CC BY
9
DOI: 10.17516/1997-1397-2022-15-3-281-291 УДК 519.21
Central Limit Theorem for Weakly Dependent Random Variables with Values in D [0,1]
Olimjon Sh. Sharipov*
National University of Uzbekistan Tashkent, Uzbekistan V. I. Romanovskiy Institute of Mathematics Academy of Sciences of Uzbekistan Tashkent, Uzbekistan
Anvar F. Norjigitov"
Academy of Sciences of Uzbekistan Tashkent, Uzbekistan
Received 03.10.2021, received in revised form 04.12.2021, accepted 09.02.2022 Abstract. The main goal of this article is to prove the central limit theorem for sequences of random variables with values in the space D [0,1]. We assume that the sequence satisfies the mixing conditions. In the paper the central limit theorems for sequences with strong mixing and pm-mixing are proved.
Keywords: central limit theorem, mixing sequence, D [0,1] space.
Citation: O.Sh. Sharipov, A.F. Norjigitov, Central Limit Theorem for Weakly Dependent Random Variables with Values in D [0,1], J. Sib. Fed. Univ. Math. Phys., 2022, 15(3), 281-291. DOI: 10.17516/1997-1397-2022-15-3-281-291.
1. Preliminaries
A central limit theorem for Banach space valued dependent random variables have been studied by many authors (see [6,11,15-17] and references therein). It is known that validity of the central limit theorem depends on the geometric structure of Banach space. One of the most difficult space in this sense is D [0,1] (the space of all real-valued functions that are right continuous and have left limits, which is endowed with the Skorohod topology) space. In this paper we will prove the central limit theorem for mixing random variables with values in D [0,1].
Let {Xn(t), t e [0,1], n > 1} be a sequence of random variables with values in D [0,1]. We say that {Xn(t), t e [0,1], n > 1} satisfies a central limit theorem if the distribution of
(Xi(t) + ... + Xn(t)) converges weakly to a Gaussian distribution in D [0,1].
n
The central limit theorem in D [0,1] is very important from applications point of view. It immediately implies asymptotic normality of empirical and weighted empirical processes. The central limit theorem for the sequence of independent identically distributed (i.i.d) random variables with values in D [0,1] were studied by many authors (see [?, 1, 2, 8,12]) and references therein).
*[email protected] [email protected] © Siberian Federal University. All rights reserved
The first central limit theorem was proved by Hahn [8]. Later the central limit theorem in D [0,1] was proved by D. JukneviCiene (1985), V. Paulauskas and Ch. Stive (1990), P.H. Bezandry and X. Fernique (1992), M.Bloznelis and V. Paulauskas (1993), X.Fernique (1994). The result of M.G.Hahn [8] can be formulated as follows.
Theorem 1.1 ([8]). Let {Xn(t), t £ [0,1], n ^ 1} be a sequence of i.i.d. random variables with values in D [0,1] such that
EXi(t) = 0, EXl(t) < to for all t £ [0,1] . (1)
Assume that there exist nondecreasing continuous functions G and F on [0,1] and numbers a > 0.5, 3 > 1 such that for all 0 ^ s ^ t ^ u ^ 1 the following two conditions hold:
E (Xi(u) - Xi(t))2 < (G(u) - G(t))a,
E (Xi(u) - Xi(t))2 (Xi(t) - Xi(s))2 < (F (u) - F (s)f . (2)
Then {Xn(t), t £ [0,1], n ^ 1} satisfies the central limit theorem in D [0,1] and the limiting Gaussian process is sample continuous.
As it was already noticed in [2], the condition (2) is connected with the fourth moments of the process Xi(t). This conditions does not allow us to apply Theorem 1.1 to a wide class of weighted empirical processes. In [2] and [3] authors obtained the following results (where a A b denotes min(a, b)):
Theorem 1.2 ([2]). Let {Xn(t), t £ [0,1], n ^ 1} be a sequence of i.i.d. random variables with values in D [0,1] satisfying the condition (1) and assume that there exist nondecreasing continuous functions G and F on [0,1] and numbers a, 3 > 0 such that for all 0 ^ s ^ t ^ u ^ 1 the following two conditions hold:
E (Xi(u) - Xi(t))2 < (G(u) - G(t))i/2 log-4'5-a (1 + (G(u) - G(t))~i) , (3)
E (|Xi(t) - Xi(s)| A 1)2 (Xi(u) - Xi(t))2 <
^ (F(u) - F(s)) log-- (1 + (F(u) - F(s)ri) . (4)
Then {Xn(t), t £ [0,1], n ^ 1} satisfies the central limit theorem in D [0,1] and the limiting Gaussian process is sample continuous.
