Proceedings of the International Congress of Mathematicians
Vancouver, 1974
Limit Theorems for Dependent Random Variables
Under Various Regularity Conditions
V. Statulevicius
1. Introduction. Let Xt91 = 0, 1, 2, •••, be a sequence of random variables defined
on the probability space {Q9<F9P) with values in Rk9 k ^ 1. In limit theorems of
probability theory asymptotic properties of the distribution P„{A) = P{Zne A}
of the normalized sum
n
y=i
are investigated most often as n -> oo, where An and B~l are a nonrandom vector
and a matrix, respectively.
The following are implied here :
(a) the determination of approximating distributions G{A) for the distribution
PJ4);
(b) the investigation of the accuracy of the approximation Pn{A) — G{A);
(c) the improvement of this accuracy by adding terms of the asymptotic expansion to G{A) ;
(d) the investigation of probabilities with large deviations when A is removed
together with n and Pn{A) -+ 0 when n -• oo.
The following classes of sets A are usually considered : a class B of Borei sets,
and a class E of convex measurable sets. The distance p{Pn9 G) between Pn and G
is also investigated in various metrics.
Similar problems arise in investigating a still more complicated distribution of a
random process Zn{t)9 0 ^ t ^ 1, formed by partial sums S 0 = 0, Si, *••, Sn
in the configuration of a random polygonal line with vertices at the points
{tk9 B-^Sk - Ak))9 k = 0, 1, •••, n910 = 0 g tx ^ ••• ^ tn = 1, or the distribution
of functionals of Z„{t)9 for example, max o<k^nSk a n d so on.
© 1975, Canadian Mathematical Congress
173
174
V. STATULEVICIUS
It stands to reason that we can speak of general theories in these problems only
when the variables X\9 •••, Xn are stochastically independent or, in some sense,
weakly dependent on one another.
At the present time, in the case of independent summands particularly great
attention is paid to the questions (b)—(d). Significant results have been obtained
here, especially in the case offinite-dimensionalXj. Due to the absence of good analytical methods it is much more difficult (excluding (a)) to discover those peculiarities which arise in the case of dependent summands, though limit theorems for the
distribution of sums of dependent random variables become more and more topical
problems in the statistics of random processes, statistical physics, additive number
theory and so on.
We present some results for problems of the types (b)—(d) under various conditions of weak dependence.
2. Conditions of weak dependence. Let SF\ — a{XU9 s ^ u ^ t} denote the
(7-algebra of events generated by the random variables XU9 0 g s ^ u ^ t. L{!F§
stands for the totality of all J^J-measurable random variables withfinitevariance.
The following conditions of weak dependence are usually considered :
(I) strong mixing (SM):
sup
sup
\P{AB) - P{A)P{B)\ = a{s) -» 0
{s -> oo),
(II) full regularity (FR):
sup E[ var \P{B\&$ - P{B)\] = ß{s) - 0
(j - oo),
(III) uniformly strong mixing (USM):
sup
'
sup
[ ^ ) - y ( J » |
;4e.^; PU)>0: B<z&~3
= y (
^
0
^
^
(
^2)
P{A)
(IV) Markov type regularity (RMT):
\P(AB\C) - P(A\C)P(B\C)\ S { ™ | C W s )
(
for all Ae^b,Ce
&§£-\ B e &?+s, where T&s) -• 0 {s -> oo), i = 1, 2;
(V) regularity of correlation functions:
Let E\Xt\k < oo for all t and Sf{tÌ9 •••,**) = r{Xtl9 •••, Xt} be a correlation
function of a random process Xt of thefcthorder, i.e., a simple semi-invariant of a
random vector {Xtl9 •••, Xt). The regularity condition lies in that S^ßfa, ••-, tk)
should be sufficiently small when max(f;- — ff-) -> oo, for instance, in the sense of
the existence of integrals J" ••• jS^{th —9tk) d¥{tÌ9 •••, tk) for all k ^ 1.
