CC BY
14
2018, Т. 160, кн. 2 С. 266-274
УЧЕНЫЕ ЗАПИСКИ КАЗАНСКОГО УНИВЕРСИТЕТА. СЕРИЯ ФИЗИКО-МАТЕМАТИЧЕСКИЕ НАУКИ
ISSN 2541-7746 (Print) ISSN 2500-2198 (Online)
UDK 519.21
ON THE ESTIMATION OF THE CONVERGENCE RATE
IN THE MULTIDIMENTIONAL LIMIT THEOREM FOR THE SUM OF WEAKLY DEPENDENT RANDOM VARIABLES FUNCTIONS
F.G. Gabbasova, V.T. Dubrovinb, M.S. Fadeevab
aKazan State University of Architecture and Engineering, Kazan, 420043 Russia bKazan Federal University, Kazan, 420008 Russia
Abstract
A refinement of estimates of the convergence rate obtained earlier in the multidimensional central limit theorem for the sums of vectors generated by the sequences of random variables with mixing is close to optimal. This has been achieved by imposing an additional condition on the characteristic functions of these sums, more accurate estimates of the semi-invariants, and using asymptotic expansions for the characteristic functions of the sums of independent random vectors. The result has been obtained using the summation methods for weakly dependent random variables based on S.N. Bernstein's idea of partition of the sums of weakly dependent random variables into long and short partial sums, as a result of which the long sums are almost independent, and the contribution of short sums to the total distribution is small. To estimate the differences between the sum distributions, we have used the S.M. Sadikova's inequality connecting the difference between the characteristic functions of random vectors with the difference between the corresponding distributions. To estimate the contribution of short sums, Markov and Bernstein's inequalities have been used.
Keywords: limit theorem, strong mixing, semi-invariants, asymptotic expansion, convergence rate
Random fields arise in the description of continuum mechanics processes for constructing physical relationships between various characteristics of the processes [1—5], plasma physics processes in calculating the velocity of plasma-chemical reactions [6-8], etc. In the statistical simulation of phenomena, limit theorems underlie the probabilistic criteria. The present paper is devoted to the use of asymptotic expansions of characteristic functions and estimates of the semi-invariants of sums of random variables to obtain estimates of the convergence rate in the limit theorem for the vector sequences generated by the sequences of random variables with mixing. We note that in [9] a new condition for the weak dependence of a sequence of random variables was introduced. In this case, the estimate of the rate of convergence in the central limit theorem for weakly dependent quantities is the same as for independent random variables. In [10, 11], the one-dimensional limit theorems (central and with large deviation) were proved and almost optimal estimates of the convergence rate were obtained. In [12-14], the multidimensional limit theorems for endomorphisms of Euclidean space were proved. In [15], the question of large deviations in the multidimensional limit theorem for trajectories generated by endomorphisms of the Euclidean space was investigated.
Let ai,a2,... be stationary sequence of random variables satisfying the strong mixing condition (introduced by M. Rozenblatt) in the narrow sense, with the coefficient
a(k) < cexp(-ak), where c> 0, a> 0 are the constants. We define random variables Cj = fi(aj, aj+i,...), CSj = E{Cij\a,j,..., a,j+s-i}, where fi are the measurable space maps of numerical sequences into a number line, E{Cij\H} is a conditional mathematical expectation with respect to a set of quantities H, i = 1, 2,... ,m, j = 1, 2,..., s = 1, 2,...
We form vectors Cj = (Cij,,..., Cmj), Cj = (Cj, Cj,..., Csmj). We denote by Pn
a distribution of the sum Cj/Vn, $r — m is the dimensional normal distribution
j=i
with covariance matrix R and the zero vector of mathematical expectations. Besides, w denotes an arbitrarily large fixed positive number. We assume that s = s(n), 1 < s(n) < w ln n.
We consider the following conditions:
1) ECi1 = 0, < B, where \ ■ \ denotes the vector length, B > 0 is a constant;
n n
2) the matrix R with elements rij = lim E^ (^^ Cjk^J /n is non-degene-
k=1 k=1
rate;
3) ECa — E{Ci1a1,..., ar}\2 < D exp(-dr), where d > 0 is a constant;
n m
4) lim sup E exp [i[t,/^Cj)/\fn) < 1, where (t,Cj) = tiCij, t =
3=1 i=1
(ti, t2, . . . , tm) .
