CC BY
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УДК 519.214
LOCAL LIMIT THEOREM FOR CONDITIONALLY INDEPENDENT R A N D O M F I E L D S
Khachatryan L.A., Nahapetian B.S.
Institute of Mathematics, National Academy of Sciences of Armenia Контакты: [email protected]
In this paper we introduce the conditions under which the local limit theorem for random fields with weakly dependent components follows from the central one. Obtained result can be applied to Gibbs and martingale-difference random fields.
Keywords: local limit theorem, conditional independence, Gibbs random field, martingale -difference random field.
1. Introduction
The validity of the central limit theorem (CLT) for random fields with weakly dependent components was considered in many works (see, for example, [1; 2], where were discussed different methods of proving the CLT, and references therein). There are a lot of results on this subject under different types of dependency conditions for random variables - so called mixing conditions. However the question of validity of the local limit theorem (LLT) for weakly dependent random fields practically was not under consideration. The possible explanation is that classical mixing conditions are probably not enough to obtain the LLT.
In the same time this problem is very important from the point of view of statistical physics, particularly concerning the problem of equivalence of ensembles. The importance of the LLT first was discussed in [3], where the LLT was proved for number of particles in the case of the ideal gas. The notion of Gibbs random field predetermined further development of the theory of limits theorems for random fields. The
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LLT for Gibbs random fields was a subject of consideration in many works (see [4^8] and [9]).
For us the work [8] of Dobrushin and Tirozzi is of special interest. In this work it was shown that the LLT for Gibbs random fields with finite-range potentials follows from the CLT. Dobrushin and Tirozzi's approach to prove LLT essentially uses the finite-range condition of interaction potential. In our paper we introduce the notion of conditionally independent random field (not necessary Gibbsian). For such random fields we present general conditions under which the LLT follows from the CLT. The result can be applied to Gibbs and martingale-difference random fields.
Let us also note the work [10] where the similar result was obtained in one-dimensional case for random processes with finite-range dependence.
2. Preliminaries
In this paper we consider random fields on d -dimensional integer lattice Zd, d > 1, with finite phase space X e Z, 1 < |X| 1, i.e. collections of random variables (<^) = (<^,seZd), each of which takes value in X .
For any S e Zd, xs e X we denote by Xs =(xx, s e S) the space of all configurations on S. If S = 0, we assume that the space X0={0} . For any S, T e Zd such that S R T = 0 and any configurations x e Xs and y e XT, we denote by xy the concatenation of x and y, that is, the configuration on S U T equal to x on S and to y on T. For any S e T, x e XT , we denote by x5 the restriction of x on S. Also we denote by W = {V e Zd :|V| < <»} the set of all finite subsets of Zd .
1 Here and below the symbol |X| is used to denote the power of the finite set X .
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Let 3Z be the a-algebra, generated by cylinder subsets of the set XZ . The distribution of a random field (&) is the probability measure
P on (XZd, 3Zd), such, that
Pr ((&, ^ e Zd )e B) = P (B), B g3z' .
For the random filed (&) and any S e Zd denote by a aalgebra, generated by &, s e S .
A random field (&) is called a homogeneous random field if for any V eW and a e Zd
P(& = Xs ,s eV) = P(£+a = ^s ,s eV), Xs e X, s eV,
and called ergodic if for any I, V eW and x e X1, y e XV the following relation holds
IP (fc = Xs, s e I} n fc+a = yr, r eV }) = P = X, s e I) P = yr, r eV), V«|aeV„
where Vn = [-", "]d, " = 1,2,.... For a given random field (&) denote SF , VeW. We say
seV
that for the random field (&) the CLT is valid if
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lim P
n^œ
^ Sv - ESV ^
■fis;,
i x
= J e-"2'2du , x g R
y
and the LLT is valid if
lim sup
( sv„=j )-'
(Sv, - ESv, )2 2DS„
= 0.
3. Main result and some of its applications
Let us introduce the following condition of weak dependence of components of a random field.
We say that a homogenous random field is conditionally independent with coefficient Pj if for any I,V,AeW such that IR V = 0 and I ,V cA, and any random variables ^, r]2 which are - and 3F -measurable correspondingly the following relation holds
E
(ъъ/3л\( / и; ))- E (ъ/
3л\(/UV )) E (%/
3
л\( / Uv )/
)<P/ (P(1 ,v))
œ
where p(I,V) is the distance between I and V, and P7 ^0 as p^
(and hence At Zd ) and I is fixed.