The first condition was introduced by M. Rosenblatt (1955, [1]), the second by
A. N. Kolmogorov, the third by I. A. Ibragimov (1959, [2]), the fourth by B. Rjauba
and V. Statulevicius (1962, [3]) and the fifth by V. P. Leonov and A. N. Sirjaev
(1959, [4]).
Analogues of conditions (I)—(III) in terms of conditional expectations were
LIMIT THEOREMS FOR DEPENDENT RANDOM VARIABLES
175
considered in the classical work by S. N. Bernstein [5].
In some works (see, for example, [6], [7]) the asymptotic behavior of PZn is investigated under elegant conditions on P{X;eA |S/-i}, j = 1, 2, •••, n. These conditions, however, are hardly verifiable and they do not separate the properties of
dependence from the individual properties of summands, though such a separation
would be quite desirable.
Instead of condition (III) one may introduce condition (III') : p{s) -• 0 {s -> oo),
where p{s) is a maximal coefficient of the correlation between the past ^ and the
future <Ff+t of the process Xt :
m—**-"*?-">
oXoY
in which the supremum is taken over all random variables l e L ( ^ ) and
YeL&ï+s), and t9 while a here denotes standard deviation. The maximal
correlation coefficient was considered in [8]—[10]. We always havep(,y) ^ 2<p1/2{s).
Similarly it is possible to define the maximal kth order correlation coefficient
where the supremum is taken over all £i e L{^0)9 ÇjeL&fy.j = 2, •••, k - 1,
£he&fi.t with E\Çj\* < co and over all * and Uj S vj9 where 0 g / < w2 ^
v2 < ••• < w*-i g vÄ_i < t + J. The symbol JFis defined as follows: The sign " ^ "
over 7 denotes the centering 7 = 7 - £ 7 ; then
^ 7 i - . . 7 r = J E?7i7 2 -. 7 r _i7 r .
It is evident that p2{s) = p{s). Theorems of large deviations for P{Zn > x} under
the condition pk{s) g k\ Lk exp { - ßn • s} with all A: ^ 2 were proved in [11] and [12].
If the random variables Y\9 •••, 7 r are related to a Markov chain x{t)9 t = 1,
2, •••, r, 7; =/(x(0) with transition probability P,(;c, y4) and initial distribution
PY{A)9 then
# 7 ? ... 7 - = J ... U^i)p(dxl)ÜMxJ){Pj{xj-l9
dxf) - Py(Äy))
y—2
where Pt{A) = P { x ( 0 e ^ } .
Let us consider the kth order correlation among the indicators IAl of sets
4 e ^ £ > J = 2> ••'» * - 1> Aie&b, Ake&?+S. Assume that a{Al9 •», ^Ä) =
EIAì '" IAi. Let us say that the condition {ak) is satisfied if
sup \a{Ai9 •••, Ak)\ = ak{s) -+ 0
(J -> oo)
<i «/éy/i At
as well as the condition {(pk) if
sup
k^i,-y,^)l
t.«^vitA«P(AU>0
P{Ai)
=
(j)
™W
^
0
(
j
V
'
There exists a positive constant ck9 depending only on k9 such that
176
V. STATULEVICIUS
Pk{s) â ckq>yk{s).
The properties of a{AÌ9 •••, Ak) were studied by Kolmogorov and Zurbenko
[13]. The conditions (a*) and {(pk) were applied in [35] in investigating the accuracy
of the approximation of Pn{A) by a normal distribution.
Note that (a2) coincides with condition (I) and {<p2) with (III). When k increases
the conditions {ak) and {<pk) become more and more restrictive, and transform into
RMT I and RMT II, respectively.