Obviously, these conditions will be valid for CSj.
Theorem 1. Let us suppose that conditions 1) -4) are satisfied. Then,
sup \Pn(M) — $r(M)\ = 0(lnm+4 n/vn), M
where M are the convex measurable sets from an m -dimensional Euclidean space 1.
Proof. Let us prove the theorem for the sequence . We present the estimates that are used in proving of the theorem.
i
..s _ T? I es
'' k=l ' ' k=l
Let us aj = E (£&)(£j)/l-
Lemma 1. If s = w ln n, then aj = rj + O(1/l).
Proof. First of all, let us consider how E(£i1£j,r+1) and E(£?1 Cjr+1) behave as r increases. We write
E iùi£j,r+i) = E{(ùi - £,a2]+1)£j,r+i} + E iéri2]+1Çj,r+i)-Applying Holder's inequality and the strong mixing property, we obtain that
\E(ùij+i)\< Er/2\jr+i\E 1/2\Ù,1 - +4B2a([r/2]),
where B is a constant.
Now, from the conditions of the theorems, we obtain that \E£n£jr+i\ < Cir-U. Here and below, Ci > 0 are the constants. It is clear that this inequality is also satisfied
for №Uir+i\ .
1Theorem 1 for m =1 was proved in [16]. The asymptotic equality from theorem 1 will also hold
for the stationary sequence .
Through the stationarity of the sequences , , we have
i
s _ T7£S (-s I \ V-I ,^I1\T?ÏÙS Ùs I ùs ùs
j + E(i - r/l)E{^ï,r+i + tSAr+1} <
r = 1
œ
rij = Eiiiiji + E E{inij,r+
Hence,
^ji
r=1
l-1 l-1
asj - rij | < e E\çi1j+1 - tsAr+11 +E E\çi1j+1 - ^+11+
oo l-i
+2 E + + ErEithjr+i + ZSAr+i}\ = o(i/i),
r=l+1 r = 1
Using Minkowski and Holder's inequalities, we have
i-i
J2E\Ziijr+i - $i8,r+i\ < C2l(Ei2iE\iii - Hi\2)1/2 < C2l-^3/4.
r=0
The lemma is proved. □
We introduce the random variable Tj = (i/|i|, j. Let xv(n) be a semi-invariant
n dV ( n \
of the v-order of the sum ^^Tj, i.e, \v(n) = dzv lnEexP \Z^2/Tj) ■
j=i j=i z
Lemma 2. There is a constant H independent of v, such that the estimate lxv(n)I < Hv(v!)2nsv-\ 1 < s < wlnn
holds.
The proof of the lemma is carried out in the same way as in [16].
k 2v
Lemma 3. If v < C2^Jk/ lnk, then El E j < C'iv(lnk)vkv(2v)!.
s j
j=1
The lemma is proved similarly to lemma 2 in [17].
In the proof of the theorem, we shall use the summation method for weakly dependent random variables. It is based on breaking up the sum of weakly dependent random variables into long and short partial sums. As a result, long sums are almost independent, and the contribution of short sums to the total distribution is small. Later, instead of , we use and, without any loss of generality, assume that the matrix Rs is a unit matrix.
n
The sum, the distribution of which we study, is Sn j .
j=\
Let Q and N be natural numbers that grow together with n and satisfy the condition \n — p(Q + N)| < p.
The sum Sn is divided as follows
p+1 Q
y0 = \/~Q(Zp + z0)
Su = VQj^yj + y^J^yj = viQ(zp + zP), yj = (1/viQ)Yl£(j-1)(.Q+N)+r,
j=1 j=1
N n
yj = (1/VQ)^2 £jQ+(j-1)N+r, y0+1 = 53 £p(Q+N )+r ■
r=1 r=1
We denote by yj , j = 1, 2, ■ ■ ■ ,p, the independent random vectors distributed in the
p
same way as yi and let zp ^^ yj . Let A be the covariance matrix of the vector yi ■ By
j=i
lemma 1, the elements of A differ from the elements of the identity matrix by the value O(1/Q). We denote by A a matrix, such that A'A = A-1, where A' is the transpose of the matrix A. Obviously, the vector Azp/^Jp has a unitary covariance matrix.