Introduced condition of weak dependence of components of random fields seems stronger than classical mixing conditions. Also it seems more suitable to use conditions of this type instead of classical ones for random fields which are described by means of their conditional distributions such as Markov, Gibbs and martingale-difference random fields.
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In the next theorem conditions under which for conditionally independent random field the LLT follows from the CLT are presented.
Theorem 1. Let (&) be a homogenous random field with phase space X. If
1. DSV =a2|V|(1 + o(1)) as V T Zd, a > 0;
2. (&) is conditionally independent with coefficient &
such that
& (p)<|I| &(p) and =
where ju(p)^ 0 arbitrarily slow as p^-<x>;
3. there exists y > 0 such that for any I e V eW
P (Sj = y/) > y for any possible value y of S7; then for the random field (&) the LLT follows from the CLT.
Let us make some remarks on possible applications of this theorem. First let us note that Gibbs random fields with finite-range R potentials are conditionally independent, since for such random fields Pj(p) = 0 as soon as p> R. Hence the Theorem 1 generalizes Dobru-
shin and Tirozzi's result.
Now let us consider martingale-difference random fields. A random field (&) is called a martingale-difference (see [11]) if for any
5 e Z d
E |£| and E (£/3z, x{,}) = 0 (a.s.).
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For homogenous ergodic martingale-difference random fields the CLT is valid (see [11]). It is easy to see that if a random field is conditionally independent than it is ergodic. Hence we can formulate the following result.
Theorem 2. Let (£) be a homogenous martingale-difference random field with phase space X, and let there exists y > 0 such that for any I ^ V eW
P (Sj = y/ ) > y for any possible value y of S7.
If, in addition, (£) is a conditionally independent with coefficient P such that
P (p)<|I| P(p) and P(p) = ^0,
where ju(p)^ 0 arbitrarily slow as p^-ro, then for the martingale-difference random field (£) the LLT is valid.
4. Proof of the main theorem
In this section we give the proof of the Theorem 1.
Proof of the Theorem 1. For any n e N set V = \~n, n~f, S„ = ^ £
seV„
and let f (t) be a characteristic function of S„. Then
1 n
P (Sn = j) = — J e-'tfn (t) dt.
2ж
-л
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ES
Denote zn;. = j ,-" . Then j = znj,jDSn + ESn, and we can write
SDS"
P (* = j )-rJe
Also we have
itznj^DSn -itESn r J n
i
-i&nj-itESn f^DSn r fn
i t ^
K^nj
dt.
42ж
w
e 4/2 =_L j e-t2/2e-ltZnndt for any z e R .
9тт j J n
—w
1 — z2/2 1
2ж
Hence we can write
2ж sup
jeZ
Щр (s.=j)~L~ ~l-(sn—ESn)2
V2iexp I 2DS,
жЛ DSr
( . \
= sup
jeZ
T
< j|e
—T
+ j
t < t| ^jDS;
T < j
—itZnj —itESn WDSn f e Jn
—itES, U DS,
E exp i
kM j
dt — j e-ltZnJ-fndt
—e
-tn
dt + j | e~
t >t
— 12
dt +
—itES„ UDSr
E exp I
JUL M
dt <
e-'tESn 4 DSn E exp
itS.
№
n e—t^2
dt + j e—ndt + j
H>t тфж^т;
^ i. S„ — ES E exp ^ it-n n
Dn
dt,
where T, 0 < T < t^Jl)Sn , can be chosen big enough for sufficiently large " .
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ж. DS
itz
nj
e
Let s>0 be fixed. If the CLT for (£) is valid, then for sufficiently large T we have
T
J
e-itESn 4DSnE exp <
itSn
4Dsn
■-e
-t / 2
dt <s
Choosing T big enough we obtain also
e dt <s.
t >T
It remains to show that
J
T <| t| <л4тп
E exp < it -
Sn-ES n
4DSn
dt <s
(1)
Let us make the sectionalization of Vn. Set p = p(n) = o(n), q = q ( n) = o ( p) = n -A( n), where A( n) = ^2/(3d n), r> 0. Set also
In (j) = [-n+jp+jq;-n+(j+!) p+jq ] =
j = 1,2,...,
2n +1
p+q.
