Moderate deviations results for martingales
using Lindeberg's method
Ion Grama
Institute of Mathematics and Informatics, Academy of Sciences
Academiei str. 5, MD-2002 Kishinev, Moldova
Erich Haeusler
Institute of Mathematics, University of Giessen
Arndtstr. 2, Giessen, Germany
In 1922 Lindeberg gave a new elegant and illustrative proof of the central limit theorem
for sums of independent random variables and in 1944 Bergström proved a result on the rate
of convergence in the central limit theorem via this idea. Since then the approach became
known under the name "method of compositions". It turned out to be a very useful tool for
establishing exact rates of convergence in central limit theorems in various situations, especially
in the multivariate case. In the present talk we shall show how Lindeberg's approach can be
used in an elementary fashion for establishing results on moderate and large deviations for
sums of dependent random variables. Contributions and renements for the case of univariate
martingale dierences can be found in Bolthausen (1982) and Haeusler (1988), Grama (1993,
1997). We shall consider martingale dierences with values in Rd ; d 1: Our exposition follows
Grama and Haeusler (1998).
For the sake of simplicity we start with the univariate case d = 1: Let (nk ; Fnk ) kn be
a square integrable martingale dierence sequence with values in R : We agree that n = 0
and Fn = f;; g: Set
k
X
n
Xk = ni; 1 k n:
1
0
0
0
i=1
Denote by hX i the quadratic characteristic of the martingale X n; i.e.
n
hX nik =
k
X
i=1
2
ani ; ank = E nk
jFn;k,1 ; 1 k n:
We also agree that X n = 0 and hX ni = 0: Set for any > 0
0
0
Ln =
n
X
i=1
E jni j2+2 ; Nn = E jhX nin , 1j1+ :
Theorem 1. Assume that 2 (0; 1=2] and > 0: Let r = r(x) be the solution of the
equation
n
o
x = (1 + r)3+2 exp r2 =2 ; x 1:
Then, uniformly in x such that 1 x (Ln + Nn), ;
1
P (Xnn r)
=
1
+
C () fx(Ln + Nn )g1=(3+2) ;
1 , (r)
where jj 1: Here C = C () is a constant depending only on .
Note that for r we have the asymptotic expansion
2
q
r = r (x) = 2 log x , 2 (3 + 2) log 1 + 2 log x + ::: ;
2
2
as x ! 1:
The proof of the Theorem 1 is based upon the following result on the rate of convergence
in the central limit theorem for martingales which is proved readily using Lindeberg's approach.
Lemma 2. Let 2 (0; ]: For any r 0 and " > 0
1
2
n
o
jP (Xnn r) , (1 , (r))j C ", , (Ln + Nn) + " exp(,r =2 + "r) ;
2
2
2
where C is a constant.
Now we turn to the multivariate case. For any integer d 1; let Rd be the space of all
column vectors x with transpose x0 = (x ; :::;Px d ); where x i 2 R ; 1 i d. Let (x; y) be
the usual scalar product
in Rd; i.e. (x; y) = di x i y i ; and let jjxjj be the usual Euclidean
q
norm, i.e. jjxjj = (x; x): Assume now that the martingale dierence sequence (nk ; Fnk ) kn
takes values in Rd : Denote by hX ni the quadratic characteristic of X n; i.e.
(1)
( )
=1
( )
( )
1
( )
0
hX n i k =
k
X
0 jF
ani; ank = E (nk nk
1 k n:
n;k , ) ;
1
i=1
We again agree that X n = 0 and hX ni = 0: Let N ;V be a d-dimensional normal random vector
with mean 0 and covariance matrix V: We require that V is non-negative denite, with at least
one positive eigenvalue. Let > ::: > p be the strictly positive eigenvalues of the matrix
V and let ; :::; p be their corresponding multiplicities. Set, for brevity, = ( ; :::; p) and
= ( ; :::; p):
The appropriate norm for the variance term hX ninn , V is the trace norm, which, for
any symmetric d d matrix U; with eigenvalues ; :::; d; is given by
0
0
0
1
1
1
1
1
d
d
p
X
X
kU ktr = tr U = ji j = sup j(ei ; Uei)j ;
2
i=1
i=1
where trA denotes the trace of the matrix A and where the supremum is taken over all orthonormal bases e ; :::; ed in Rd: Set for any > 0
1
L =
n;d
n
X
i=1
E knik
2+2
; Nn;d = E khX nin , V ktr :
1+
The following result shows that the range of moderate deviations in the multivariate case
depends heavily on the largest eigenvalue of the matrix V and its multiplicity :
Theorem 3. Assume that 2 (0; 1=2] and > 0: Let r = r(x) be the solution of the
equation
( )
,1
x = (1 + r)
exp 2r ;
x > 1:
1
1
2
4+2
1
Then, uniformly in x; such that 1 < x (L + N ), ;
P (kXnnk r) = 1 + C (; d; ; ) nx(Ln;d + N n;d)o =
P (kN ;V k r)
n;d
n;d
1
1 (3+2 )
0
;
where jj 1: Here C = C (; d; ; ) is a constant depending only on its arguments.