Let Gp be the distribution of zp /^Jp, GA be the distribution Azp /^fp, fp(t), fp(t) be the characteristic functions Azp / ^Jpj, Azp / ^/p, respectively. If N = 2[o>1 ln n], 1 < s < u1 ln n, then
|fp(t) - fp(t)\< C4p/n"■ (1)
This inequality follows from the relations (7), (8) of [16]. We denote
V
gvp(t) = exp(-|t2|/2)(l + Y, Pr (it)(1/Vp)r),
r = 1
where Pr (it) depends on the semi-invariants of (Ay1, it) of order at most r + 2 :
r-1 r-1 jl-1 j3-1
Pr (it)= Xr+2(it)/(r + 2)+^^ ]T ■ ■■£ X
l = 1 jl=ljl-l=l-1 j2 = 2
^ (r - j1)(jl - j-1) ■ ■ ■ (j2 - j1)Xr-ji+2(it)Xji-ji-1 +2 ■ ■ ■ Xji+2 (it)
X (r - ji + 2)!( ji - jl-1 +2)! ■■■ (j2 - j1 + 2)!( j 1 +2)! ■
3\ = 1
For the characteristic function of the sum Azp/y/p, the following asymptotic expansion holds: for |t| < ^p/(8(E\Ay1\V+3)1(v+1)) = Tvp takes place
fp(t)= gvp(t) + O^2^3 exp(-|t|2/4)/TVp+1), (2)
(see [18], relations (7) and (8)). It follows from lemma 2 that
Xr (it)| = O((r!)2Hr s^1^/Q(r-2)/2),
(3)
H ■ - r"" ■ -|t|' --/Q^- 1 ■ - I ■
l=1
\Pr (ii)| = o( è Hr+2lT2r+2l\t\r+2lsr+2l-1/Q(l+r-1)/^ .
We use Sadikova's inequality [19], which relates the difference between the characteristic functions with the difference between the corresponding distributions
\G£(M) - $(M)| < cmr(m-1)/2[h + 2V2I2 + 4V2I3/T]+
+ (S) + 2P{\n\ > r} + 4P{\$\ > ¿1,
where n is the normal random vector with distribution $ is a vector with characteristic function g(t) = exp(-a2\t\2/2) and density (2na2)-m/2 exp(-\x\2 / 2a2),
12 =J\t\-2\fP(t) - exp(-\t\2/2)\dt, l2 = j \fp(t) - exp(-\t\2/2)\dt,
|t|<1 1<\t\<T
= J lh(t)l2dt, cm = \mS(Oi).
\t\>T
Here, O\ is the ball of unit radius with the center at the origin, S(Oi) is the surface of
* X -1
the sphere, Xk = (2n)m+1 I s sinm ada I , (S) is the least upper bound over all
v° . /
convex sets M of the probability of hit of a random vector n in the S-neighborhood of a boundary M.
The technique of using Sadikova's inequality is described in [18].In our case, it is necessary to choose
T = C4^/QTup, r ln T, a = ymmTVjWT/T, S = ar.
From lemma 3, we obtain that
m(v+3)/2 < E\Ayi \ < C!v+3(v + 3)!(ln Q)(v+3)/2,
Ce-TT; < Tvp < C7 VP, Cr^f^ <T<C9 VPQ. (4)
v ln Q v ln Q
In the same way as in [20], we obtain
I3 = O(1/T), cmr(m-1)/2 = O((ln T )(m-1)/4), ^(S) = O(lnT/T)), P{\n\ > r} = P|\?\ > a} = O(1/T). The integrand in Ii and I2 is estimated as follows
\fp(t) — exp( — \t\2/2)\ < \fp(t) — fp(t)\ + \fp(t) —
— gvp(t)\ + \gvp (t) — exp( — \t\2/2)\ = Fi(t) + F2(t) + fs(t) The integrals Ii and I2 are estimated by the sums of three integrals
(5)
3
/
ii < E
1/2
¡F2(t)/ltl2 dtl = Ei(l),
\\t\<1 ' 1
3
i/2
I2 < E
■^(t)/ltl2 dt\ = E
\l<\t\<T ' ''_1
Using (1), we get l1 ' = O(n w^Jp/Q). According to relations (2) and (4),
l12 = O(3v+2T+1) = O(3v+2vv lnv Q/p1v+1)/2). From relations (3), we obtain
2 \1/2
if] = f (EHrsrr2r/(pQ)r/2)2d^ )= O(1/yÇQ).