2 n+1
_ p+q
In = U In (j) ,
j=0
-T
2 Here and below by [ z ] we denote the integer part of z .
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and let Id be a Cartesian product of d copies of In. Then Id is a union
of к =
2n +1
P + q.
d -dimensional cubes A^ with side-length p, which
are numerated in some way: Idn = ^A\n).
j=1
Denote by 3„ = 3F ^ =*(£, s eV„ \ Idn ). We have
у n \ n 4 /
( (
Eexp\itSn n !> = E
4Dsn
E
v v
к SIÍ _ ESIÍ i к ^ ^^ _ ^ \I.¿ exp < it —n --n !> • exp < it -
jDSn
4DSn
(
= E
к SV Mi " ESV \i- I r exp< it —n n ,-n — !> • E
4DSn
I ^ - ESI¿ l /
exp r^r V3'
//
'3„
/У
Hence
E exp < it
Sn - ES„
< E E V
Г Sn_-ESiil L
exp Г
= E
Sj- ESj
0°*f Щ
I3„
where
Sj = . Denote Ъ =
S, - ES, j j and consider
4DSn
E J = E J-^E (e'^ / 3„ ) + ;QE (ej 3„ ) .
Using the following relation (see, for example, Lemma 3.3.1 in [7])
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d
(n)
* [n^/^J-nE ( e"/ =
= £ff())U(fie"/ I-E()fE(^^^^^^
r=2 V m=r+1
we obtain
E If
ZI П E(\e'"~\/Zj) I. E[ Пem> / |-E{е»'/Зя)ПE(e" /) <
r=2 \m=r+1 y V /=1 / У j=1
Г(e"/)| + t(/,(AJ,/„rf\Aij)))<n|E(e"/)| + ^^^9).
For any j = 1, ки we can write
E ( e"
(
exp < it
- E (/3 )! L E (/3 )-eS
JDSJ [' exp Г" VDSJ
<
<
exp <j it
f
exp <j it
E ( /3 j )-ES/
JdsJ
Sj -E ( /3 J )
A
E
exp < it
S/ -E(/^jУ -JDSJ
yfDSJ
Further,
E
exP И-1 '
= 1--*-D(S/3 )-
2DS, ( '' j)
DS„
Let 7 be the set of all possible values of S.. Using the third condition of the theorem, we can write
+
<
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D ( sj/) = £( y - E ( S ))2 ( S = y/>уХ( y - E ( S))2 y«
yeY
ye!
where « > 0. Hence
a
£
exp < ^^^
S —E ( S /3 » ) ^
< 1 -
ya
O (t2 ) -t2 + —^-i
2DS DS„
On the other hand
e 2DS»' = 1--« 12+У —
ТПС -¿—I 7, I
1 | —ya
2DS., k=2 k Ц 2DS.,
From here it follows that there exists S> 0 (which is independent of j) such that for any |t| < Stt^JDSn
f
E
exp < it
yfDs„
< e
y« . 2 2 DS»
Then
П|е ( e" V 3„ )<nexp
j=1 J=
2DS,
y« 12l = exp 1—У«.-^-. 12^< е~с"* 2 ,
2 DS„
where, due to the first condition of the theorem,
ya
2» + 1
P +1 .
C
(1 + О1))» (p + q)a
and C is a positive constant.
2
У« .2
d
Cn =
2
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Hence for the integral in left-hand side of (1) we can write
i
T < t| <^DSn
E exp <j it
S - ES
n_n
n
^ J n
Sn^DSn <|t|<n^DSn j =
dt < J e~Cn*2dt + J knpdP(q)dt +
T <|t| <Sx^jDSn T <|t| <X^DSn
' Л
Г S -E(S. /3 )
exp<itS-j/
< VdS_
dt.
Let us show that for sufficiently big T and n and sufficiently small S each summand on the right-hand part of the obtained inequality can be made smaller then s/3. By doing this we conclude the proof. It is obvious for the second summand. For the first one taking into account the second condition of the theorem we can write
< С
J knPdP(q)dt = 2knPdP(q)(*JdS~„-T): , /"(n)•
2n +1
p+q
• pd •^M • C 'd/2 <
P 3d/2 C n -q
„3d/2
A3d/2 (n )• n3d/2
= С' 1ЗЛ/2 (n),
where C is a positive constant. Since (n)^ 0 as n ^ro, for sufficiently large n
J knpdp(q)dt
<
T < t <»./DS,
Consider the third summand. We have
i Й
SжлfDS^-1DSn j=
E
exp < it
S - E (S1/ ^) 1 >/DS_ I
dt J f[\E(eiS/Sn)
S7r<tj=1
dt.
