LARGE DEVIATIONS FOR PARTIAL SUMS U-PROCESSES IN ...

LARGE DEVIATIONS FOR PARTIAL SUMS U-PROCESSES IN DEPENDENT CASES PETER EICHELSBACHER Abstract. The large deviation principle (LDP) is known to hold for partial

sums U {processes of real-valued kernel functions of i.i.d. random variables Xi . We prove an LDP when the Xi are independent but not identically distributed or ful ll some Markov dependence or mixing conditions. Moreover we give a general condition which suces for the LDP to carry over from the partial sums empirical processes LDP to the partial sums U -processes LDP for kernel functions satisfying an appropriate exponential tail condition.

1. Introduction Let fXn; n 2 N g be a sequence of random variables on a probability space ( ; A; P ) with Polish state space S . Dembo and Zajic [3] proved an LDP for

[nt] X 1 (1.1) Ln (t) = n Xi ; t 2 [0; 1]; i=1 where the Xi ful ll Markov dependence or mixing conditions. They observed that the special conditions, which guarantee the LDP to hold for the empirical measures of Polish space valued random variables suce for the LDP to carry over to the corresponding partial sums processes Ln (
). One application of these results considers the calculation of how large delays are built up in a single server queue in the presence of dependent customer interarrival times. For other applications see [3] and [19].

In [13] the LDP is proved for partial sums U -processes, i.e. X h(X ; : : : ; X ); t 2 [0; 1]; U (t) = ? 1
n

n m Cm[nt]

i1

im

(1.2)

where the Xi are i. i. d. random variables and h is a measurable, R d -valued function, called kernel function, which is symmetric, i.e. invariant under permutation of its arguments. For k; m 2 N Cmk denotes the set f(i1 ; : : : ; im) : 1  i1 <


< im  kg. The LDP holds in L1 ([0; 1]; (Rd ; k
k)), the space of almost surely bounded, measurable R d -valued functions on [0; 1], where [0; 1] is equipped with 1991 Mathematics Subject Classi cation. 60F10 (primary). Key words and phrases. Large deviations, partial sums, U-processes, Markov chains, hypermixing, strong mixing. 1

2

P. EICHELSBACHER

Lebesgue measure and the underlying topology is the supremum norm topology. As a payo, applications in the linear case were extended to higher order statistics. Thus LDP results for multi-server queues or cost functions with several variables were obtainable. The LDP for U -statistics of degree m 2 N , i.e. X U := U (1) = ? 1
h(X ; : : : ; X ); n

n

i1

n m Cmn

im

with h and Cmn as above, is proved in the i. i. d. case for R d -valued h and Polish space valued Xi in [12] and under weaker moment conditions in [10]. In [13] the LDP for fUn (
)gn2N is proved under the same exponential tail conditions for the kernel function as for U -statistics. In this paper we discuss the LDP for the partial sums U -processes for dependent Xi as well as the LDP for another class of processes, the so called V -processes or von-Mises processes, given by X h(Xi ; : : : ; Xim ); t 2 [0; 1]; (1.3) Vn (t) = n1m nt 1

Dm[ ]

where Dmn := f(i1 ; : : : ; im) : 1  i1 ; : : : ; im  ng and h is as above. The corresponding statistics are given by the V -statistics or von Mises-statistics given by X Vn = n1m h(Xi ; : : : ; Xim ): Dn m

1

Due to the symmetry of h, Un and Vn mainly dier by the diagonal terms, where at least two of the indices i1 ; : : : ; im are equal. For weakly dependent sequences fXn ; n 2 N g invariance principles and the weak convergence of appropriately normalized sample path U -processes and von Misesprocesses to Brownian motion or limit processes expressible as multiple Wiener integrals were studied in [6]. Notice that U -processes indexed by a class of measurable functions are studied for example in [1]. We will proceed as follows: as in the i.i.d. situation rst we will prove that the m-fold product L n m(
) of Ln (
) satis es a large deviation principle in a suitable topology, if Ln (
) satis es an LDP for the underlying sequence Xi . Next we will apply a generalized version of the well-known contraction principle using an approximation of the kernel function worked out in [13]. On the other hand a special moment condition (2.4, stated in Section 2) has to be veri ed to guarantee that the partial sums U -process with the approximating kernel has the same large deviation behaviour as the partial sums U -process we are interested in. In our examples we will check Condition 2.4 for a suitable modi cation of Un (
). The remaining part of the proof is then to verify that these modi cations have a