We shall not linger on the conditions for convergence of the distributions
PZn and PZtt(-) to limit ones. We shall only direct the reader to the papers [1] —
[41], where one can also find a rather complete bibliography. We shall only illustrate an interesting result by Ibragimov [14]. Assume that Xt is a strictly stationary sequence with EXt = 0 and p{s) -> 0 as s -> oo. Then either s\xpna2S„ < oo or
o2Sn = n-h{n)9 where h{n) is a slowly varying Karamata function. If in addition
E\Xt\2+d < oo for some ö > 0 and o2Sn -> oo {n -> oo), then
lim P {SJaSn < x} = 0{x)9
»—oo
—oo
3. Large deviations.
THEOREM
1. If the condition RMT II with
r&) ^ e~r''s>
(1)
?n > 0,
is satisfied, Xj e R9 EXj = 0, and
E{\XPj\ l ^ r 1 } ^p\LKP~2a2XJ9
j = 1, - . , « ,
with probability 1 for all p ^ 3, then for Zn = SJBn9 B2 = o2Sn9 the relation of
large deviations
(2)
P n
i- 0(X)}
= ex
P«*3/4,)W4,)} (i + eH.3 ^+A)
holds when 0 ^ x S S-An9 ö < öff9 where 0ff9d is bounded by a constant depending
only on H and 59
oo
1
m^zhj",
* = o,i,-.
I^I=IHW'
Here
â
=
CKL-T„-B„
H=CKr
L
? = 1 °2X'
The positive constants cKtL and CKtL depend only on K and L9 and 5H > 0 is a
maximal root of the equation 6HdH/{l - dH)z = 1 (estimation for 0S,H can be
found in [15]).
in
REMARK 1. If \Xj\ g C \j = 1, ••-,«, with probability 1, then in this case
LIMIT THEOREMS FOR DEPENDENT RANDOM VARIABLES
177
(2) holds when An = {Tn-Bn)l{HrC™)9
H = 2Hh where H1 > 0 and H2 > 0
are absolute constants included by the estimate for the semi-invariant /7Ä{5'B} of
the M L order of the sum Sn:
(3)
I^JliMiag!^,
, = 3,4,..,
REMARK 2. If Xt91 e (— oo, oo), is a strictly stationary process, E\ Xt | * < oo for
all & ^ 1 and
for allfc^ 3, then for P { Z r > x}9 where Z T = (Cr - E^T)/(T^T9 Çr = lïXt dt>
the relation of large deviations (2) is valid when âT = cTl/2/H^9 H = H^jc29
îîOKT'Z
CT (see [11]).
The method of proving such theorems is as follows. The kth order semi-invariant
of the sum
r„{sn}= s
r{xtl,-,xu}
l£tu"',U£n
may be exactly expressed in terms of EXth ••• XUm9 m ^ k. When \Xj\ ^ C (w) ,
j = 1, ••-,#, and (1) is true the estimation
\ÊXtll - Xtlm\ <; C-C<->«-» exp{-r„-(* y . - g } r f ( A . ^ y ,
holds if tjt ^ ••• ^ (,-„, where C is an absolute constant. Hence we get estimates of
the type (3) for rk{s'H}. Further, if \rk{Z] | g k\ H\Ak~2 for all k^39EZ
= 0,
a 2 Z = 1, then (2) is valid for P{Z > x} with the parameters H and J (see [15]).
Estimates for r{Xh9 •••, Xtt} under the RMT II conditions were obtained by
I. G. Zurbenko [27], [28] as well.
4. Rate of convergence and asymptotic expansion.
2. IfXj e R*9 EXf = 0,y = 1, ••«, n9 and RMT II wiYA ï2{s) S l/{T%-sa)
is satisfiedfor all I ^ s ^ nandsomea > 39T„> 0, iS? | A"y | 3 < 00,7 = 1, ---9n9 then
THEOREM
(4)
Sup\P{SttzA}-0sXA)\£C(a,k)
sup ^
"
?
f
feffl^>.
Here (#, >>) denotes the scalar product, and C{a9 k) depends only on k and a.
