JOINT DISTRIBUTION OF THE NUMBER OF VERTICES AND THE AREA OF CONVEX HULLS GENERATED BY A UNIFORM DISTRIBUTION IN A CONVEX POLYGON Текст научной статьи по специальности «Математика»

DOI: 10.17516/1997-1397-2021-14-2-230-241 УДК 519.21

Joint Distribution of the Number of Vertices and the Area of Convex Hulls Generated by a Uniform Distribution in a Convex Polygon

Isakjan M. Khamdamov*

National University of Uzbekistan named after Mirzo Ulugbek

Tashkent, Uzbekistan

Zoya S. Chayt

Tashkent University of Information Technologies named after M. al-Khwarizmi

Tashkent, Uzbekistan

Received 05.07.2020, received in revised form 13.12.2020, accepted 20.01.2021 Abstract. A convex hull generated by a sample uniformly distributed on the plane is considered in the case when the support of a distribution is a convex polygon. A central limit theorem is proved for the joint distribution of the number of vertices and the area of a convex hull using the Poisson approximation of binomial point processes near the boundary of the support of distribution. Here we apply the results on the joint distribution of the number of vertices and the area of convex hulls generated by the Poisson distribution given in [6]. From the result obtained in the present paper, in particular, follow the results given in [3, 7], when the support is a convex polygon and the convex hull is generated by a homogeneous Poisson point process.

Keywords: convex hull, convex polygon, Poisson point process, binomial point process, central limit theorem.

Citation: I.M. Khamdamov, Z.S. Chay, Joint Distribution of the Number of Vertices and the Area of Convex Hulls Generated by a Uniform Distribution in a Convex Polygon, J. Sib. Fed. Univ. Math. Phys., 2021, 14(2), 230-241. DOI: 10.17516/1997-1397-2021-14-2-230-241.

Introduction

This paper is devoted to the study of properties of convex hulls generated by independent observations over a random vector that has a uniform distribution in a convex polygon. Convex hulls are very complex objects from the analytic point of view. Therefore, studying the properties of the simplest functionals of convex hulls, such as, the number of vertices or the area, is not an easy task. This explains the fact that, prior to obtaining the central limit theorem for the number of vertices of a convex hull by P. Groeneboom, the main achievement was considered to be the study of asymptotic expressions for the mean values of similar functionals (see, for example, [4, 5,16]); the problems on asymptotic expressions for the variance remained unsolved until the appearance of the studies by C. Buchta [1,2] and J. Pardon [14,15].

It should be noted that P. Groeneboom, using the well-known property of homogeneous binomial point processes, which is that near the boundary of the support, it is almost indistinguishable

* [email protected] tchay1526@ mail.ru © Siberian Federal University. All rights reserved

from a homogeneous Poisson point process, and using such powerful techniques as strongly mixing stationary processes and martingales, has proved the central limit theorems for the number of vertices of a convex hull in the case when the support of the original uniform distribution is either a convex polygon or a unit disk. The modified P. Groeneboom technique was applied in [3] to prove limit theorems for the area and perimeter of a convex hull in a polygon, and in [9], to prove a limit theorem for an area outside a convex hull in a disk.

Similar results were obtained later by J. Pardon [16,17] without imposing any regularity conditions on the support boundary. In the present work, there is no need for using martingales, strongly mixing stationary processes, etc.; the approach used is a modification of the methods presented in [7,10-13]. The results obtained by Sh. K. Formanov, I. M. Khamdamov in [6], are applied here; a joint limit distribution for the number of vertices and the area of the convex hull generated by a Poisson point process in a cone was obtained by elementary analytical and direct probabilistic methods.

1. Statement of problem and results

Let Xj, j = 1,2,... ,n be the independent observations over a random vector having a uniform distribution in a convex polygon A with r sides. A matrix Xn is called a sample, the j-row of which is formed by the components of the vector Xj. Let us denote the convex hull generated by vectors Xj, j = 1, 2,... ,n by Cn = Cn(Xn).