LARGE DEVIATIONS FOR PARTIAL SUMS U -PROCESSES

3

large deviation behaviour like Un (
). In the i.i.d. case this is done via the wellknown Hoeding decomposition of a U -statistic into a dependent mean of i.i.d. means. For dependent sequences this is not as easy. We will prove a general Theorem which allows us to transfer the LDP for fLn (
); n 2 N g with respect to a suitable topology to the LDP for modi cations of fUn (
); n 2 N g (on L1 ([0; 1]; (Rd ; k
k))), whenever Condition 2.4 can be veri ed and the kernel function satis es appropriate exponential tail conditions. After that we have to assume some more tail conditions to get the LDP for the partial sums U -processes as well as for the partial sums V -processes. This paper is divided into four sections. In Section 2 we present the main results. In Section 3 the main Theorem will be proved. In Section 4 we will prove a LDP for some independent but not identically distributed sequences or Markov dependence or mixing conditions. To this end we will apply some techniques of [16] which are already developed to get LDP results for U -statistics for weakly dependent sequences. 2. Statement of the results Let fXn; n 2 N g be a sequence of random variables on a probability space ( ; A; P ) with Polish state space S . Denote by S the Borel -Algebra on S . We denote by M(S ), M+ (S ) and Mm(S ), respectively, the sets of Borel measures on S which are signed, positive and positive having total mass m, respectively. These spaces are equipped with the topology of weak convergence. The topological dual of M(S ) is identi ed withR the set Cb (S ) of continuous and bounded mappings from S to R via hf; i = S fd. M+(S ) becomes a Polish spaces when equipped with the metric topology induced by

(;  ) := supfj

Z

S

fd ?

Z

S

fd j : f 2 Cb (S ); jjf jj1 + jjf jjL  1g;

where jj
jj1 denotes the supremum norm, jjf jjL := sup jf (xd)(x;? yf)(y)j ; x6=y and d denotes the metric on S (see [3], Lemma A.1.). Let us denote by D([0; 1]; (M+(S ); )) the space of all maps on [0; 1] with values in (M+ (S ); ) continuous from the right and having left limits equipped with the uniform metric topology induced by d1 (y(
); z(
)) := sup (y(t); z(t)): t2[0;1]

We will assume that the underlying sequence fXn; n 2 N g has the property that the partial sums processes fLn (
); n 2 N g satisfy the LDP in D([0; 1]; (M+(S ); )) with a convex rate function I ( (
)). Notice that in all cases of interest a nice representation of I ( (
)) is given on its support AC 0 , where AC 0 is de ned as the

4

P. EICHELSBACHER

set of all maps  : [0; 1] ! M+ (S ) that are absolutely continuous with respect to the variation norm jj
jjvar, satisfy  (t) ?  (s) 2 Mt?s(S ) for all t  s, while  (0) = 0, and possess a weak derivative for almost all t (the latter means that for almost every t the expression hf;  (t + h) ?  (t)i=h converges to a limit hf; _ (t)i for every f 2 Cb (S )). Following [3], the sequence fLn (
); n 2 N g satis es the LDP in D([0; 1]; (M+(S ); )) with rate function I ( (
)), if the following assumption is ful lled.

Condition 2.1. Fix m 2 N and 0 = t0 < t1 <


< tm  1. Denote by Ln = (Ln(t1 ); Ln(t2) ? Ln (t1 ); : : : ; Ln (tm) ? Ln (tm?1 )). Then fLn; n 2 N g satis es the LDP in D([0; 1]; (M+(S ); ))m with the good rate function Jm (z) =

m X

?
(ti ? ti?1 )J t ?zit ; i i?1 i=1

where z = (z1 ; : : : ; zm ) and J (
) is the convex good rate function associated with the LDP of fLn (1); n 2 N g.