THEOREM 3. IfXj e R9 EXS = 0, E\ Xj\* < 00,7 = 1, •••, n, and RMT II with
T2{s) ^ e~r"'sis satisfied, then there exists an absolute constant Ci such that
sup
'{^r^Hirj-'"*
0
1 max. ^ • ^ e s s s u p J g d Z y l 3 ! ^ - 1 } ,
B2 =
2S
SC,TnBn minj
i ^ e s s i n f ^ l ^ f 1 }
In fact condition RMT II is necessary in order to obtain the theorems of large
deviations such as Theorem 1 (there is an example indicating that relation (2) is not
178
V. STATULEVICIUS
valid without RMT). To obtain (4), however, it is not required. Further, let
XjGR9j = 1, •-•, n9 and
L - L%i^supE{\Xj\r\^^}
1
THEOREM
B2 =
<J2S
D
n
n
4. If the condition (p3) with
(5)
f&) * yj-r,
l*s
Zn,
is satisfied for some a > 3, E\ Xj\* < oo9j = 1, •••, n9 then there exists a constant
Ca depending only on a such that
sup|P{Z B < x} - 0{x)\ £ Ca-L3n.
x
THEOREM
5. //I^y] ^ C(w), j = 1, •••, n9 with probability 1 and the condition
(0:3) with
(6)
a&) i 1/rj-J-,
1 ^ s ^ n,
is satisfiedfor some a > 3 then there exists a constant C'a such that
sup \P{Z„ < x} - 0(X) I * C
a ( x n
^\
|x|y
If we want to get an asymptotic expansion for P {Zn < x} — 0{x) with the help
of Cebysev-Hermite polynomials
P{Zn <x} = 0{x) +
(2J)1/2
e-*l/2r£pvn{x) + 6rLrn
with as simple a structure of the remainder term as that of Lrn9 the condition {<pr)
will be needed here if £r|Ar/|r < 00 and {ar) when \Xj\ g Cin),j = 1, •••, n (see
[35]). Note that in Theorems 2-5 the conditions for T2{s)9 <pr{s)9 ar{s) are not best
possible; they can be weakened. And in general in asymptotic expansions one can
get estimates of the order (log n)/{nl/2)r"2 by imposing stronger restrictions on
<pm{s), m <r9 but it will be very difficult to describe the structure of the remainder
term exactly. For instance, if <p{s) = <p2{s) ^ C3e~b's9 E\Xj\3 g C4, j = 1, •••, n,
then it is possible to obtain the estimate
P{Zn <x} = 0{x) + 0{{logn)/n"2)
when a2 {St — Sk} M / - k9 1 ^ k < I ^ n. The rate of convergence under the
condition (III) was dealt with in [22], [32], [29].
In the remainder terms of Theorems 2-5 the conditional moments E {\Xj |3| ^f1}
can be replaced by absolute ones 23ÌX,|3 by multiplying the remainder terms by
log n. The method of proof rests on the accurate investigation of logarithmic
derivatives
r»{Smt} = -£• lOg/5.«
of the characteristic function of the sum S„ when \t\ £ Lj£/(r~~2'iJ%1.
LIMIT THEOREMS FOR DEPENDENT RANDOM VARIABLES
179
We shall present here one more theorem which employs the regularity conditions
(V).
Let Xt be a strictly stationary random sequence with EX( = 0 and E\X$ \ < co
for all/? ^ 1. Let F{X) = $f{X) dX be a spectral function of the process Xt with
bounded density sup^/(^) ^ L. Let
be the estimation for F{X).
THEOREM 6. If
2 - ZW('i, -,'*-i,0)| ^ k\H^m~29
k—i
then
sup p { T i n
X
<?"§ ^TW ~ F^\ -
ffT x
' }~ p{ IUP K')l ^ *}
< c
iog(i + r)
where w(j) is a Wiener process, a\ = 2% \1\2f{l)dX.
Estimations of such a type for the Gaussian sequence were obtained by T. Arak
[37].
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INSTITUTE OF PHYSICS AND MATHEMATICS, VILNIUS STATE UNIVERSITY
VILNIUS, U.S.S.R.