CC BY
10
DOI: 10.17516/1997-1397-2020-13-3-275-284 УДК 519.21
On Limit Theorem for the Number of Vertices of the Convex Hulls in a Unit Disk
Isakjan M. Khamdamov*
Tashkent University of Information Technologies
Tashkent, Uzbekistan
Received 12.02.2020, received in revised form 06.03.2020, accepted 03.04.2020 Abstract. This paper is devoted to further investigation of the property of a number of vertices of convex hulls generated by independent observations of a two-dimensional random vector with regular distributions near the boundary of support when it is a unit disk. Following P. Groeneboom [4], the Binomial point process is approximated by the Poisson point process near the boundary of support and vertex processes of convex hulls are constructed. The properties of strong mixing and martingality of vertex processes are investigated. Using these properties, asymptotic expressions are obtained for the expectations and variance of the vertex processes that correspond to the results previously obtained by H. Carnal [2]. Further, using the properties of strong mixing of vertex processes, the central limit theorem for a number of vertices of a convex hull is proved.
Keywords: convex hull, Poisson point process, Markovian jump process, martingales, Central limit theorem.
Citation: I.M.Khamdamov, On Limit Theorem for the Number of Vertices of the Convex Hulls in a Unit Disk, J. Sib. Fed. Univ. Math. Phys., 2020, 13(3), 275-284. DOI: 10.17516/1997-1397-2020-13-3-275-284.
Introduction
The functionals of convex hulls are complex objects in analytical aspect. Therefore, studying the properties of even the simplest functionals of convex hulls such as the number of vertices or the area, has for a long time remained a difficult task. This explains the fact that such well-known researchers as in [2, 3,15] and others, limited their interests to studying the average value of the number of vertices, the area, and the perimeter of a random polygon. For many years, due to the lack of valid research methods, the attempts to develop this area have not been successful.
In paper [4] has made a significant progress in this field. He managed for the first time to obtain the limit distribution for the number of vertices of a convex hull in the case when the support of initial uniform distribution is either a convex polygon or an ellipse. His research method is based on the original idea of using the Poisson approximations of a binomial point process near the boundary of the support of initial distribution. Then he applied powerful methods such as martingales, mixing of stationary processes and others. Based on this method, in [1] have established the limiting distribution for the area of the convex hull when the support of initial distribution is a convex polygon. In [6] proved the limit theorems for the area outside a convex hull when the support is a unit disk. These results in a more general form, for the
* khamdamov.isakjan@gmail.com https://orcid.org/0000-0002-7464-8358 © Siberian Federal University. All rights reserved
joint distribution of the vertex number, area, and perimeter, were obtained by [12] using the idea of [4] on Poisson approximations of a binomial point process near a polygon boundary. In [7] has developed this problem for the case when the convex hull is generated by distributions with exponential tails, including, in particular, the normal distribution.
The approach used in this paper is a modification of the methods proposed by [4, 5,12] and adapted to a wider class of initial distributions.
1. Statement of the problem and results formulation
Let the support of initial distribution A be a unit disk with a center at a point(0,1). Suppose that random points (ri; ai) are given in the polar coordinate system (with pole (0,1)) in a disk A, where ri and ai are independent and a^s uniformly distributed in [—n, n] and
P (ri > 1 — x) = x3, 0 < x < 1, P > 1, (1)
where L(x) is the slowly varying function in the Karamata sense given by
e(t)
L(u) = exp | J^ ^Tdt^j , e(t) ^ 0, t ^ œ.
t
Next, assume that Xi = ri sin ai, 1 — Yi = ri cos ai and denote by Cn the convex hull generated by random points (Xi, Y1), (X2,Y2),..., (Xn, Yn), and denote by vn, sn and ln the number of vertices, the area and the perimeter of the Cn, respectively. Denoting the largest root of the equation by bn
nx-(3+1 )L(x) = 1. (2)
In this case, in [2] obtained asymptotic expressions for the expectations of Evn, Esn and Eln. In the one-dimensional case, in [13] studied the role of the extreme summands in the sums, when the tail of the distribution of the initial random variable is (1) regularly varying. This paper is a continuation of [9,10,13] in the multidimensional case. According to P. Groeneboom's remark, we consider vn for the case when L(x) = 1. Then from (2) we get
2
bn = n wi. (3)
The basic theorem of the present paper is given.