\t\<
Thus, we estimated i .
Similarly to the estimates of il1 and l13, we obtain
I(1) = O(pTm\/p/Q/nu), I13 = O(1/\fpQ). 12)
The estimate of the integral I2 is somewhat more complicated. We divide the domain of its integration into two sub-domains: 1 < \t\ < Tvp and Tvp < \t\ <T. By virtue of relation (2), the integral I221 of the first sub-domain is estimated as
O(CV+2/TV+1) = O(C[+2v! lnv Q/p1 v+1)/2).
(22)
The integral I2 of the second sub-domain is also divided into two integrals and we estimate each of them separately
, \ 1/2 ( \ 1/2
T (22) T2
\fp(t)\2dt I +
J \gvp(t)\dt
\Tvp<\t\<T ) \TvP<\t\<T
( \1/2
J pm\f(At)\2pdt\ + O(exp(-Tvp/4)).
\TvP/VP<\t\<T/VP J
According to condition 4) of the theorem, there exists a positive constant a, such that \f (At)\ < exp(-a). Therefore, we get that
I{22) = O((pQ)m/2 exp(-ap)+exp(-C 1l^p/ lnv Q)). The obtained estimates make it possible to write down that
sup\G£(M) - $(M)\ = O(lnm/4(pQ)((pQ)1m+1)/2/n^ +
+ p1m+3)/2Q1m-1)/2/n» + C1+2vv lnv Q/p1 v+1)/2 +
+ (pQ)m/2 exp(-ap) + exp(CnVp/v/lnQ) + 1/^Q) + vln2(pQ)/VpQ)- (6)
Since the distance sup\GA(M) - $(M)\ is invariant under non-degenerate linear M
transformations, then the same estimate holds for sup \Gp(M) - $a(M)\. The ele-
M
ments of the matrix A differ from the elements of the identity matrix by O(1/Q). Therefore, the estimate sup \Gp(M) - $(M)\ differs from the previous estimate
M
of (6) by O(1/Q). Following this, we proceed from the estimate of the distribution Gp(M) = P(zp/^p e M) to the estimate of the distribution P((zp + z^)/
_ n
^/p e M) = P(1/%/pQ J2 Cj e M) in the same way as in [18, p. 94], but using
j=1 _
lemma 3. We obtain a difference of order O(^N ln N/Q + 1/nu). We proceed similarly
n
to the estimate of the distribution Pe M) and obtain the difference of
j=1
order O((N +1)/vQ + c27u lnv n(2v)!nv/(Q2vpv)). Choosing p = O(u5 ln5 n), v = lnn, Q = O(n/ ln5 n) we obtain a statement of the theorem for the sequence j
To complete the proof, we need to show that the error from the replacement of by Cj goes to the remainder term.
From Minkowski's inequality and condition 3) of the theorem it follows that
nn
E\ ^^Cij - CSj\2 = O(1/nw). Then, using the Markov's inequality operating in
j=1 j=1
the same way as in [18], we obtain the assertion of the theorem. □
Acknowledgements. The work is partially supported by the Russian Foundation
for Basic Research (project no. 17-41-160-277).
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11. Dubrovin V.T. Large deviations in the central limit theorem for endomorphisms of Euclidean space. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2011, vol. 153, no. 1, pp. 195-210. (In Russian)
12. Dubrovin V.T., Gabbasov F.G., Chebakova V.Ju. Multidimensional central limit theorem for sums of functions of the trajectories of endomorphisms. Lobachevskii J. Math., 2016, vol. 37, no. 4, pp. 409-417. doi: 10.1134/S1995080216040053.