If Condition 2.1 is ful lled, the rate function I ( (
)) has the form

I ( (
)) = where

Z 1

0

J (_ )dt if  2 AC 0 ; Z

J ( ) = sup f f d ? (f )g; f 2Cb (S ) S

and

(2.2) (2.3)

n Z X 1 (f ) := nlim !1 n log E [exp( i=1 S f dXi )]:

Condition 2.1 especially implies that for any xed t 2 [0; 1] the sequence fLn (t); n 2 N g satis es the LDP with rate function t J (
=t). For m  2 we denote by D([0; 1]; (M+(S m); m)) the space of all (M+ (S m); m)valued cadlag-functions. S m as well as M+ (S m) are Polish spaces, since S is Polish. We take m to be the metric on M+ (S m) de ned analogously to . Again we equip the space D([0; 1]; (M+(S m ); m)) with the uniform metric topology induced by dm1 (y(
); z(
)) := supt2[0;1] m (y(t); z(t)). Denote by jj
jj the Euclidean norm on R d and by AC 0 ([0; 1]; (Rd ; jj
jj)) the space of all absolutely continuous R d -valued functions  on [0; 1] with (0) = 0. We consider the following conditions for a sequence fRngn2N  Mt(Mt (S m)), t 2 [0; 1]. Let B (S m) denote the space of all bounded, real-valued, S m -measurable functions on S m .

LARGE DEVIATIONS FOR PARTIAL SUMS U -PROCESSES

5

Condition 2.4. There exist constants ; M 2 [1; 1) and a reference measure  2 M1(S ) such that the inequality sup

n2N

Z

 Z

exp n



' d Rn (d ) m

1=n

M1 (S m ) S holds for all ' : S m ! [0; 1) with ' 2 B (S m).

M

Z

Sm

exp( ') d m

Remark 2.5. Condition 2.4 describes the amount of dependence of the underlying process under which a LDP for the laws of the partial sums empirical measure is preserved under the transformation leading to the partial sums U -process. The condition is also involved in transferring a LDP for empirical measures to a stronger topology on the set of probability measures (c.f. [7], Lemma 3.2.19 and Theorem 3.2.21) as well as transferring a LDP for empirical measures to products of empirical measures and U -empirical measures, respectively (c.f. [16]).

Let h : S m ! R d be a measurable and symmetric function and de ne the corresponding partial sums U -process Un (
) and the partial sums V -process Vn (
) as in (1.2) and (1.3). We will prove the LDP for fUn (
); n 2 N g assuming

Condition 2.6. For every  > 0, Z

Sm

exp(khk) d < 1;

where  is the reference measure of Condition 2.4.

Proving the LDP for fVn (
); n 2 N g we will assume Condition 2.7. For every  > 0, Z

Sm

exp(kh   k) d < 1

for every map  : f1; : : :; mg ! f1; : : :; mg, where  : S m ! S m is de ned by  (s) = (s(1); : : :; s(m)) for every s = (s1; : : : ; sm) 2 S m, and  is the reference measure of Condition 2.4.

In the applications we will check Condition 2.4 for a suitable modi cation Mn (t) of L n m (t) for any xed t 2 [0; 1]. The remaining part is to verify that these modi cations have a large deviation behaviour as L n m (t). Recall that for xed t 2 [0; 1] the sequences fL n m(t); n 2 N g and fMn(t); n 2 N g are called exponentially equivalent with respect to the metric m if for each  > 0 lim sup n1 log P ( m (L n m (t); Mn(t)) > ) = ?1: n!1 In all dependent situations we have in mind, that the modi cations Mn (
) are "partial\ sums of L n m . To be more precise de ne X (Xi (!);:::;Xim (!)) ; (2.8) Mn (t) := jA 1 j m;[nt] (i ;:::;im )2Am; nt 1

1

[

]

6

P. EICHELSBACHER

where Am;n  Dmn has the property nm ? jAn;mj = 0; lim n!1 nm fn (
) be the polygonal approximation of Mn (
): if n ! 1. Finally let M X fn (t) := Mn (t) + (nt ? [nt]) 1 M jAn;mj Am? ; nt (Xi ;:::;Xim? ;X nt ) : 1

1 [

1

[

(2.9)

]+1

]

fn (
) 2 C0 ([0; 1]; (M+(S m ); m )), the space of all continuous By de nition we get M maps equipped with the uniform metric topology induced by dm 1 . Now our main result can be stated as follows:

