Berry-Esseen bounds for the number of maxima in planar regions

We derive the optimal convergence rate $O(n^{-1/4})$ in the central limit theorem for the number of maxima in random samples chosen uniformly at random from the right equilateral triangle with two sides parallel to the axes, the hypotenuse with the slope $-1$ and consituting the top part of the boundary of the triangle. A local limit theorem with rate is also derived. The result is then applied to the number of maxima in general planar regions (upper-bounded by some smooth decreasing curves) for which a near-optimal convergence rate to the normal distribution is established.


Introduction
Given a sample of points in the plane (or in higher dimensions), the maxima of the sample are those points whose first quadrants are free from other points.More precisely, we say that Ô ½ ´Ü½ Ý ½ µ dominates Ô ¾ ´Ü¾ Ý ¾ µ if Ü ½ Ü ¾ and Ý ½ Ý ¾ ; the maxima of a point set Ô ½ Ô Ò are those Ô 's that are dominated by no points.The main purpose of this paper is to derive convergence rates in the central limit theorems (or Berry-Esseen bounds) for the number of maxima in samples chosen uniformly at random from some planar regions.As far as the Berry-Esseen bounds are concerned, very few results are known in the literature for the number of maxima (and even in the whole geometric probability literature): precise approximation theorems are known only in the ÐÓ -class (regions for which the number of maxima has logarithmic mean and variance); see Bai et al. (2001) for more precise results.We propose new tools for dealing with the Ô Ò-class (see Bai et al. (2001)) in this paper.Such a dominance relation among points is a natural ordering relation for multidimensional data and is very useful, both conceptually and practically, in diverse disciplines; see Bai et al. (2001) and the references cited there for more information.For example, it was used in analyzing the 329 cities in United States Figure 1: The maxima-finding problem can be viewed as an optimization problem: minimizing the area in the shaded region (between the "staircase" formed by the selected points and the upper-right boundary).
selected in the book Places Rated Almanac (see Becker et al., 1987).Naturally, city A is "better" than city B if factors pertaining to the quality of life of city A are all better than those of city B. The same idea is useful for educational data: a student is "better" than another if all scores of the former are better than those of the latter; also a student should not be classified as "bad" if (s)he is not dominated by any others.While traditional ranking models relying on average or weighted average may prove unfair for someone with outstanding performance in only one subject and with poor performance in all others, the dominance relation provides more auxiliary information for giving a less "prejudiced" ranking of students.
To further motivate our study on the number of maxima, we mention (in addition to applications in computational geometry) yet another application of dominance to knapsack problems, which consists in maximizing the weighted sum È ½ Ò Ô Ü by choosing an appropriate vector ´Ü½ Ü Ò µ with Ü ¾ ¼ ½ , subject to the restriction È ½ Ò Û Ü Ï , where Ô , Û and Ï are nonnegative numbers.Roughly, item dominates item if Û Û and Ô Ô , so that a good heuristic is that if Û Û Ô Ô then item can be discarded from further consideration; see Martello and Toth (1990).A probabilistic study on the number of undominated variables can be found in Johnston and Khan (1995), Dyer and Walker (1997).Similar dominance relations are also widely used in other combinatorial search problems.
Interestingly, the problem of finding the maxima of a sample of points in a bounded planar region can also be stated as an optimization problem: given a set of points in a bounded region, we seek to minimize the area between the "staircase" formed by the selected points and the upper-right boundary; see Figure 1 for an illustration.Obviously, the minimum value is achieved by the set of maxima.
Let be a given region in Ê ¾ .Denote by Å Ò ´ µ the number of maxima in a random sample of Ò points chosen uniformly and independently in .
It is known that the expected number of maxima in bounded planar regions exhibits generally three different modes of behaviors: Ô Ò, ÐÓ Ò and bounded (see Golin, 1993;Bai et al., 2001).Briefly, if the region contains an upper-right corner (a point on the boundary that dominates all other points inside and on the boundary), then ´ÅÒ ´ µµ is roughly either of order ÐÓ Ò or bounded; otherwise, ´ÅÒ ´ µµ is of order Ô Ò.
The ÐÓ Ò-class was studied in details in Bai et al. (2001), where the analysis relies heavily on the case when is a rectangle Ê (or a square).Basically, Å Ò ´ µ can be expressed (in case when has an upperright corner) as ½ • Å ÁÒ ´Êµ, where the distribution of Á Ò depends on the shape of .While rectangle is prototypical for ÐÓ Ò-class, we show in this paper that the right triangle Ì of the form plays an important role for the Berry-Esseen bound of Å Ò ´ µ when the mean and the variance are of order Ô Ò.