Theorem 1.3 ([2]). The statement of Theorem 1.2 remains true if conditions (3) and (4) are replaced by
E (Xi(u) - Xi(t))2 < (G(u) - G(t))i/2 log-2'5-« (1 + (G(u) - G(t))~i) , (5)
E (Xi(t) - Xi(s))2 (Xi(u) - Xi(t))2 < (F(u) - F(s))log-— (1 + (F(u) - F(s))-) . (6)
Theorem 1.4 ([3]). Assume p, q ^ 2. Let f,g be nonnegative functions on [0, which are nondecreasing near 0 and let F, G be increasing continuous functions on [0,1]. Let X(t) be a random process with mean 0, finite second moment, and sample path in D satisfying
E (IX(s) - X(t)| A IX(t) - X(u)|f < f (F(u) - F(s)),
E \X(s) - X(t)\q < g (G(t) — G(s)), for 0 ^ s ^ t ^ u ^ 1, u — s small and
I f 1/p(u) • u-1-1/pdu< x, I g1/q(u) • u-1-1/(2q)du < x.
J 0 J 0
Then {Xn(t), t £ [0,1], n ^ 1} satisfies the central limit theorem in D [0,1] and the limiting Gaussian process is sample continuous.
2. Main results
The main goal of this article is to prove the central limit theorem for mixing sequences of random variables with values in space D [0,1].
Below, we give the definitions of mixing coefficients for a sequence of random variables with values in a separable Banach space B. In Definition 2 it is assumed that B is an infinite-dimensional space.
Definition 1. For a sequence {Xn(t), t £ [0,1], n ^ 1} the coefficients of p, a-mixing are defined by the following equalities.
\E(g - Eg)(n - En)\ . ^ LF, E2 (g - Eg)2E2 (n - En)2
P(n) = suH 1 ,2 : g G L2(Fkk), n G L2(F?+k), k G N
a (n) = sup {\P(AB) - P(A)P(B)\ : A G F*, B G F^+n, k G N} .
where is the a-algebra generated by random processes Xa(t),... ,Xb(t) and L2(F%) is the space of all square integrable random variables measurable with respect to
Definition 2. For the sequence {Xn(t), t G [0,1], n ^ 1} the coefficients of pm (n)-mixing and am (n)-mixing are defined by the following equalities
Pm (n) = supsup{ JE(g - Eg)(n- Enl : g G L2(Fk(Rm)),n G L2(F^+k(Rm)), k G n} , Rm {E 2 (g - Eg)2E 2 (n - En)2 )
am (n) = sup sup {\P (AB) - P (A)P (B)\ : A G Ff(Rm), B G F^+n (Rm), k G N} ,
Rm
-»m\
where Fb(Rm) is the a-algebra generated by random processes \\m Xa(t), ■ ■ ■, üm X(t) and Urn is a projection operator B in m-dimensional subspace Rm i.e. YIm : B ^ Rm■ A sequence is called p-mixing (or pm—, a—, am— mixing ) if
p(k) ^ 0 as k ^ <, (7)
pm(k) ^ 0 as k ^ < and m = 1, 2, ■ ■ ■, (8)
a(k) ^ 0 as k ^ <, (9)
am(k) ^ 0 as k ^ < and m = 1, 2, ■ ■ ■ (10)
respectively.
As the example given in Zhurbenko [13] shows, in general (8) does not imply (7), though (7) always implies (8), the same is true with (9) and (10). In (8) and (10) it is actually required that all finite-dimensional projections of the sequence {Xn(t), t e [0,1], n > 1} satisfy the mixing condition and these conditions are weaker than the conditions (7) and (9).
Set Sn(t) = (Xi(t) + • • • + Xn(t)) and in what follows ^ denotes weak convergence. n
Now we formulate our theorems.