We are interested in the joint limit distribution of the following functionals of Cn: the total number of vertices vn and the area Sn. It is clear that Cn, and, consequently, the indicated functionals, are uniquely determined by the set of vertices Wn. If the principle of vertex labeling is chosen, then it can be represented as a Vn x 2 matrix. It is easy to show that this matrix has the property of sufficiency with respect to the boundary of the set A — the support of original distribution. The latter circumstance is of interest from the point of view of statistics of uniform distributions.

Before formulating the main results, we introduce some notation. Let S be the area of the polygon A. Then we assume that

Dn = S — Sn

and let

2r log n t San I 27 , I 27 \[5nbn

an = , an = ~ bn = V 10r log n, bn = n\ 28rS2 log n = T^

We denote by w a vector having a two-dimensional normal distribution with a zero vector of mean values, unit variances and a correlation coefficient \J5/14.

Let us state the main theorem.

Theorem 1. Under our assumptions, a random vector with components bn (vn — an) and b'n (Dn — a'n) converges in distribution to w.

Let us make the necessary explanations of the notation. The symbols 4-, A, A a.s. denote convergence in distribution, in probability, and almost sure, respectively. f (e) x g(e) means that there are positive constants c1,c2,e0 such that c1f (e) < g(e) < c2f (e) for any 0 < e < e0. Generally op (1) is used for a sequence of random variables converging in probability to zero. The notation = Op(1) means that supn^x P (|£n| > t) a 0 as t a to. Everywhere c, ci, c2,... are the positive constants whose values might be changed from line to line and c(@), c1(fi), c2(ft)

are the positive constants, depending on the specified arguments. Further, £d=C means that the random variables £ and Z have a common law of probability distribution.

2. The Poisson approximation

In this section, we present the key idea of [7] about the Poisson approximation of a homogeneous binomial point process (h.b.p.p) Bn(A) generated by n independent observations of a random variable having a uniform distribution with support A in a slightly different way. Here we consider the more general case, assuming that A is an arbitrary bounded convex set in R2.

Let rA be the boundary of the set A. For each z e rA, consider an open sphere S(z,e) of radius e centered in z. It is easy to see that the set A£ = A — |JzEFa S(z, e) is a strip along the border rA. Let us denote Be = A — A£ and assume that A(A) = 1, where A(-) is the Lebesgue measure.

Let Wn, as before, be the set of vertices of the convex hull Cn generated by Bn(A). The next lemma is a simple modification of Lemma 2.1 and its Corollary 2.1 given in [7].

Lemma 1. There is a sequence of positive numbers en converging to zero such that the probability that at least one of the vertices Cn laying in Be, converges to zero in e > en.

Proof. It is easy to see that the event E = {Wnf) Be = 0} coincides with the event "there is a pair of neighboring vertices w1 and w2 such that w1 e Be ". Let the straight line (p,z — w\) =0 pass through the point w2. Since wi e Be, then this line divides A into two parts, the measure

of each is no less than some value of c(e) > 0 such that lime^0 c(e) = 0. Therefore at n > 2

n( n — 1) n 2

P(E) =-2--pn-2 {n — 2 the sample points Xn lie on one side

of the straight line (p, z — w1) = 0} dw1dw2 ^ n2 (1 — c(e))n . It remains to assume that

f 3 log n 1

en = if e : c(e) > —n— \ . (1)

The lemma is proved. □

Note that the rate of decrease c(e) at e ^ 0 depends on the smoothness rA. In particular, if A is a sphere, then c(e) x e 2; if A is a polygon, then c(e) x e2 and etc.

Since we are not interested in the estimates of the rate of convergence in the theorems given below, we will not worry about optimizing the choice of the strip containing Wn.

Let now nn(-) be a homogeneous Poisson point process (h.p.p.p.), the intensity of which is equal to nX(-).

Consider the narrowing nn(A) of this process to the set A. We denote by C'n the convex hull generated by it, and the set of its vertices we denote by W'n.

Lemma 2. The probability that at least one of the vertices C'n laying in Be, converges to zero, as n ^ to uniformly in e > en, where en, is determined by relation (1).