Theorem 1. Let the conditions At n ^ œ the following ratio is true
i
Vn - a\(P, n)bn d
a2(fi,n)bn
■N(0,1).
d
Here ^ means the weak convergence, N(0,1) denotes the standard normal distribution with parameters (0,1), a1 (P,n), a2(P,n) are positive constants determined from relations (12) and (13).
In particular, if P =1 Groeneboom's result [4] follows. Corollary. If condition (3) is satisfied, then
vn — C1 (P)bn 4 N(0,1),
C2(P)bn
where c1(P), c2(P) have an explicit form, and c1(1), c2(1) coincide with the corresponding constants in the [4].
2. Preliminaries
In this section we give a modification of the key approach by [4,12] on the Poisson approximation of binomial point process (b.p.p.) B„(-) generated by n random sample from the distribution (1) in the unit disk. Assume that
Se = \(x,y):l - e < /x2T(l—W < l) ,
(4)
Kp(A) = P ((Xi,Yi) e A).
Consider a convex hull C'n generated by a nonhomogeneous Poisson point process (n.h.p.p.p.) n„(Se) with intensity of nKp (■).
Lemma 1. Let Bn (Se) be the n.h.b.p.p. with parameter(n, Kp (■)). Then there is the n.h.p.p.p. nn (Se) with intensitynKp (■) such that
P (Bn (Se) = nn (SE)) < 2Kp (Se), P (Cn = C'n) ^ 0,
at n ^ e ^ 0.
To formulate Lemma 2, we need some notation.
Let R(S)= ^(x,y): y< l, l - y < /XX^+Il-yf < l, j—y <tgS^ ,
( x2 X2 S2 1
R*(S) = }[(x,y): |x| < S, — < y < y + y] ,
where S = Sn = ^\f\ognfbn. For any set of forms
f x2 A = \(x,y) : a < x < b, — < y1(x) < y < y2(x)
introduce a measure
np fb f v(x) / x2N
np r\ fy2(xu x2\
a''a>= 2p j. dxlm (y " s)
- tt dy>
K(B)=0, if B C ^(x,y): y< .
Then assume that
K(B) = 0, at B C {(x,y) : x2 + (l - yf > l} . Lemma 2. There is nn (Se) the n.h.p.p.p. with intensityk*n() such that for any e > 0
(logp+2 n\ P (K(SE)=n(SE)) = M-n^l.
We denote by Cn the convex hull generated by the realization of n.h.p.p.p. nn (D), where
D =l(x,y): x2 < y < lj , S*(e) = j(x,y): x^ < y < y + e, y ^ 4 .
Then assume that
Introduce the following measure
Rn
/ \
(x,y): 2bn ^ y
An (A)
(y - t) dxdy,A C Rn
A
(5)
0, at A ^ Rn.
Then denoting by nn(-) n.h.p.p.p. with intensityAn(■) it is easy to see that
nn(-)=n*n(b-1-).
(6)
Now the whole circle is divided by mnparts, where mn = log n. Each section is 2n\fbnlog n long, with central angle 2n log n/vbn. Disk section corresponding to the circumference section
((log n)(2k — 1),^v/b~(log n)(2k + 1)
is denoted by Ik,n,(k = 0, 1, ..., mn — 1).
From Lemma 2, Poisson's processes n*n(b—1-) and ni(-) are almost similar. So consider Poisson's process ni(-) in Ik,nonly.
Following [4], consider the statement of Poisson point process in each sector Ikn separately. The vertex process Wn(a) = (Xn(a),Yn(a)) for any a £ (a-,a+) is such a point (Xk,Yk) of n.h.p.p.p. realization n,(-), for which Yk — aXk takes the minimum value, where a- = = —n log n/yfbn, a+ = n log n/sf&n.
It is easy to understand from the definition that, Wn(a) is a non-stationary Markov jump process.