Theorem 2.1. Let {Xn(t), t e [0,1], n ^ 1} be a strictly stationary sequence of pm-mixing random variables with values in D [0,1] such that
EX1(t) = 0, E |X1(t)|2 < то for all t e [0,1] .
Assume that there exists a nondecreasing continuous function F on [0,1] such that for all 0 ^ s ^ t ^ 1 and £ > 0 the following hold:
E (Xi(t) - Xi(s))2 < (F(t) - F(s)) log-(3+£) (1 + (F(t) - F(s))-1) , (11)
lim E (X1 + • • • + Xn)2 = то for all t e [0,1],
п^ж
n
(2k) < то, m = 1, 2,... .
k=i
Then {Xn(t), t e [0,1], n ^ 1} satisfies the central limit theorem i.e.
Sn(t) ^ N(t) as n ^ то and the limiting mean-zero, sample continuous Gaussian process has the covariance function: F (ti,t2)= lim ESn(ti)Sn(t2), ti,t2 e [0,1].
Theorem 2.2. Let {Xn(t), t e [0,1], n ^ 1} be a strictly stationary sequence of pm-mixing random variables with values in D [0,1] such that
EXi(t) = 0, E |Xi(t)|2+e < то, for all t e [0,1] and some £> 0.
Assume that there exists a nondecreasing continuous function F on [0, 1] such that for all 0 ^ s ^ t ^ 1 and the following hold:
E |Xi(s) - Xi(t)|2+e < (F(s) - F(t)) log-(3+2e) (1 + (F(s) - F(t))-i) , (12)
(2k) < то, m = 1, 2,....
k=i
Then {Xn(t), t e [0,1], n ^ 1} satisfies the central limit theorem i.e.
Sn(t) ^ N(t) as n ^ то and the limiting mean-zero, sample continuous Gaussian process has the covariance function: F (ti,t2)= lim ESn(ti)Sn(t2), ti,t2 e [0,1].
Theorem 2.3. Let {Xn(t), t £ [0,1], n ^ 1} be a strictly stationary sequence of am-mixing random variables with values in D [0,1] such that
EXi(t) = 0, E |Xi(t)\2+S < to, for all t £ [0,1] and some S> 0.
Assume that there exists a nondecreasing continuous function F on [0,1] such that for all 0 ^ s ^ t ^ 1 and £ > 0 the following hold:
2 + e
(e |Xi(t) - Xi(s)|2+^ 2+5 < (F (t) - F (s)) log-(3+2£) (1 + (F (t) - F (s))-1) , (13) 22am+s (k) < to, m = 1, 2,....
Then {Xn(t), t £ [0,1], n ^ 1} satisfies the central limit theorem i.e.
Sn(t) ^ N(t) as n ^ to and the limiting mean-zero, sample continuous Gaussian process has the covariance function: F (ti,t2)= lim ESn(ti)Sn(t2), ti,t2 £ [0,1]. Theorems 2.1-2.2 improve the results of [11].
3. Preliminary results
The proofs of the theorems are based on the following lemmas.
Lemma 1 ([2]). Let Xi(t),X2(t),... ,Xn(t),... be random variables with values in D [0,1]. Assume that there exist a nondecreasing continuous function F on [0,1] and positive numbers Yi, ci, £i such that for all X > 0 and 0 ^ s ^ t ^ u ^ 1.
P (\Xn (t) - Xn (s)| A \Xn (u) - Xn (t)| > X) < ciX-2^327l + i+£l (F(u) - F(s)),
where gp(u) = u |log u|-p ,p> 0. If for all ti ,...,tk £ [0,1], k = 1, 2,...
(Xn(ti),...,Xn(tfc)) ^ (X (ti),...,X (tfc)) as n ^to
and
P (X (1) = lim X (t)) = 1.
Then Xn =>■ X as n to.