Theorem 2.10. Assume that the sequence fXn ; n 2 N g satis es Condition 2.1. Let fMn(t); n 2 N g be exponentially equivalent (with respect to m ) to L n m (t) for every t 2 [0; 1]. Assume that Condition 2.4 holds for a reference measure  2 M1 (S ) and Mn (t) for every t 2 [0; 1]. Assuming 2.6 holds for h with respect to the reference measure  then the seR R fn (
); n 2 N g satisfy the LDP in quences f Sm h d Mn (
); n 2 N g and f Sm h d M L1 ([0; 1]; (Rd ; k
k)) with the good rate function

I1 () = inf f

Z 1

0

I (_ )dt;  2 AC 0 \ K1 and

Z

Sm

hd m (
) = (
)g

(2.11)

and speed n, if  2 AC 0 ([0; 1]; (Rd ; jj
jj)) and I1 = 1 otherwise. Here

K1 :=

[

f (
) : I ( (
))  Lg:

L0

Remark 2.12. In Section 4 we will formulate the moment conditions under which the LDP for the partial sums U -processes and V -processes can be deduced from Theorem 2.10. Remark 2.13. Under the conditions mentioned in Remark 2.12 we can extend the results to the laws of

V" (t) = "m

X

Dmt="] [

and

U" (t) = ?11="


h(Xi ; : : : ; Xim ); t 2 [0; 1];

X

m Cm[t="]

1

h(Xi ; : : : ; Xim ); t 2 [0; 1]; 1

for " > 0. More precisely we get the LDP for the family fU" (
); " > 0g in L1 ([0; 1]; (Rd ; k
k)) with the good rate function I1 (
) given by (2.11) and speed "?1 whenever it can be established for Un (
). The same is true for fV" (
); " > 0g.

LARGE DEVIATIONS FOR PARTIAL SUMS U -PROCESSES

7

Finally we want to show that for m = 1 and h(x) = x we get the same rate function as given in [4]. Let us assume that the assumptions of Theorem 2.10 are ful lled, thus we get the LDP for n X 1 Sn (t) = n Xi ; t 2 [0; 1]; i=1 with rate function Z 1

inf f

0

I (_ )dt;  2 AC 0 \ K1 and

Z

S

xd (
) = (
)g:

(2.14)

We will show that this is the usual rate function for fSn (
); n 2 N g. Let  (x) := sup2Rd fh; xi ? ()g, x 2 R d , where n 1 log E [exp(X () := nlim h; Xii)]: !1 n i=1

Corollary 2.15. Whenever inf fJ ( (1));  (1) 2 \K1;1 and S

where K1;1 = L0 Z 1

0

f% 2 M1(S ) :

 (_ (t))dt = inf f

Z 1

0

Z

S

xd (1) = (1)g =  ((1));

J (%)  Lg, we get

I (_ )dt;  2 AC 0 \ K1 and

Z

S

xd (
) = (
)g:

3. Proof of the Theorem First we will collect some preliminaries. The following Lemma is an easy observation proved in [14], Example 2.1.

Lemma 3.1. Let S be Polish and dv the metric induced by the total variation norm k
kv on M1(S ). Then a LDP for a sequence of probability measures on M1(S )

with respect to the weak topology is transfered to every dv -exponentially equivalent sequence of probability measures on M1 (S ).

Lemma 3.2. Suppose the sequence fXn ; n 2 N g satis es Condition 2.1. Let us de ne

L n m (
) := n1m

and

Lmn(
) := n 1

X

Dmn
] [

(Xi ;::: ;Xim ) 1

X

(m) (i1 ;::: ;im )2Im;[n
]

(Xi ;::: ;Xim ) ; 1

?1 (n ? k) and where I  f1; : : : ; ngm contains all m-tuples where n(m) := km=0 m;n with pairwise dierent components. Then the sequences fL n m (
); n 2 N g and fLmn(
); n 2 N g satisfy the LDP in D([0; 1]; (M+(S m); m)) with the good rate function I m( (
)) = I (1(
)) if 1m (
) =  (
) and 1 (
) 2 AC 0 and I m( (
)) = 1 otherwise. Q