The following lemma gives the types of distributions Wn(a) which correspond to various situations.
Lemma 3. Let s = y — ax + a2bn/2.
Then
1)P (Wn(0) G (dx,dy))
2)P (Wn(0) G (dx,dy)) =
P l y
exp < —
P+2
2nVK
B[P + fy — x-
2
2 \ p-i
2b
dxdy;
P
exp
2nVK
3)P (Wn(a) = Wn(0)/Wn(0) = (x,y)) = r ^
J x — abn
sp+ 2 / 1 \ I ( x~
72nB{P + 1;2H [y — dxdy;
n
2 \ P-l
= exp
s--;— I du —
2b J .
/2b~y
,2 \ P
y — 2bn dy).
Proof. Let v = a(u — x) + y be a straight line passing through points (x, y) with angular coefficient a, A(a, x, y) is the set of points in the domain bounded by lines v = a(u — x) + y and v = u2/(2bn).
It is easy to see that if ui and u2 are the roots of equation
u2/ (2bn) = a(u — x) + y,
then ui 2 = abn + y/2bns.
2
Calculate Kn (A(a,x,y)) (see (5)). Considering
u
y + a(u - x) - —— = y - ax +
2bn (u - abn)2
2b „
we get
K0n (A(a,x,y)) =
2-Ky/bn Jm 72sp+1 r1
u
y + a(u - x) - —
p+ i
2bn
p
du =
(u - abn)
2bn
/2bns
/2bns
2p
2b n
du =
I (1 - „2)<* du = L^B(p +^.
Next, let d = \J(Ax)2 + (Ay)2, v = au + c-, v = au + c+ be two straight lines parallel to v = a(u - x) + y and passing at distanced from below and above, respectively.
By A- (a, x, y) and A+(a, x, y)denote the sets bounded by lines v = au + c-,
V(2bn)
and v = au + c+, v = u2/ (2bn) respectively. Assume that Axy = [x,x + Ax] x [y,y + Ay].
It follows from the definition Wn(a) that if n(A) is the number of points in A realization of n.h.p.p.p. nn(-), then
P (Wn(a) e Ax,y) < P (n(Ax,y) > l, n (A-(a,x,y)) = 0) . (7)
On the other hand, it is easy to see that
P (Wn(a) e Ax, y) > P (w(Ax,y) = l, n (A+ (a,x,y) - Ax,y) = 0) . (8)
Considering the property of the Poisson process (the independence of increments) and from inequalities (7) and (8), using (5) at d ^ 0 we obtain the first relation of the lemma. Similarly, the other relations of the lemma are obtained. Assume that
Rn (a) = Xn (a) - abn, Sri (a) = Yn (a) -
Xn (a) , R2n (a)
2bn
+ Tn (a) = (Rn (a) ,Sn (a))
2bn
Obviously that Tn(0) = Wn(0) a.s. and therefore
P (Tn(0) e (dr, ds)) = P (Wn(0) e (dr, ds)).
Lemma 3 leads to the following lemma. Lemma 4. Tn(a) is a stationary Markov jump process and
l) P (Tn(0) e (dr,ds))
exp{-B (P + l;l)} (s - i) drds;
2) P (Tn(a) = (ri, si)/Tn(0) = (ro, so))
= exp
l
V2nL (bn)
sP+2 | ri (l - t2)P dt - sp0 +11 ^ (l - t2)P
sj 2bnsl \/ibns0
dt
,/ib.
where r1 = r0 - abn, s1 = s0 - ar0 + a?bn/2;
3) P (Tn(a) e (dri, dsi)/Tn(0) = (ro, so)) = P (Tn(a) e (dru dsi)),
2
a
=s
2
2
s
v=u
if abn - V2bnSi > v2bnSo;
4) P (Tn(a) G (dr2, ds2)/Tn(0) = (ri, si)) =
„ß+ 2
exp
V2n
„ß+2
(1 - t2)ß dt-
sl-s2 + abn
a V 2bns2 V2bns2
1
-t2)ß dt
sl -s2 + abn
is1 V2bns1
2 \ P-i
S2 - 2b" ' dr2ds2.