13. Gabbasov F.G., Dubrovin V.T., Kugurakov V.S. On the multidimensional limit theorem for endomorphisms of the Euclidean space. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2015, vol. 157, no. 1, pp. 25-34. (In Russian)
14. Gabbasov F.G., Dubrovin V.T. Estimation of the rate of convergence in the multidimensional central limit theorem for endomorphisms of Euclidean space. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2013, vol. 155, no. 2, pp. 33-43. (In Russian)
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Received October 4, 2017
Gabbasov Farid Gayazovich, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Applied Mathematics
Kazan State University of Architecture and Engineering
ul. Zelenaya, 1, Kazan, 420043 Russia E-mail: [email protected]
Dubrovin Vyacheslav Timofeevich, Candidate of Physical and Mathematical Sciences, Associate Professor of the Department of Mathematical Statistics Kazan Federal University
ul. Kremlevskaya, 18, Kazan, 420008 Russia E-mail: [email protected]
Fadeeva Maria Sergeevna, Student of the Institute of Computational Mathematics and Information Technologies Kazan Federal University
ul. Kremlevskaya, 18, Kazan, 420008 Russia E-mail: [email protected]
УДК 519.21
Об оценке скорости сходимости в многомерной предельной теореме для сумм функций от слабо зависимых случайных величин
Ф.Г. Габбасов1, В.Т. Дубровин2 , М.Е. Фадеева2
1 Казанский государственный архитектурно-строительный университет, г. Казань, 420043, Россия 2 Казанский (Приволжский) федеральным университет, г. Казань, 420008, Россия
Аннотация
Проведено близкое к оптимальному уточнение полученных ранее оценок скорости сходимости в многомерной центральной предельной теореме для сумм векторов, порожденных последовательностями случайных величин с перемешиванием. Этого удалось
достичь за счет наложения дополнительного условия на характеристические функции этих сумм, более точных оценок их семиинвариантов и использования асимптотических разложений для характеристических функций сумм независимых случайных векторов. Результат получен с использованием методов суммирования слабо зависимых случайных величин, основанных на идее С.Н. Бернштейна разбивать суммы слабо зависимых случайных величин на длинные и короткие частичные суммы, в результате чего длинные суммы почти независимы, а вклад коротких сумм в общее распределение мал. Для оценки разностей между распределениями сумм используется неравенство С.М. Садиковой, связывающее разность между характеристическими функциями случайных векторов с разностью между соответствующими распределениями, а для оценки вклад коротких сумм -неравенства Маркова и Бернштейна.
Ключевые слова: предельная теорема, сильное перемешивание, семиинварианты, асимптотическое разложение,скорость сходимости
Поступила в редакцию 04.10.17
Габбасов Фарид Гаязович, кандидат физико-математических наук, доцент кафедры прикладной математики
Казанский государственный архитектурно-строительный университет
ул. Зеленая, д. 1, г. Казань, 420043, Россия E-mail: [email protected]
Дубровин Вячеслав Тимофеевич, кандидат физико-математических наук, доцент кафедры математической статистики
Казанский (Приволжский) федеральный университет
ул. Кремлевская, д. 18, г. Казань, 420008, Россия E-mail: [email protected]
Фадеева Мария Сергеевна, студент Института вычислительной математики и информационных технологий
Казанский (Приволжский) федеральный университет
ул. Кремлевская, д. 18, г. Казань, 420008, Россия E-mail: [email protected]
For citation: Gabbasov F.G., Dubrovin V.T., Fadeeva M.S. On the estimation of the convergence rate in the multidimentional limit theorem for the sum of weakly dependent random variables functions. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Ma-tematicheskie Nauki, 2018, vol. 160, no. 2, pp. 266-274.
Для цитирования: Gabbasov F.G., Dubrovin V.T., Fadeeva M.S. On the estimation / of the convergence rate in the multidimentional limit theorem for the sum of weakly \ dependent random variables functions // Учен. зап. Казан. ун-та. Сер. Физ.-матем. науки. - 2018. - Т. 160, кн. 2. - С. 266-274.