8

P. EICHELSBACHER

Proof of Lemma 3.2. In view of the contraction principle [5], Theorem 4.2.23, we have to check that  (
) 7!  m (
) maps D([0; 1]; (M+(S ); )) into D([0; 1]; (M+(S m); m)) and is continuous with respect to the topologies of uniform convergence. This is already proven in [13], Lemma 3.6. Since moreover for every xed t 2 [0; 1] jjL n m(t) ? Lmn (t)jjvar  nCm with a constant C , which does only depend on m, we get the LDP for fLm n (
); n 2 N g applying Lemma 3.1 Remark 3.3. Since for every xed t 2 [0; 1] fMn (t); n 2 N g is exponentially equivalent to fL n m(t); n 2 N g by the assumption of Theorem 2.10, we get

m lim sup 1 log P (dm 1 (Ln (
); Mn (
)) >  ) = ?1:

n Therefore the sequence fMn (
); n 2 N g satis es the LDP in the same space with the same rate I m( (
)). Since fn (t)jjvar  C jjMn(t) ? M nm fn (
); n 2 N g with a constant C , which only depends on m, we get the LDP for fM applying Lemma 3.1. Note that f (
) : I m( (
)) < 1g  C0 ([0; 1]; (M+(S m); m )) fn (
) 2 C0 ([0; 1]; (M+(S m ); m ))) = 1. Thus the LDP for fM fn (
); n 2 N g and P (M + m m holds in the space C0 ([0; 1]; (M (S ); )) by [5], Lemma 4.1.5. n!1



m := S m Lemma 3.4. With the notations of Lemma 3.2 let K 1 L0  (
) : I ( (
))  R L . If  (
) 2 K1m \AC 0 then hd (
) 2 AC 0 ([0; 1]; (Rd ; jj
jj)) for all h : S m ! R d , satisfying Condition 2.6. Thus I1 () < 1 implies  2 AC 0([0; 1]; (Rd ; jj
jj)). If R  (
) 2 K1m \ C0 ([0; 1]; (M+(S m); m)), then hd (
) 2 C0 ([0; 1]; (Rd ; jj
jj)) for all h : S m ! Rd , satisfying Condition 2.6.

The proof of Lemma 3.4 is given after the proof of Theorem 2.10. Proof of Theorem 2.10. 1.Step: R As in the proof of Theorem 2.4 in [13], we de ne a function F by F (%(
)) = h d%(
). We want to prove a LDP for F (Mn (
)). By Remark 3.3 the sequence fMn(
); n 2 N g satis es a LDP. Therefore we would like to apply the contraction principle (cf. [5], TheoremR 4.2.23, and [18], Theorem 2.1). Since for arbitrary kernel functions h the integral hd%(t) is generally not de ned for all %(t) 2 M+(S m ), rst we have to m we get I m (%(
)) = I ( (
))  K describe the images under F carefully. If %(
) 2 K1 for some K  0 and  m (
) = %(
). For any xed t 2 [0; 1] the sequence fLn (t); n 2 N g satis es the LDP in M+ (S ), endowed with the weak topology, with the good rate function t J (=t),  2 Mt (S ). Thus by assumption fMn (t); n 2 N g satis es the LDP in M+ (S m), endowed with the weak topology, too. The rate function has

LARGE DEVIATIONS FOR PARTIAL SUMS U -PROCESSES

9

the form t J m (%) := t J ( ), if ( t )m = %. Since Mn (t) satis es Condition 2.4, we get by [7], Lemma 3.2.7 and Lemma 3.2.19, that t H (%jm)  (t J m (%) + log(2M )); % 2 M1(S m); (3.5) and , M and  as in Condition 2.4. Since by Lemma 3 in [3] we get k X  (ti?1 ) ); (3.6) I ( (
)) = sup (ti ? ti?1 )J (  (tit) ? i ? ti?1 0=t 0. Actually, we need this condition in the proof of Theorem 4.4 only for the elements (i1 ; : : : ; im ) in the sets Cn (; r). In comparision to condition (U^ ) in [7], the stronger assumption for the U -processes LDP seem to be a natural generalization. Nevertheless, in the multivariate case the assumptions get a little more involved. The proof of the LDP for the partial sums U -processes now is a simple exercise. We only have to prove that the additionalR assumptions suce to get the exponential equivalence of fUn (
); n 2 N g and f hdMn (
); n 2 N g in L1 ([0; 1]; (Rd ; k
k)), i.e. Z ?
1 lim sup n log P sup kUn (t) ? hdMn (
)k >  = ?1: n!1 t2[0;1] Applying the Chebyshev-Markov inequality, the proof of Lemma 3.2 in [13] gives us the result. Remark 4.13. By the same arguments we can transfer the LDP to the partial sums V -processes, assuming that Condition 2.7 holds uniformly in  2 S for  de ned in (4.2). We get the LDP for the laws of V" (
) and U" (
) analogously (see Theorem 2.6 in [13] and its proof).