Proof. We assume that

E' = { WnH Be = 0}

and let pn() be the random counting measure corresponding to nn(A). By the formula of total probability we have

P(E') = Y] P (Pn(A) = k) P (E' f^(A) = k). (2)

k=0

Since the conditional distribution nn(A) under the condition pn(A) = k coincides with Bn(A), according to Lemma 1 for k ^ 3 we have

P (E' fpn(A) = k) < k2 (1 - c(e))k-2 . (3)

Taking into account (2) and (3), we write

P(E') ^ Y, k2 (1 - c(e))k-2 P (pn(A) = k)+ P (\^n(A) - n\ > n)=Sl + S2- (4)

\k-n\< n

Using the Chebyshev inequality, we have

S2 < 16n-1. (5)

Further on, for sufficiently small e > 0

Si < kma< n k2 (1 - c(e))k-2 ^Y (1 - c(e))"-2 . \k-n\<n V 4 J

It is easy to see that

sup S1 = o(1). (6)

Combining (4)-(6), we arrive at the assertion of the lemma being proved. The lemma is proved. □

Let Ce be the convex hull constructed from the part of the sample Xn in Ae.

Lemma 1 implies that

sup P (Cn = C£) ^ 0 as n ^ <x>. (7)

Let Bn(Ae) be the narrowing of the h.b.p.p. Bn() on Ae. According to Lemma 2.2 in [7], nn(Ae) and Bn (As) can be defined on one probability space in such a way that

P CMAe) = Bn(Ae)) < 2A(Ae). (8)

Let us denote the convex hull generated by nn(Ae) by C'E. Then from Lemma 2 it follows that

lim sup P (Cn = C'F) = 0. (9)

n^OO

From (7)-(9) it follows that as n ^ <

P (Cn = Ce) ^ 0. (10)

Remark. Let fi, i = 1, 2,... ,k be a certain finite number of functionals defined on the set of convex polygons. If the joint distribution of random variables fi (Cn), i = 1, 2,...,k converges to some distribution G, then it follows from (10) that fi (Cn), i = 1, 2,...,k also has this property. Thus, the problem of the limit distribution of the functionals vn and Sn, introduced in Section 1, is reduced to the study of vn and S'n are the corresponding characteristics of convex hulls generated by the h.p.p.p.

3. Convex hulls generated by the h.p.p.p.

3.1. Some properties of the h.p.p.p. Let К be a cone formed by two rays ¡i = = (z : z = tei, t > 0), i =1, 2, where e1 and e2 are the unit vectors. Without loss of generality, we assume that ei and e2 are the orthonormal vectors

e1 + e2

eo = —2— • (11)

Let further П(-) be a h.p.p.p. with intensity A(-). We denote the narrowing on K by П(К). Consider the convex hull C generated by К by П(К) and the set of its vertices Z. Let us denote the vertex by z0 e Z for which (e0, z — z0) > 0 for all z e Z. It is obvious that z0 is determined unambiguously almost sure. The straight line

(e0,z — z0 ) = 0 (12)

is the supporting line for C'.

Consider a triangle formed by rays ¡i, i = 1,2 and a supporting line (12). We denote the set

of interior points of this triangle by S0, and the area is denoted by £0. It is easy to see that

2

C0 = f, (13)

where x0 = y0 = u0 + v0 and z0 = (u0, v0). Assume that

П0 = ^ • (14)

x0

Then from (13) and (14) it is easy to obtain

U0 = (1 — П0)л/20, V0 = n0V/2C0- (15)

Let us label the vertices C', going around the boundary counterclockwise. Since z0 is defined, each of the vertices gets its own number j, —то < j < то. Let us choose on the ray ¡1 a sequence of points xj, j > 1, lying on the intersection of ¡1 and the lines passing through the vertices zj-1 and zj, respectively. Likewise, on the ray ¡2, points yj, j < —1, are obtained as a result of intersections of ¡2 and the lines passing through zj, zj+1, respectively.

Let Sj, j = 0; the set of interior points of a triangle with vertices zj-1, (xj-1,0), (xj, 0), if j > 1, and vertices zj+1, (0, yj+1), (0, yj), if j < —1. We denote the vertices of the triangle by (x0,0), (0, y0), the set of interior points by S0. The third vertex of this triangle is the point (0,0). The figures are taken from [6] (see Fig. 1).

