v2)>2r implies Vv1
\
Vv2=;, so by (10) we may take b=maxv |Vv(2r)|. (11)
For example, forp∈ {1, 2, . . .}andn≥5 consider the latticeV ={1, . . . ,n}p moduloninZpand
E={{v,w}:Ppi=1|vi−wi|=1}; in this casedis theL1norm. To consider the case where we call
vertexva local maximum if Cvexceeds the values Cw over the immediate neighborsw of v, we
taker=1 and obtain
Vv=Vv(1) and that |Vv(1)|=1+2p,
the 1 accounting forvitself, and 2pfor the number of neighbors at distance 1 fromv, which differ fromvby either+1 or−1 in exactly one coordinate.
Lemma 3.1, (11), and|Xv| ≤1 yield
|Ys−Y| ≤max
v |Vv(2)|=1+2p+
2p+4
p 2
=2p2+2p+1, (12)
where the 1 countsvitself, the 2pagain are the neighbors at distance 1, and the term in the paren-thesis accounting for the neighbors at distance 2, 2p of them differing in exactly one coordinate by+2 or−2, and 4 p
2
of them differing by either+1 or−1 in exactly two coordinates. Note that we have used the assumptionn≥5 here, and continue to do so below.
Now letting Cv have a continuous distribution, without loss of generality we can assume Cv ∼
U[0, 1]. As any vertex has chance 1/|Vv|=1/(2p+1)of having the largest value in its
neighbor-hood,µ=EY satisfies
µ= n
2p+1. (13)
To begin the calculation of the variance, note that whenvandware neighbors they cannot both be maxima, soXvXw=0 and therefore, ford(v,w) =1,
Cov(Xv,Xw) =−(EXv)2=−
1 (2p+1)2.
If the distance betweenvandwis 3 or more,XvandXware functions of disjoint sets of
Whend(w,v) =2 there are two cases, as vandwmay have either 1 or 2 neighbors in common, and
EXvXw=P(U>Uj,V >Vj,j=1, . . . ,m−k and U>Uj,V >Uj,j=m−k+1, . . . ,m),
wheremis the number of vertices over whichvandware extreme, som=2p, andk=1 andk=2 for the number of neighbors in common. Fork=1, 2, . . ., letting Mk=max{Um−k+1, . . . ,Um}, as
the variablesXvandXw are conditionally independent givenUm−k+1, . . . ,Um
E(XvXw|Um−k+1, . . . ,Um) = P(U>Uj,j=1, . . . ,m|Um−k+1, . . . ,Um)2
= 1
(m−k+1)2(1−M
m−k+1
k )
2, (14)
as
P(U>Uj,j=1, . . . ,m|Um−k+1, . . . ,Um) = Z1
Mk
Zu
0
· · ·
Zu
0
du1· · ·dum−kdu
=
Z1
Mk
um−kdu
= 1
m−k+1(1−M
m−k+1
k ).
SinceP(Mk≤x) =xkon[0, 1], we have
E Mm−k+1
k = k
Z1
0
xm−k+1xk−1d x= k
m+1 and E(Mm−k+1
k )
2 = k
Z1
0
x2(m−k+1)xk−1d x= k 2m−k+2.
Hence, averaging (14) overUm−k+1, . . . ,Umyields
EXvXw=
2
(m+1)(2(m+1)−k).
Forn≥3, whenm=2p, fork=1 and 2 we obtain
Cov(Xv,Xw) =
1
(2p+1)2(2(2p+1)−1) and Cov(Xv,Xw) =
2
(2p+1)2(2(2p+1)−2), respectively.
Forn≥5, of the 2p+4 p2
in common withv, while the remaining 4 p2
of them share 2 neighbors. Hence, σ2 = X given by (13), (15) and (12), respectively, that is,
A= (2p+1)(4p+1)(2p
In the classical urn allocation modelnballs are thrown independently into one ofmurns, where, fori=1, . . . ,m, the probability a ball lands in theithurn isp
i, with Pm
i=1pi=1. A much studied
quantity is the number of nonempty urns, for which Kolmogorov distance bounds to the normal were obtained in[6]and[21]. In[6], bounds were obtained for the uniform case wherepi=1/m for alli =1, . . . ,m, while the bounds in[21]hold for the nonuniform case as well. In[19]the author considers the normal approximation for the number of isolated balls, that is, the number of urns containing exactly one ball, and obtains Kolmogorov distance bounds to the normal. Using the coupling provided in[19], we derive right tail inequalities for the number of non-isolated balls, or, equivalently, left tail inequalities for the number of isolated balls.