It is easy to see that r2 admits the asymptotic expansion
q
r2 = r2(x) = 21 log x , 21 (4 + 2 , 1 ) log 1 + 21 log x + ::: ;
as x ! 1: This formula shows that the behavior of the range of moderate deviations in
Theorem 3 is dierent from that in the one dimensional
case. It is somewhat surprising that,
p
when 1 > 4 + 2; the range becomes larger than 21 log x:
Although we consider the case of sums of martingale-dierences, our results give new
asymptotics of moderate deviations for the case of sums of independent random variables, as
well. Let us illustrate the case of independent identically distributed random variables 1; :::; n;
taking values in the space Rd; 1 d < 1; and having means 0 and covariances Ek k0 = I;
where I is the d-dimensional identity matrix. Assume that E kk k3 < 1: Set Sn = 1 + ::: + n:
Then, for d > 2; it follows from the more general results in Theorem 3 that
!
d=2,1
q
1 + O log1 n
P p1n kSnk q log n = 2d=2,11,(d=2) (q logpnn)q
!!
;
where ,() is the gamma function and q is a real number satisfying 0 < q < 1; if d < 13 and
q 1; if d 13: This result is an extension of the univariate results in Rubin and Sethuraman
(1965) and Amosova (1972), and to the best of our knowledge is new in the multivariate case,
even for independent identically distributed random variables. It shows, however, an essential
dierence in comparison with the univariate case: If the dimension of the space Rd is large
enough, i.e. d 13; one can take q = 1: This is explained by a dierent behavior at the
boundary of the moderate deviations range. Theorem 3 gives an exact answer for this case as
well. Assume that d > 2 and q 13 , d: Set, for n large enough,
s
q
rn = log n , 2q log 1 + log n :
Then
!
p
q
1 + log n rnd,2 (
p
1+O
!)
1
1
:
d=
2,1
n
2 ,(d=2)
n
log n
This shows that,pif d = 13 (and, thus, one can take q = 0), then the boundary ofpthe range is
exactly of order log n; and, if d > 13; then this boundary becomes larger than log n:
The crucial point in the proof of our multivariate results is the following lemma where for
brevity we say, that for the symmetric matrices A and B the relation A B holds, if A , B is
non-negative denite and the relation A ' B holds, if A B and B A:
Lemma 4 (on sequential projectors). Let V and a1 ; :::; an be (non-random) non-negative
denite matrices. Set Ak = a1 + ::: + ak ; for k = 1; :::; n: Then there exist a sequence of moments
1 1 ::: d n and, corresponding to them, a sequence S1 ::: Sd of subspaces of
Rd; such that, with P1; :::; Pn; dened by Pk = PS ; for i k < i+1 (where 0 = 1; d+1 = n;
S0 = Rd), the following statements hold true, for k = 1; :::; n :
a. Abk P1a1 P1 + ::: + Pk ak Pk V ;
b. Abk ' Ak ; on the subspace k = fPk x : x 2 Rd g;
c. Abk V , k ; on the subspace ?k ; where k = maxfjja jj : j kg:
P p1 kSnk rn =
i
j
With the notation Ab0 = 0; the moments 1 ; :::; d and the subspaces S1; :::; Sd can be
dened inductively: for k = 0; :::; d , 1; the moment k+1 is the lowest index i 2 f1; :::; ng for
which the matrix Abk ,1 + ak + ::: + ai , V has a strictly positive eigenvalue and the subspace Sk
is generated by all eigenvectors of the matrix Abk ,1 , V + k I; corresponding to non-positive
eigenvalues.
We apply our lemma on sequential projectors to prove the following multivariate analog
of Lemma 2.
Lemma 5. Let 2 (0; 21 ]: For any r 0 and " > 0
(
!)
2
r + "r
n;d
jP (kXnnk r) , P (kN0;V k r)j C (d; ; ) ",2,2 (Ln;d
+ N ) + " exp ,
21
1
;
where C (d; ; ) is a constant depending on its arguments only.
References
Amosova, N. N. (1972) On limit theorems for probabilities of moderate deviations. Vestnik
Leningrad. Univ. 13, 5-14 (in Russian).
Bergström, H. (1944) On the central limit theorem. Scand. Aktuarietidscr. 27, 139153.
Bolthausen, E. (1982) Exact convergence rates in some martingale central limit theorems.
Ann. Probab. 10, No. 3, 672688.
Grama, I. (1993) On the rate of convergence in the central limit theorem for d-dimensional
semimartingales. Stochastics and Stochastics Reports. 44, No. 3-4, 131-152.
Grama, I. (1997) On moderate deviations for martingales. Ann. Probab. 25, 152-183.
Grama, I and Haeusler, E. (1998) An asymptotic expansion of moderate deviations for
multivariate martingales. Manuscript. 22 pp.
Haeusler, E. (1988) On the rate of convergence in the central limit theorem for martingales
with discrete and continuous time. Ann. of Probab. 16, No. 1, 275299.
Lindeberg, J. W. (1922) Eine neue Herleitung des Exponentialgesetzes in der Wahrscheinlichkeitstheorie. Math. Z. 15, 211225.
Rubin, H. and Sethuraman, J. (1965) Probabilities of moderate deviations. Sankhya.
A37, No. 2-4, 325346.
Resumé français
Nous montrons que la methode Lindeberg peut être appliqué pour obtenir les résultats de
déviations grandes pour les variables aléatoires dependantes. Nous considérons le cas multivarié
où une technique spéciale est nécessaire.