We assume that

Then it is easy to obtain

j = A(Sj )•

1 (x, — x*-1)/2, if j ^ 1 j = { j , (16) uj+1(yj — yj+1)/2, if j < —1

vj

I uj+

where zj = (uj,v,). If we assume that

p. = uj uj -1, (17)

vj 1 — vj

then

v2 1

6j = j1(Pj - Pj-i). (18)

Now we define the boundary functionals

6t = inf {j : xj * T} and 6'T = inf {-j : yj * T} , (19)

where T > 0.

We assume that

a(T) = ^, (0 = 1°^. S = { «1 + 62 + ... + W if St * 1 and ST = { " + 6"2 + - + if >'t * 1 . (ai)

T 1 0 if 9t =0 T y 0 if eT =0 v 7

We present the following theorem with corollaries obtained in [6], which play the key role in this article (see Theorem 1, Corollaries 1, 2, 3 [6]).

Theorem 2 (Formanov and Khamdamov). Under our assumptions, as T ^ <, we have

(P (T))-1 (St - a(T), St - a(T)) A N (0, B) with B = ^ 1 14/5 ^ .

Here N(0, B) is a normally distributed random vector with a zero vector of mean values and a covariance matrix B.

Corollary 1 (Formanov and Khamdamov). In our case EeT = a(T) + o(P(T)) and VareT = P2(T)(1 + o(1)) as T

Corollary 2 (Formanov and Khamdamov). Let 0 < T1 < T2 such that c1T1 < T2 < c2T1 for some c1 > 0, c2 > 0. Then eT2 - eTl = op(P(T1)) as T1 ^ <.

Corollary 3 (Formanov and Khamdamov). Let 0 < T1 < T2 such that c1T1 < T2 < c2T1 for some c1 > 0, c2 > 0. Then (ST2 - STl) /P(T1) converges in probability to zero as T1 ^ <.

It is easy to see that at min{T1,T2} ^ to the random vectors (6Tl,STl) and (6'T ,ST2) are asymptotically independent. Moreover, the statements of Theorem 2 and its Corollaries 1-3 hold for S2) .

4. Proof of Theorem 1

The reasoning here is completely elementary. Generally, a verbal description of geometric objects is somewhat lengthy.

In accordance with the conclusions obtained at the end of Section 2 from Lemmas 1 and 2, it is sufficient to obtain the limit distribution for the number of vertices v'n and the area S'n of the convex hull C'n generated by the narrowing of the nn(A) h.p.p.p. nn(-) on the set A. The scheme of further reasoning is as follows. First, we divide the boundary C'n into 2r conditionally independent parts in such a way that each of the r angles of the polygon A corresponds to two elements of this partition. Thus, each of the functionals of interest to us v'n and S'n is represented as a sum of 2r random variables. Then, using the properties of the h.p.p.p. stated in Section 3, the asymptotic independence and normality of these random variables are established.

Thus, the general principles for studying the problem are the same as in [7], although their implementation is completely different.

4.1. Dividing the boundary into conditionally independent parts. We denote the vertices of an r-gon of the support A of the initial uniform distribution by a(i), i = 1, 2,... ,r. Let further, for some e > 0

where S (z,e) is a disk of radius e centered at z. Let us denote the narrowing nn(-) to a cone Ki with the vertex a(i) and generating rays li1 and li2 by nni(■), i = 1,2,.. .,r, passing through a(i+1) and a(i-1) respectively. It is clear that a(-1) = a(r), a(r+1) = a(1).

Let e0i play the same role with respect to Ki as played by the vector with respect to Ki in Section 3. Note that e0 is determined by the equality (11). More precisely,

We denote the convex hull as Cni generated by nni(-). Let us agree to denote the set of vertices C'n by Zni. Recall that the set of vertices C'n in Section 2 is denoted by W'n. We select in Zni and W'n the elements z0i and w0i, that possess the property that the straight lines (e0i, w — z0i) =0 and (e0i, w — w0i) = 0 are the supporting lines for Cni and C'n, respectively.

Assume that

(21)

Y = {n : z0i = w0i, i = 1, 2,.. .,r}

(22)

and

T2 = {n : z0i e Bi, i = 1, 2,.. .,r} ,

where n is the implementation of nn(-), and Bi is determined by equality (21). It is easy to understand that as n ^ to

(23)

P (Yi) ^ 1, i = 1,2.