Fori=1, . . . ,nletXi denote the location of balli, that is, the number of the urn into which balli
lands. The numberY of non-isolated balls is given by
Y =
We first consider the uniform case. A construction in [19] produces a coupling of Y to Ys,
having theY size biased distribution, which satisfies |Ys−Y| ≤2. Given a realization of X=
{X1,X2, . . . ,Xn}, the coupling proceeds by first selecting a ballI, uniformly from{1, 2, . . . ,n}, and
withN∼Bin(1/m,n−1). Now letJ be uniformly chosen from{1, 2, . . . ,n} \ {I}, independent of all other variables. Lastly, ifB=1, move ballJinto the same urn asI. It is clear that|Y′−Y| ≤2, as at most the occupancy of two urns can affected by the movement of a single ball. We also note that ifMI=0, which happens when ballI is isolated,π0=1, so thatI becomes no longer isolated after relocating ballJ. We refer the reader to[19]for a full proof that this procedure produces a coupling ofY to a variable with theY size biased distribution.
For the uniform case, the following explicit formulas forµandσ2can be found in Theorem II.1.1 of[13], Hence, in this limiting caseAandBsatisfy
A≍ 2(1−e
−α)
e−α−e−2α(α2−α+1) and B≍ 1
png(α).
In the nonuniform case similar results hold with some additional conditions. Letting
||p||= sup
yieldB, we find that (3) of Theorem 1.1 holds with
A=24, 495γ2e2.1γ and B=1.5
3.4
An application to coverage processes
torus of volumen, that is, the cubeCn= [0,n1/d)d ⊂Rd with periodic boundary conditions, let
V denote the total volume of the union of thenballs of fixed radiusρcentered at thesenpoints, andSthe number of balls isolated at distanceρ, that is, those points for which none of the other n−1 points lie within distanceρ. The random variablesV andSare of fundamental interest in stochastic geometry, see[12]and[18]. Ifn→ ∞andρ remains fixed, bothV andSsatisfy a central limit theorem[12,17,20]. TheL1distance ofV, properly standardized, to the normal is studied in[4]using Stein’s method. The quality of the normal approximation to the distributions of both V andS, in the Kolmogorov metric, is studied in[10]using Stein’s method via size bias couplings.
In more detail, for x ∈Cn and r >0 let Br(x) denote the ball of radius r centered at x, and Bi,r=B(Ui,r). The covered volumeV and number of isolated ballsSare given, respectively, by
V=Volume(
We will derive concentration of measure inequalities for V andS with the help of the bounded size biased couplings in[10].
Assumed ≥1 andn≥4. Denote the mean and variance of V byµV andσ2
It is shown in[10], by using a coupling similar to the one briefly described for the urn allocation problem in Section 3.3, that one can constructVswith theV size bias distribution which satisfies
whereµV andσ2
V are given in (18) and (19), respectively. Similarly, withY =n−Sthe number
of non-isolated balls, it is shown that YswithY size bias distribution can be constructed so that
|Ys−Y| ≤κd+1, whereκd denotes the maximum number of open unit balls inddimensions that can be packed so they all intersect an open unit ball in the origin, but are disjoint from each other. Hence (2) of Theorem 1.1 holds forY with
AY =
(κd+1)(n−µS) σ2
S
and BY =
κd+1 2σS . To see how theAV,AY andBV,BY behave asn→ ∞, let
Jr,d(ρ) =dπd Z r
0
exp(−ρdωd(t))td−1d t,
and define
gV(ρ) = ρdJ2,d(ρ)−(2dφ+φ2)e−2φ and
gS(ρ) = e−φ−(1+ (2d−2)φ+φ2)e−2φ+ρd(J2,d(ρ)−J1,d(ρ)).
Then, again from[10], lim
n→∞n −1µ
V = lim n→∞(1−n
−1µ
S) = 1−e−φ,
lim
n→∞n −1σ2
V = gV(ρ)>0, and
lim
n→∞n −1σ2
S = gS(ρ)>0.
Hence,BV andBY tend to zero at raten−1/2, and
lim
n→∞AV =
φ(1−e−φ)
gV(ρ) , and nlim→∞AY =
(κd+1)(1−e−φ) gS(ρ) .
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