where ¾ ´¾ ÐÓ ¾ ½µ Ô and AE´¼ ½µ is a normal random variable with zero mean and unit variance.The mean and the variance of Å Ò satisfy See also Neininger and Rüschendorf (2002) for a different proof for (1) via contraction method.
We improve (1) by deriving an optimal (up to implied constant) Berry-Esseen bound and a local limit theorem for Å Ò .Let ¨´Üµ denote the standard normal distribution function.
Note that the same error terms in (4) and ( 5) hold if we replace Ò or Ò in the left-hand side by their asymptotic values Ô Ò and Ò ½ , respectively.The Berry-Esseen bound and the local limit theorem are derived by a refined method of moments introduced in Hwang (2003), coupling with some inductive arguments and Fourier analysis; the technicalities are quite different and more involved here.Roughly, We start the approach by considering the normalized moment generating functions Ò ´Ýµ ´ ´ÅÒ AE´ Ò ¾ Ò µµÝ µ.We next show that ´Ñµ Ò ´¼µ ´ÅÒ AE´ Ò ¾ Ò µµ Ñ Ñ Ñ Ò Ñ ´Ñ ¼µ for a sufficiently large .This is the hard part of the proof.Such a precise upper bound suffices for deriving the estimate uniformly for Ý Ó´Ò ½ ½¾ µ.We then use another inductive argument to derive a uniform estimate for ´ ´ÅÒ Òµ Ý Ò µ for Ý Ò and conclude (4) by applying the Berry-Esseen smoothing inequality (see Petrov, 1975) and ( 5) by the Fourier inversion formula.
Although the proof is not short, the results (4) and ( 5) are the first and the best, up to the implied constants, of their kind.Also the approach used (based on estimates of normalized moments) can be applied to other recursive random variables; it is therefore of some methodological interests per se.While the usual method of moments proves the convergence in distribution by establishing the stronger convergence of all moments, our method shows that in the case of a normal limit law the convergence of moments can sometimes be further refined and yields stronger quantitative results.
The result (4) will also be applied to planar regions upper-bounded by some smooth curves of the form where ´Ùµ ¼ is a nonincreasing function on ´¼ ½µ with ´¼• µ ½, ´½ µ ¼ and Devroye (1993) showed that if is either convex, or concave, or Lipschitz with order 1, then Our result says that if is smooth enough (roughly twice differentiable with ¼ ¼), then the number of maxima converges (properly normalized) in distribution to the standard normal distribution with a rate of order Ò ½ Ò ÐÓ ¾ Ò, where Ò is some measure of "steepness" of defined in (28); see Theorem 3 for a precise statement.While the order of Ò can vary with , it is bounded or at most logarithmic for most practical cases of interests.The method of proof of Theorem 3 is different from that for Theorem 1; it proceeds by splitting into many smaller regions and then by transforming in a way that Å Ò ´ µ is very close to the number of maxima in some right triangle Ì Ò .Then we can apply (4).The hard part is that we need precise estimate for the difference between the number of maxima in and those in the approximate triangle Ì Ò .
The proofs of Theorems 1 and 2 are given in the next section.We derive a Berry-Esseen bound for Å Ò ´ µ for nondecreasing in Section 3. We then conclude with some open questions.
Results related to ours have very recently been derived by Barbour and Xia (2001), where they study the bounded Wasserstein distance between the number of maxima in certain planar regions and the normal distribution.Since the bounded Wasserstein distance is in general larger than (roughly of the order Î Ö´Å Ò ´ µµ ½ ) the Kolmogorov distance, our results are stronger as far as the Kolmogorov distance is concerned.On the other hand, our settings and approach are completely different.Their approach relies on the Stein method using arguments on point processes similar to those used in Chiu and Quine (1997).The latter paper studies the number of seeds in some stochastic growth model with inhomogeneous Poisson arrivals; in particular, since the point processes are assumed to be spatially homogeneous in Chiu and Quine (1997), Theorem 6.1 there can be translated into a Berry-Esseen bound for Å Ò (maxima in right triangle) with a rate of order Ò ½ ÐÓ Ò; see Barbour and Xia (2001) for some details.See also Baryshnikov (2000) for other limit theorems for maxima.While it is likely that Stein's method can be further improved to give an alternative proof of (4), it is unclear how such an approach can be used for proving our local limit theorem (5).
Notations.Throughout this paper, we use the generic symbols , and (without subscripts) to denote suitably small, absolute and large positive constants, respectively, whose values may vary from one occurrence to another.For convenience of reference, we also index these symbols by subscripts to denote constants with fixed values.The abbreviation "iid" stands for "independent and identically distributed".The convention ¼ ¼ ½ is adopted.