Here assume that
(ri} si) G £ = {(r, s) : s > (r2) / (2b„)} , ab„ - ^ y/2bnsu
S2 + (a2b„) /2 + ar2 > si > S2 - (a2b„) /2 + ari. Consider the following a-algebras generated by process Tn(a):
c>0 _
= a {Tn(c) : c < 0} ,
Qa+ _
= a {Tn(c) : c > a} .
From Lemmas 3 and 4 it is easy to prove the properties of strong mixing of the process Tn(a) Lemma 5. For any A £ ^ and B £ %n+ P (ARB) — P(A)P(B)\ < Tn(a), where
' -M- \ 3+ 2 ^
Tn(a) < 4exp < -
1
V2n
a bn 8
B\ß + i;2
Lemma 6. If a > (ane*n) /bn, then under the conditions of Lemma 5 we have Y1 (Tn(a)Y < œ
n=l
___ 2ß — S— 1
for any T > 0, where an = ^2bn log n, £*n = (log n) 2(2ß+1) , 0 <5 < 1. Proof. If a > (a,ne*n) /bn, then from Lemma 5 we get
¡.(a) < 4exp{-c(logn)i+ .
This immediately implies the statement of Lemma 6. Now introduce notations
M(k) (t; R2) =
ß
/2bns
2nyfbn Jr
ß , V^-r
2nVKJ o
(u - r)M s - 2b-
2 \ ß-i
du =
Uk Is- (U + r)2
2 ß-i
2bn
du,
where t = (r, s). Lemma 7. Processes
and
f a
N (a) - M(i) (T (b); R2) db
0
fa
N2(a) - [2N(b) + 1] M(i) (T(b); R2) db
0
are martingales with respect to a-algebra 9[o,a] = a {T(c) : 0 ^ c ^ a} .
□
i
1
2
i
i
n
Proof. We have
E {N (a + h) - N (a)} = E {N (a + h) - N (a)/T (a)}
Hence, due to stationary nature of the process T(a)
E {N(a + h) - N(a)/T(a) = (r, s)} = E {N(h) - N(0)/T(0) = (r,s)} -
- En(A*(h; r,s)) - K0 (A*(h; r, s)), where W(A* (h; r, s)) is defined in the previous paragraph and
A*(h; r,s) = A0(h; r, s) U Ai(h; r,s).
Further, by the definition K°(-)of a measure (see (5)), it is easy to show that at small h K0n (A0(h; r,s)) = o(h) and
bbn + V 2bnsi
K0n ('A(i)(a,b,x0,y0)) < C2hp — J du = O (hp+i-E) . (9)
abn + V2bnso
From the latter, again using definition (5) at small h, we have
K0n (A*(h; r,s))=K0n (A0(h; r,s)) + o(h) = hM(t; R2) + o(h). (10)
By virtue of (9) and (10), we obtain the proof of the first statement of the lemma. Proceed to the proof of the second statement of Lemma 6. We have
E {N2 (a + h) - N2 (a)/T(a) = (r, s)} = = E {(N (a + h) - N (a)) (N (a + h) - N (a) + 2N (a)) /T (a) = (r, s)} = = E{ (N (a + h) - N (a))2/T (a) = (r, s^ + 2N (a)E{N (a + h) - N (a)/ - T (a) = (r, s)} = = E{N(a + h) - N(a)/T(a) = (r,s)} + o(h) + 2N(a)E{N(a + h) - N(a)/T(a) = (r, s)} = = (2N(a) + l) E{N(a + h) - N(a)/T(a) = (r,s)} + o(h) = = (2N(a) + l) hM(i) (t, R2) + o(h).
So, Lemma 7 is completely proved. □
Using these Lemmas, calculate the asymptotic behavior of the moments N (a) and N2(a)at fixed a and at n .