S
3
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Denote a =
E (ej3 я )
. We can write
IÎ a = exp |ln ГТa j = exp j11 ln a21 « exp | \1 ( a2 -1)
Further for any j we have
a —1 =
j
E (e'tSj/3n ) —1 = 1(cos t (x — y) —1) P (Sj = x/3» ) P ( Sj = y/3» ) =
x, yeY
2 1 sin2 ^ P ( Sj = x/3» ) P ( Sj = y 13» )<— 2У 1 sin2
x,,eY
x,ye Y
where we used the third condition of the theorem. Hence
П E(e"Sj/3n) < exp(— k»y2 1 sin2^^ = exp{—kn.g(t)},
j=1
x, yeY
where g(t) = -^2 ^ sin2t^x y^ is continuous positive function. Using
x, yeY
the mean value theorem for integral, we obtain
_ kn _
JDS» j ЦЕ (e'tSj/3n ) dt j e^dt <
ôn<\t\<n j=1
for any fixed 5, some t0 e[5;r;^] and n big enough. The theorem is proved.
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In conclusion let us note that for homogenous martingale-difference random fields the proof of the theorem can be simplified in some way. Indeed, for such fields one have
ESn = 0 and DSn = ESn = E#-VJ> 0.
To estimate the integral in (1) we can write
E exp \ USK3L ^ = EeitSn
4Щ,
t2
= 1+^=es,„--es2 +
it
2DS„
DS„
Hence there exists S> 0 such that for any < Sx^DSn
EeitSn /JDS„ < e-t2/2
Therefore
T <| i| <Sn^DSn
E exp -j it
S.
4Dsn
dt < j e
T
-t /2
dt <■
for T and n sufficiently large. It remains to show that
Sn^DSn <|t|DSn
E exp -j it
S.
yfDsn
dt <
which can be done in the same way it was done above.
s
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References
1. Doukhan P., Mixing. Properties and examples. Lecture notes in statistics 85, 1994.
2. Nahapetian B.S. Limit theorems and some applications in statistical physics. Teubner-Texte zur Mathematik 123 (B.G. Teubner Verlagsgesellschaft, Stuttgart.Leipzig), 1991.
3. Khinchin A.Ya. Mathematical foundations of statistical mechanics. New York, Dover Publications, 1949.
4. Arzumanyan V.A., Nakhapetyan B.S., Pogosyan S.K. Local limit theorem for the particle number in spin lattice systems. Theoretical and Mathematical Physics 89, 1991, 1138-1146.
5. Campanino M., Capocaccia D., Tirozzi B. The local central limit theorem for a Gibbs random field. Commun. Math. Phys. 70, 1979, 125-132.
6. Campanino M., Del Grosso G., Tirozzi B. Local limit theorem for a Gibbs random field of particles and unbounded spins. J. Math. Phys. 20, 1979, 1752-1758.
7. Del Grosso G. On the local central limit theorem for Gibbs processes. Commun. Math. Phys. 37, 1974, 141-160.
8. Dobrushin R.L., Tirozzi B. The Central Limit Theorem and the Problem of Equivalence of Ensembles. Commun. math. Phys. 54, 1977, 173-192.
9. Minlos R.A., Khalfina A.M. Two-dimensional limit theorem for the particle number and energy in the grand canonical ensemble. Izv. Akad. Nauk SSSR 34, 1970, 1173-1191.
10. Kazanchyan T.P. The local limit theorem for the sequences of dependent random variables. Izvestia NAN Armenii 39, 2004, 33-42.
11. Nahapetyan B.S., Petrosyan A.N. Martingale-difference Gibbs random fields and central limit theorem. Ann. Acad. Sci. Fennicae, Ser. A. I. Math., Vol. 17, 1992, 105-110.
12. Nahapetian B. Billingsley-Ibragimov Theorem for martingale-difference random fields and its applications to some models of classical statistical physics. C. R. Acad. Sci. Paris, Vol. 320, 1995, 1539-1544.
ЛОКАЛЬНАЯ ПРЕДЕЛЬНАЯ ТЕОРЕМА ДЛЯ УСЛОВНО НЕЗАВИСИМЫХ
СЛУЧАЙНЫХ ПОЛЕЙ Хачатрян Л.А., Нахапетян Б.С.
В статье приводятся условия, при которых локальная предельная теорема для случайных полей со слабо зависимыми компонентами следует из центральной предельной теоремы. Полученный результат может быть применен к гиббсовским и мартингал -разностным случайным полям.
Ключевые слова: локальная предельная теорема, условная независимость, гиббсовское случайное поле, мартингал -разностное случайное поле.
Дата поступления 18.11.2015.
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