As a Corollary we obtain the result of [3], Theorem 3(b), the R d -valued case for Markov chains which ful ll (4.1):

Corollary 4.14. If the Markov-chain ful lls Condition 4.1 and sup

Z

2S S

exp(kxk)(; dx) < 1

for all  > 0 (condition U^ in [7]), then the sequence f;n ; n 2 N g,  2 S , of P ] laws of Zn (t) = 1=n i[nt =1 Xi , t 2 [0; 1], satis es for each  2 S the LDP in L1 ([0; 1]; (Rd ; k
k)) with the good rate function

I () =

Z 1

0

 (_ (t)) dt if  2 AC ([0; 1]; (Rd ; jj
jj)); (0) = 0;

I () = 1 otherwise. Here, for x 2 R d ,  (x) := sup2Rd fh; xi ? ()g, where n X 1 () := nlim !1 n log E [exp( i=1 h; Xi i)]:

Proof. With Exercise 2.1.20 (i) and Exercise 4.1.56 in [7], we get

inf fJ1( (1));  (1) 2 \K1;1 and S

Z

S

xd (1) = (1)g =  ((1));

where K1;1 = L0f% 2 M1 (S ) : J1 (%)  Lg. Thus we are done by Corollary 2.15.

LARGE DEVIATIONS FOR PARTIAL SUMS U -PROCESSES

17

4.2. Stationary, hypermixing sequences. Let fXi; i 2 N g be a stationary sequence of random variables which take values in S . We assume that the hypermixing conditions (H1) and (H2) of [5] are ful lled. We will treat only the case m = 2 and will suppose that an additional assumption is ful lled:

Assumption 4.15. There exist l2 2 N and ; 2 [1; 1) such that for all i 2 N and f 2 B (S 2; [0; 1)), EP

bi=Y 2?l2 c

j =1



f (Xi?j ; Xj )  EP

bi=Y 2?l2 c

j =1

f (Xi?j ; Xej )

1=

;

where the process fXej gj2N is an independent copy of fXj gj2N . Remark 4.16. There are some nice examples (see Exercise 6.4.19 in [5] and the continuous time Ornstein-Uhlenbeck process discussed in [7]), where (H1), (H2) and (4.15) are ful lled, but (4.1) not. Notice moreover, that there are only a few results concerning U -Processes with mixing increments, see for example [2], where a central limit theorem for -mixing sequences is studied.

For a xed l 2 N de ne A2;n := f (i; j ) 2 f1; : : :; ng2 j ji ? j j  l for all i 6= j g: (4.17) Denote by Mn (
) the corresponding partial sums process with respect to A2;[nt]. We will prove:

Theorem 4.18. Assume that Condition 2.6 holds for h and L(X1), then the meaR sures fP  ( hdMn (
))?1; n 2 N g and fP  (Un (
))?1; n 2 N g satisfy a LDP on L1 ([0; 1]; (Rd ; k
k)) with the good rate function I1 (), de ned as in (2.11), where I ( (
)) = and J is de ned by (2.3).

Z 1

0

J (_ )dt if  2 AC 0 ;

(4.19)

Proof. By Theorem 4 in [3] the partial sums process fLn(
); n 2 N g satis es the LDP in (D([0; 1]; (M+(S ); )); d1) with rate function (4.19), when condition (H-2) is ful lled. We obtain kL n 2(t) ? Mn (t)kvar ! 0 as n ! 1, therefore Mn (t) has the property (2.9) and the measures fP  (L n 2(t))?1 g[nt]2l are exponentially equivalent to fP  (Mn(t))?1g[nt]2l . We have to check Condition 2.4. Using (H1) and Assumption 4.15, by the proof of Theorem 3.24 in [16] we get the following estimate:    X 1 log E exp n '(X ; X )

n

P

jA[nt];lj (i;j)2A nt ;l ?