Lemma 2 ([9]). Let {Xi, i ^ 1} be a strictly stationary sequence of real valued random variables with p-mixing and
EXi = 0, EX2 < to, lim E (Xi + • • • + Xn)2 = to,
n
Ep(2fc) < to.
k=i
Then
-^(Xi + ••• + Xn) ^ N (0,CT2) as n ^x,
where N (0, ct2) Gaussion random variable with zero-mean and variance
a2 = lim -E (X1 + ••• + Xn)2 > 0.
n^TO n
Lemma 3 ([14]). Let {Xi, i ^ 1} be a sequence of real-valued random variables with p-mixing and for some q ^ 2
EX1 =0, E\X1 \q < x,
n
£p_(2") < x.
k=1
Then there exists a constant K such that the following inequality holds:
/ q
E\ X1 + ••• + Xn \q < Kin _ max (e\ Xi \2) 2 + n max E\ Xi \q
Lemma 4 ([13]). Let {Xi, i ^ 1} be a stationary sequence of random variables with a-mixing and
EX1 =0, E\X1 \2+3 < x,
to _+_
y^a2+_ (k) < X,
k=1
for some 6 > 0. Then
ct2 = EX2 + 2 £ E(X1Xj) < x when ct2 > 0,
j=2
(X1 + ••• + Xn) ^ N (0,1) as n ^x.
Lemma 5 ([4]). Let {Xi, i ^ 1} be a strictly stationary sequence of random variables with a-mixing and
EX1 =0, E\X1 \2+3 < x,
to
Et_. _+_-t n2 1a _+_ (k) < X,
k=1
for some 0 <6 ^ x and 2 ^ t < 2 + 6. Then
E
E(Xk — m)
k=1
< cn _ E\ X1 \ 2+sy+s
CTa n
4. Proof of Theorems
Proof of Theorem 2.1. We will use the method developed in the papers [2,8] and [12]. It follows from Lemma 1 that it suffices to prove
P (|Sn (t) - Sn (s)| A \Sn (u) - Sn (t)| > A) < c1\-(2+£)g3+e (F (u) - F (s)),
where A G (0,1], 0 < s < t < u < 1.
It is easy to see that the following inequality holds for A G (0,1].
P (|Sn (t) - Sn (s)| A \Sn (u) - Sn (t)| > A) < P (|Sn (t) - Sn (s)| \Sn (u) - Sn (t)| > A2) . We have
J = Sn (t) - Sn (s)||Sn (u) - Sn (t)| =
n-(Xk (t) - Xk (s))
2
k=1
\ 2
1 i 1 _ „ , 1
n-(Xk (u) - Xk (t))
k=1
<
^ 2 2 £ (Xk (t) - Xk (s))J 2 E (Xk (u) - Xk (t))J = Ji + J2.
In what follows we denote by C the constants (possibly depending on different parameters) which can be different even in the same chain of inequalities. We have
2) 1 2 1 2
P (J > A2) < p(ji > IA2^ + pJ > 2•
2" j ' ' 2'
We evaluate each of the summands individually. Using the Markov inequality and Lemma 3, we obtain
2
2
(14)
P Ji > A2) = P ^ (^n-2 ]=1 (Xk (t) - Xk (s))J > A2J < < A-2E |n-1 E (Xk (t) - Xk (s)) j < A-2CE (Xi (t) - Xi (s))2 ,
P ( J > A2) < A-2CE (Xi (u) - Xi (t))2 • (15)
(u
From (14) and (15) we get
\2\ ^ f+\ v /„W2 i \—2
P(J > A2) < A-2CE (Xi (t) - Xi (s))2 + A-2CE (Xi (u) - Xi (t))2 • From the conditions of Theorem 2.1
P (K (t) - Sn (s)| \Sn (u) - Sn (t)| > A2) < < A-2CE (Xi (t) - Xi (s))2 + A-2CE (Xi (u) - Xi (t))2 < < A-2C (F(t) - F(s)) log-(3+e) (l + (F(t) - F(s))-i) +
+A-2C (F(u) - F(t)) log-(3+e) (l + (F(u) - F(t))-i) < < 2A-2C (F(u) - F(s))log-(3+e) (l + (F(u) - F(s))-i) < 2CA-2g3+E (F(u) - F(s)).
Above we used the inequality
for
log-1 (1 + (F(u) - F(s))-1) < 2 | log (F(u) - F(s))|
F(u) - F(s) < 0.25.
1
(16)
Now, to complete the proof of the theorem, it remains to prove the convergence of the finite-dimensional distributions Sn (t). The convergence of finite-dimensional distributions follows from Lemma 2 and the Cramer-Wold device [5]. Thus, Theorems 2.1 is proved. □
Proof of Theorem 2.2.