Right triangle
Theorems 1 and 2 are proved by first establishing the following two estimates.Define holds uniformly for Ý Ò ½ ½¾ .Proposition 2. Uniformly for Ý Ò and Ò ¾, The hard part of the proof is the locally more precise bound (6).

Moment generating function.
Our starting point is the recurrence for the moment generating function (see Bai et al., 2001) for Ò ½ with the initial condition È ¼ ´Ýµ ½, where the sum is extended over all nonnegative integer triples ´ µ such that Recurrences.We first prove the estimate (6), starting by defining the normalized moment generating function
Estimates.We derive some estimates that will be used later.
Proof.Applying Lemma 1 and the inequality

¾Ö
we obtain the first inequality in (17).The second inequality in (17) follows from the inequality ´¾Öµ Ö ¾ Ö
where is independent of Ô and Õ.
Estimate for Ò ¿ .We first determine the order of Ò ¿ .
Substituting the estimates ( 17) and ( 18) derived above into (21) yields since, by Stirling's formula, the sum is bounded for all

Proof of Proposition 1.
With the precise estimate (20) available, we now have Ò ½ ½¾ .This proves (6).We also need an estimate for ³ Ò ´Ýµ for larger values of Ý .Note that from (6) we have for some ¾ ¼ and Ý ½ Ò ½ , where ½ ¾ ¼ are constants satisfying Here ½ is chosen so small that the right-hand side is positive for all Ò ¾ (we may take ½ ½ ´½¾ ¾ µ).
Proof of Proposition 2. We now show, again by induction, that the same estimate (24) holds for Ý , provided the constants ½ and ¾ are suitably tuned.Note that since the span of Å Ò is 1 (by induction, È ´ÅÒ µ ¼ for ½ Ò), ´ Ý µ for ¾ Ò Ò ¼ , where Ò ¼ is a sufficiently large number (see ( 27)). [Numerically, ¿ ½ suffices.]This gives another condition for ½ and ¾ : By induction using ( 8) and (24), say.By (17), we obtain (with the notation of Lemma 4) ´ We now estimate ¼ . ´ where is independent of ½ .The partial sum ½ is estimated similarly, giving and, consequently, where is independent of ½ .
The remaining terms are easy.
Thus there exists a constant such that for Ò ¾ for Ò ¾, where is independent of ½ .
Berry-Esseen smoothing inequality.We now apply the Berry-Esseen smoothing inequality (see Petrov, 1975), which states for our problem that where Ì is taken to be Ò ½ .By the two estimates ( 6) and ( 7), we obtain Local limit theorem.By the inversion formula where Ò • Ü Ò , we deduce, by splitting the integral similarly as above, the local limit theorem (5).
Define a measure of "steepness" or "flatness": Note that the ranges Ù ½ Ù alone are not sufficient for our proof.In particular, in (36) and (38) we use the ranges Ù ½ Ù , but Ù ½ Ù •½ are needed in (44).Also the index range ¾ ¿ is chosen for technical convenience. where Although Ò may diverge with Ò even under the stronger assumption that ¾ ½ , it is small compared to Ò ½ in most cases.
This example indicates that the error term in (29) may not in general be optimal in view of (4).