(24)

As follows from (22)-(24), with probability close to 1, the boundary of each hull Cni has a non-empty intersection with C Note that the points wQi, i = 1, 2,... ,r divide the boundary Cn into r parts. We split each of them into two more parts. Let w(i) be the vertex Wn C Cn, for which the straight line (pi,w - w(i)) = 0, where pi ± (a(i+1) - a(i)) is a supporting line to C It is easy to see that w(i) is the closest vertex to the ray li1 from the vertices W'n. Note that as the n vertex w(i) grows, it approaches this ray indefinitely, i.e., (pi,w(i) - a(i)) ^ 0. Since the conditional distribution on the section of the supporting line (pi,w - w(i)) = 0 lying in A, under the condition (pi,w(i) - a(i)) = t is uniform, we have

lim lim inf P I w(i) e Pi B, I =1. (25)

e^Q n^Q \ 1 1 '

Hence it follows that

lim lim inf p \wi ef| B | = 1 (26)

j=1

where wi is the base of the perpendicular drawn from wi to li1. Consider

T3 = \wi e H Bj, i = 1,2,..., j=1

As follows from (25) and (26), for any e > 0 one can find such N > 0 that, for all sufficiently large n > N, the following inequality holds

P (T3) * 1 - e.

In what follows, without specifying, we consider only those implementations of nn(-) that are contained in p|3=1 Yj. For such implementations w(i\ i = 1, 2,... ,r lie between wQi and wQ(i+1). Thus, the boundary Cn is divided into 2r parts. It is easy to see, that these parts are conditionally independent for the given wQi, w(i), i = 1, 2,... ,r.

4.2. Choice of approximating functionals. Let us consider the section of the boundary C'n between the vertices wQ1 and w(i). The section between w(r) and wQ1 is studied in a similar way. Let us label the vertices Cn, going around the boundary counterclockwise, starting from wQ1. As a result, on the considered section of the boundary, we obtain wj, j = 0,1,2,..., where wQ = wQ1, wM = w(1). We perform a similar operation with the vertices 2 e Cn1 , obtaining Zj, j = 0,1, 2,..., where, in view of (22) and (24) zQ = wQ1 = wQ.

In order to use the h.p.p.p. properties described in Section 3, we need to proceed from n(-) to nn(•). In such transition, the linear characteristics xj,yj,uj,vj, change to xj = n"2xj, yj = n"1 yj, uj = n"1 Uj, vj = n" 2 Vj respectively, while the area 6j of the triangle 5j becomes 6j = n"16j. Dimensionless quantities nj,Tj, Pj remain unchanged in such transition. We denote the images Zj of such a transformation by zj.

Let T = e^Jn, T1 = n where h is the length of the side A connecting the vertices a(1) and a(2). In accordance with (19), we assume that

e = eT and x = eTl.

It is clear that

e = inf {j : xj * e} and x = inf {j : xj * h} .

Note that xj and xj are constructed on the vertices zj-1,zj and zj-1,zj, respectively. Note

that wj = zj, at least for 0 < j < x — 1.

Let further

and

P = £1 + £2 + ••• + S'e,

q = £1 + £2 + ••• + £X.

Assume that

e-a

p1

np — a

'

where

a = 1 log n, A

^ * ==i

14 log n 27 .

From (20), (27), (28) and Theorem 2 it follows that

(e*,p) A w,

where w is determined from Theorem 1. Now we assume that

x — a nq — a

X =— q =—.

According to Corollaries 1-3, in view of (28) and (30), we have

e — x ^ 0 n(p — q) ^ 0

A

From (29)-(33) follows that

P1

(x*,q*) Aw.

(27)

(28)

(29)

(30)

(31)

(32)

(33)

(34)

Similar characteristics e',p' and x',q' constructed along the section of the boundary C'n between the vertices w(r) and w01 = w, also have properties (31) and (34). It is important that they are asymptotically independent of e, x, p and q. And no less important is the fact that e, e', p and p' are completely determined by the narrowing of nn(-) to B1. It follows that similar characteristics ei, ei, pi, pi for the boundary sections corresponding to the angles with the vertices a(i), i = 1, 2,... ,r are independent. By analogy with (29) and (32), we define

. © — 2ra . nP — 2ra

©* =-;=- and P* =

(35)

where

© = £> + ei), p = £> + pi).

i=1

i=1

Due to independence of (ei + ei,pj + pj), i,j — 1,2,... ,r, from (31) we obtain

(©,P) A w.