We will prove Theorem 2.2 by the same method as Theorem 2.1. It follows from Lemma 1 that it suffices to prove
P ( |S„ (t) - S„ (s) | A | Sn (u) - Sn (t) | > A) < C1 X-(2+E)g3+2s (F (u) - F (s)),
where A e (0,1], 0 < s < t < u < 1.
It is easy to see that the following inequality holds for A e (0,1].
/ ) / 2 + e 2 + e „
P( Sn (t)-Sn (s)| A Sn (u)-Sn (t)| > A < P( Sn (t)-Sn (s)| ~ Sn (u)-Sn (t)|~ > A2+i We have
2 + e 2 + e
I = Sn (t) - Sn (s)|— Sn (u) - Sn (t)|— =
1
< -2
(Xk(t) - Xfc(s))
* k=1 n
-n^ (Xk(t) - Xk(s))
2+e
2 + e 2
(Xk (u) - Xk (t))
* k=1 1n
-n^ (Xk(u) - Xk(t))
2 + e 2
<
2+e
I1 + I2.
We have
P (I > A2+e) < P (h > 2A2+^ + P (l2 > 2A2+e) .
Using the Markov inequality and Lemma 3, we obtain
1n
P(I1 > A2+^ = P^
< A-(2+e)E
(Xk(t) - Xk(s))
2+e
^ A2+M <
2+e
<
(Xk(t) - Xk(s))
< CA-(2+e)n-(2+e)l2n(2+e)'2 (E |X1(t) - X1 (s) |2) '^^^ + +CA-(2+e)n-(2+e)/2nE X1(t) - X1(s)f+e < < A-(2+e)C (e X1(t) - X1(s)|2)(2+e)/2 + A-(2+e)Cn-e/2E X1(t) - X1(s)|2+e < < 2CA-(2+e)E X1(t) - X1(s)|2+e . P (I2 > A2+e) < 2CA-(2+e)E X^y) - X1(t)|2+e .
From the conditions of Theorem 2.2 and using (16) we have
/ 2 + e 2 + e \
P (|Sn (t) - Sn (s)|— \Sn (u) - Sn (t)|~ > A2+£J <
< CA-(2+e) (F (t) - F (s)) log-(3+2e) (i + (F (t) - F (s))-1) + +CA-(2+£) (f (u) - F (t)) log-(3+2e)( i + (F (u) - F (t))-1) <
< 2CA-(2+e) (F (u) - F (s)) log-(3+2e) (i + (F (u) - F (s))-1) < 2CA-(2+e Ws (F (u) - F (s)).
To complete the proof of the theorem, it remains to prove the convergence of the finite-dimensional distributions Sn (t). The convergence of finite-dimensional distributions follows from Lemma 2 and the Cramer-Wold device [5]. Thus, Theorems 2.2 is proved. □
Proof of Theorem 2.3.
To prove Theorem 2.3, we estimate I as in the proof of Theorem 2.2 by I1 and I2. Using the Markov inequality and Lemma 5, we have (where e + e1 = S, e1 > 0)
P (h > A2+e) = P
(Xk(t) - Xk(s))
V k = l
2+e
^ A2+e I <
1 " 2+e < Ca (k) A-(2+e^-1+TE
2 + e
n 2
<
E (Xk(t) - Xk(s))
k=1
2 + e
< ca-(2+£) (e\X1(t) - X1(s)|2+e+^ 2+e+e1 <
2 + e
< CA-(2+e) (e |X1(t) - X1(s)|2+^ 2+' .
2 + e
P (I2 > A2+s) = CA-(2+s) (e |X1(u) - X1(t)|2+^ 2+* .
// \ 2+e / s 2+e \
P(I > A2) < (e |X1(t) - X1(s)|2+^ 2+* + (E |X1(u) - X1(t)|2+^ 2+^ .
From the conditions of Theorem 2.3 and using (16) we have
/ 2 + e 2 + e „ \
P (>„ (t) - Sn (s)|~ Sn (u) - Sn (t)|^ > A2+eJ <
< CA-(2+£) (F (t) - F (s)) log-(3+2e) (i + (F (t) - F (s))-1) + +CA-(2+£) (F (u) - F (t))log-(3+2e) (i + (F (u) - F (t))-1) <
< 2CA-(2+e) (F (u) - F (s)) log-(3+2e) (i + (F (u) - F (s))-1) < 2CA-(2+e)ff3+2£ (F (u) - F (s)).