Let „ ^
c0 = (b (2p +l, IV2j
2__
N3-
2) I ) \B (2p + l, ^y V 2p + l
where B( ■, ■) and r( ■) are the known beta and gamma functions, respectively. Lemma 8. We have
EN (a) = aXn^y/bn, DN (a) = aXff y/bn, as n ^ x,
where Xni] = TL + o(l), X{2) = c0 + o(l). \j2~k
Proof. We use Lemma 7. Since the process T(■) is stationary, we have
f a
EN (a) = E M (i)( T (b); R2) db = aEM(i) (T (0); R2) . (11)
0
By definition of M(i) (t; R2 , after some identical transformations we have
EM(i) (T (0); R2) = ^e[ " u(s - P i du
y (); ' 2nVK J0 V 2bn J
P -E < f u (s — (—+—— | du > =
2nVK 1./0 V 2bn
B (P +l, i)
V2b~p2 f"
U. sp+i
s2p-i ds I x
y1 | (l - r2 u (l - (u + r)2)P-i du} dr =
(12)
x I |(l - r f J u (l - {u + r
X 4)3+1
VKP2 ( \ ^ r f_ l
2n2(4p +l)\ B(p + l; 2)/ V 2p +l
. 2, r-i /■ i — r
/i f i-r
(l - r2 )'-i dr J u (l - (u + r)2)
— (u + r)2]' i du.
-i
It is easy to calculate that
/>- (u + r)2)'-' du = p ¡' = mn. (13)
From relations (11)-(13), the proof of the statement of the first part of Lemma 8 follows.
The second part of Lemma 8 is easy to prove, using Lemma 7 and the first part of Lemma 8, and Lemma 2.6 considering in [4]. □
3. Proof of the theorem
Assume that N. is the number of vertices of the convex hull in I. mn — "big block" and Nk*mnis the number of vertices of the convex hull in I.*mn — "small block", where sectors I. mn and I .*mn correspond to
(nVb~(logn)(2k - l)+ ^n,^Vb~(\ogn)(2k + l) - £n)
and
(nv7^(logn)(2k + l) - £n, nvbt(\ogn)(2k + l) + of the part of disk, respectively and
£n = nVb~(log n)i-s, 0 < S < l/[2(2p + l)].
Hence
mn-i mn-i
'n =53 Nk,mn +53 NKm,n. k=0 k=0
X
By the principle of construction of sectors 1* mn and I** *mn, N** *mn is insignificant relative to N** . Further, from Lemmas 5 and Theorem 17.2.2 in [8], we can apply the classical central limit theorem for the summs of random variables N**mn + N*mn + • • • + Nm-1mn and N**'*mn + +N1*mn + • • • + Nm*n-1 ,. Therefore, we get that
- mn (EN;,mn + EN»mn) = mg N;,mn - ENjk,m„ +
mn-1 N * * EN * * \/DN£ *
+ 1 y^ N;, mn - EN;;'• ■- +
. Prn~ < J '------'---
;=o ^DNt*mn x/DNl
1 "g1 N*mmn - EN**mmn , 1 "g1 N-;:mn - ENk
;=0 JDN*mn ;=0 DN;,m
+ V ;,mn • 0(1) + o(1) A N(0,1).
The theorem is proved.
References
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О предельной теореме для числа вершин выпуклых оболочек в единичном круге
Исакжан М. Хамдамов
Ташкентский университет информационных технологий
Ташкент, Узбекистан
Аннотация. Данная статья посвящена дальнейшему исследованию свойства ряда вершин выпуклых оболочек, порожденных независимыми наблюдениями двумерного случайного вектора с регулярными распределениями вблизи границы носителя, когда он является единичным диском. Следуя П. Гренебуму [4], биномиальный точечный процесс аппроксимируем пуассоновским точечным процессом вблизи границы опоры и строим вершинные процессы выпуклых оболочек. Исследованы свойства сильного перемешивания и мартингальности вершинных процессов. Используя эти свойства, получаем асимптотические выражения для ожиданий и дисперсии вершинных процессов, которые соответствуют результатам, ранее полученным Н. Карнала [2]. Далее, используя свойства сильного перемешивания вершинных процессов, доказываем центральную предельную теорему для ряда вершин выпуклой оболочки.
Ключевые слова: выпуклая оболочка, пуассоновский точечный процесс, скачкообразный марковский процесс, мартингальность, центральная предельная теорема.