Corollary 2. Assume ¾ ¾ .If ¼¼ ´Ùµ ¼ and there exists a sequence Ò ¼ such that If behaves very flatly or steeply near the origin or the unit, say The introduction of the Poisson process has the advantage of simplifying the analysis.
(iii) The next step is to show (quantitatively) that most maxima appear near the boundary: where is region close to the boundary defined below (see (40)).
(iv) We then introduce a mapping on that basically transforms into Ì and show that the numbers of maxima under the Poisson process model and the iid uniform distribution model are very close; more precisely, where Å ´ µ is the number maximal points of ´ ½ µ ´ Ò µ that also lie in .Here is a mapping on ; see (39).
(v) The final step is to construct mappings ½ and ¾ such that Å ½ ´ µ Å ¾ ´ µ Å ´ µ, and The reason for introducing these mappings is that the dominance relations for some parts of may be changed by the mapping .So we need to "fine-tune" the number of maxima.
Proof of (30).It suffices to prove that the total number of sections of the unit interval satisfies Proof of (31).From (30), there is where AE Ò ´ µ is the number of points in .Applying Theorem 1 (conditioning on fixed number of points in Ì ), we have To prove (32)-( 35), we define first a dissection and then a transformation on .

Dissection of .
We split the region into several smaller regions as follows.Let Let Ì be the triangle formed by the vertices ´Ù ½ ´Ù ½ µµ, ´Ù ´Ù µµ and ´Ù ½ ´Ù µµ and ´É Ì µ ´Ì É µ.Denote by Then is the disjoint union of É Ê and Ï , and is the difference of and the polygon Ì Ê Ï .The area of satisfies Note that ´Ì µ is the right triangle region formed by ´ ½ µ ´ µ and ´ ½ •½ µ, and Basically the main effect of is to transform into Ì .
By construction, the mapping is piecewise linear and preserves measure.Thus ´ ½ µ ´ Ò µ are Ò iid random variables uniformly distributed in ´ µ.

´É Êµ
(40) The proof for (32) consists of two parts: we first show that Å Ò ´Ì ´ µµ is negligible; and then we estimate the difference between the number of maxima in and that in ´Ì Êµ.
To show (41), we need to estimate ´Ï µ Ï .Define (see Figure 4) Here Ð is the difference between the height of ´Ì •½ µ and that of ´Ê µ.
We now estimate the difference between the number of maxima in the region and that in ´Ì Êµ.  .Also the region Ô Ô ¾ ´Ê µ and ´Ô Ì µ ¾ ÐÓ Ò Ò is negligible (by a similar argument).Thus (46) also holds and this proves (32).Note that the proof can be largely simplified if is known to be convex.
Then Å ´ µ Å Ò ´ µ when conditioning on AE ´ µ AE Ò ´ µ.It follows that where the last estimate holds by the usual Poison approximation of binomial distributions: for small Ô; see Prohorov (1953).
then tedious calculations yield Ò Ç´ÐÓ ¾ Òµ.Main steps of the proof of Theorem 3. LetÕ Ò ÐÓ Ò .Choose ¼ Ù ¼ Ù ½ ¡ ¡ ¡ Ù ½ Ù ½ as above.Let Ì be the right triangle region formed by ´¼ ¼µ ´ ¼µ and ´¼ µ.Let be a Poison Process on the plane with density Ò. Denote by Å Ò ´ µ the number of maxima of .From (30), we next deduce that the number of maxima of the Poisson process satisfies