Finally, by analogy with (35), we introduce

X* = X — 2ra Q* = Q — 2ra

(36)

*

*

e

p

where (compare with (35))

x = + xi), q = £q + qi)•

Note that (xi + xi,qi + qi), i = 1, generally speaking, are independent. However, in

view of (33) and (34), we can assert that

(X*, Q*) 4u.

(37)

It is the functionals X* and Q* that give us the required approximation for v'n and S^. 4.3. Estimation of the approximation accuracy. Let s be the area of the figure bounded by the section of the boundary Cn between the vertices w0 = w01 and wM = w(1), the segment of the ray /11 between the points w1 and x'0e11 and the supporting line (eo1,w — w01) = 0. Here, the points w0,wM,w1 are defined in Sections 4.1 and 4.2,

eii =

a(2) -||a(2) - a(1)||

and x'0 corresponds to x0 when going from n(-) to nn(-).

Let us construct similar left characteristics v' and s' in the section of the boundary between the vertices w(r) and w01.

In what follows, we denote Vi,vi, si and si, the analogs of v,v', s and s', corresponding to the angle with the vertex a(i). It is easy to see that v'n is the total number of vertices Cn and can be represented as

v'n =53 V + Vi) •

(38)

And area A — Cn can be represented in the form

A (A — Cn) = $>i + si)+ $

(39)

where C'0i is the area of the triangle cut by the supporting line Note that

<1 = Co = O(1),

, w — woi) = 0.

where £0 has an exponential distribution (see for example [6]). Similarly

nC0i = C0 = O(1), i = 1, 2,...,r.

(40)

(41)

As an approximation for si and si, we use xi,xi,qi and qi, introduced in Section 4.2.

In this case, it is enough to evaluate the proximity of (^1, s1)'==s(^, s) + op(1) and (x1 ,q1)==s(x, q). The remaining pairs of vectors are matched similarly.

To complete the proof of the theorem, it suffices to show the proximity of s and q, i.e.

n(s — q) p v ' p 0 at n^oo

and proximity of v and x , i.e.

VTôgi

V — X pk n ,

—> 0 at n oo •

A/TÔgi

(42)

We obtain the relation (42) from Corollary 3, and relation (43) from Corollary 2. The obtained relations (42) and (43) with the relations (36), (37)-(41) allow us to assert that a random vector

v'r, - 2ra n(1 - Si - 2ra

with components -and -=- converges in distribution to w. Taking into

P1 2r P2 2r

account Remark given at the end of Section 3, we obtain the assertion of the theorem. The

theorem is proved.

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Совместное распределение числа вершин и площади выпуклых оболочек, порожденных равномерным распределением в выпуклом многоугольнике

Исакжан М. Хамдамов

Национальный университет Узбекистана им. Мирзо Улугбека

Ташкент, Узбекистан

Зоя С. Чай

Ташкентский университет информационных технологий

Ташкент, Узбекистан

Аннотация. Рассматривается выпуклая оболочка, порожденная выборкой, равномерно распределенной на плоскости для случая, когда носитель распределения представляет собой выпуклый многоугольник. Доказывается центральная предельная теорема для совместного распределения числа вершин и площади выпуклой оболочки с использованием пуассоновской аппроксимации биномиальных точечных процессов вблизи границы носителя распределения. Здесь применяются результаты [6] совместного распределения числа вершин и площади выпуклых оболочек, порожденных пуассоновским распределением. Из результатов, полученных в настоящей статье, в частности, следуют результаты [3,7], когда носитель представляет собой выпуклый многоугольник, а выпуклая оболочка порождается однородным пуассоновским точечным процессом.

Ключевые слова: выпуклая оболочка, выпуклый многоугольник, пуассоновский точечный процесс, биномиальный точечный процесс, центральная предельная теорема.