Again as in the proof of previous theorems, to complete the proof of the theorem, it remains to prove the convergence of the finite-dimensional distributions Sn (t). The convergence of finite-dimensional distributions follows from Lemma 4 and the Cramer-Wold device [5]. Thus, Theorems 2.3 is proved. □
References
[1] P.H.Bezandry, X.Fernique, Sur la propriete de la limite centrale dans D [0,1], Ann. Inst. H. Poincare, 28(1992), 31-46.
[2] M.Bloznelis, V.Paulauskas, On the central limit theorem in D [0,1], Stat. Prob. Lett., 17(1993), 105-111.
[3] M.Bloznelis, V.Paulauskas, A note on the central limit theorem for stochastically continuous processes, Stochastic Processes and their Applications, 53(1994), 351-361.
[4] D.Bosq, Linear processes in function spaces. Lecture Notes in Statistics, Vol. 149, New York, Springer-Verlag, 2000.
[5] P.Billingsley, Convergence of Probability Measures, Wiley, New York, 1968.
[6] H.Dehling, O.Sh. Sharipov, M.Wendler, Bootstrap for dependent Hilbert space-valued random variables with application to von Mises statistics, Journal of Multivariate analysis, 133(2015), 200-215.
[7] X.Fernique, Les functions aleatoires cadlag, la compacite de laurs lois, Liet. Mat. Rink., 34(1994), 288-306.
[8] M.G.Hahn, Central limit theorems in D [0,1], Z. Wahrsch. Verw. Gebiete., 44(1978), 89-101.
[9] I.A.Ibragimov, A Remark on the Central Limit Theorem for Dependent Random Variables, Theor. Prob. Appl., 20(1975), 134-140.
10] D.Jukneviciene, Central limit theorem in the space D [0,1], Lithuanian Math. J., 25(1985), 293-298.
11] A.F.Norjigitov, O.Sh.Sharipov, The central limit theorem for weakly dependent random variables with values in the space D [0,1], Uzbek Math. J., 2(2010), 103-112 (in Russian).
12] V.Paulauskas, Ch.Stieve, On the central limit theorem in D [0,1] and D([0,1] ,H), Lithuanian Math. J, 30(1990), 267-276.
13] I.G.Zhurbenko, On the conditions for mixing random processes with values in Hilbert spaces, Dokl. AN USSR., 278(1984), no. 4, 792-796.
14] Q.M.Shao, Maximal inequalities for partial sums of p-mixing sequences, Ann. Prob., 23(1995), 948-965.
15] O.Sh.Sharipov, On probabilistic characterization of Banach space by weakly dependent random elements, Uzbek Math. J., 2(1998), 86-91 (in Russian).
16] O.Sh.Sharipov, A central limit theorem for weakly dependent random variables with values in co space, In: Modern problems of Probability theory and Mathematical statistics, Tashkent. "FAN", 2000, 193-197 (in Russian).
17] O.Sh.Sharipov, A.F.Norjigitov, Central limit theorem for weakly dependent random variables and autoregressive processes with values in C [0,1], Uzbek Math. J., 2(2016), 149-157 (in Russian).
Центральная предельная теорема для слабо зависимых случайных величин со значениями в Б [0,1]
Олимжон Ш. Шарипов
Национальный университет Узбекистана Ташкент, Узбекистан Математический институт им. В. И. Романовского АНУз
Ташкент, Узбекистан
Анвар Ф. Норжигитов
Математический институт им. В. И. Романовского АНУз
Ташкент, Узбекистан
Аннотация. Основной целью настоящей статьи является доказательство центральной предельной теоремы для последовательностей случайных величин со значениями в пространстве Б [0,1]. Мы предполагаем, что последовательность удовлетворяет условиям перемешивания. В статье доказаны центральные предельные теоремы для последовательностей с сильным перемешиванием и рт-перемешиванием.
Ключевые слова: центральная предельная теорема, последовательность с перемешиванием, пространство Б [0,1].