[

]

i

j

 log + log EP exp(16 k'(X1; Xe1))


18

P. EICHELSBACHER

for every ' : S 2 ! [0; 1], bounded and measurable and the theorem is proved. For the sake of completeness again we will give a sketch of the calculations. Without loss of generality ' : S 2 7! [0; 1) is a bounded and symmetric function. Remark that jA2;nj  n2 =4 for n  8l and thus njA(n?;nlj)  4 for n  8l. Now, for n  8l we get    X n '(Xi ; Xj ) EP exp jA j n;l (i;j )2An;l 2

n?l X n ( n ? l ) '(Xn+1?j ; Xj )  EP exp jA j n;l j =l 

1=n?l



minfX i?1;ng n 2( n ? l ) '(Xi?j ; Xj )  EP exp jA j n;l j =maxf1;i?ng i=l+2 i6=n+1  2Y n?l





 1f(i;j) : ji?2jjlg 

 nY +1

i=l+2

 X i?1



EP exp 8

j =1

1=2(n?l)

'(Xi?j ; Xj )1f(i;j) : ji?2jjlg

1=n?l

;

where we have applied Holder's inequality and used the stationarity of the sequence fXigi2N . We obtain by Assumption 4.15 for l2 ? l 



EP exp 8

i?1 X j =1 

'(Xi?j ; Xj )1f(i;j) : ji?2jj>lg 

 EP exp 16

i?l c bX 2

j =1

1=

'(Xi?j ; Xej )

For each k with k ? 1  l we obtain ?
EP exp('(X1; Xek ) + '(Xk ; Xe1))



Z

Z

S

S2



Z

S

exp('(x1; xek )1 (dx1)

exp('(xk ; xe1)k (dxk )



:

1=

1=

(dxe1 ; dxek )

by Assumption (H1), where 1 and k , respectively, denote the marginal distribution on the rst and k-th component, respectively, and (dxe1; dxek ) is the two dimensional marginal distribution on the rst and k-th component. Applying Assumption (H1) again, we get ?




EP exp('(X1; Xek ) + '(Xk ; Xe1)  EP ?exp('(X1; Xek ))
1= EP ?exp('(Xk ; Xe1))
1=:

LARGE DEVIATIONS FOR PARTIAL SUMS U -PROCESSES

19

Thus with k 2 N and m > 0 such that km = b i?2 l c we get 



EP exp 16

i?l c bX 2

j =1

'(Xi?j ; Xej )

1=

 EP exp(16 k'(X1 ; Xe1)) m= : ?




We arrive at    X 1 log E exp n P n jAn;l j (i;j)2An;l '(Xi; Xj ) ?
n?l?  n1 log n?l + n1 log EP (exp(16 k'(X1; Xe1))) k ?
 log + log EP exp(16 k'(X1; Xe1)) : 2

1

As a corollary we obtain Theorem 5 in [3], the R d -valued case for (H1)- and (H2)mixing sequences.

Corollary 4.20. Suppose that the stationary sequence fXi ; i 2 N g, taking values in R d , satis es the mixing conditions (H1) and (H2) and E (exp(kX k)) < 1 for P ] all  > 0. Then the sequence fn ; n 2 N g of laws of Zn (t) = 1=n i[nt =1 Xi , t 2 [0; 1], d satis es the LDP in L1 ([0; 1]; (R ; k
k)) with the good rate function I () =

Z 1

0

 (_ (t)) dt if  2 AC ([0; 1]; (Rd ; jj
jj)); (0) = 0;

I () = 1 otherwise, where  is de ned as in Corollary (4.14). Proof. We have to check

inf fJ ( (1));  (1) 2 \K1;1 and S

M1(S ) :

Z

S

xd (1) = (1)g =  ((1));

where K1;1 = L0f% 2 J (%)  Lg. To this end we can actually go along the proof of Theorem 5 in [4]. There the identi cation is checked. Here we only have to apply the rst part of Proposition 1 in [4]. The proof is completed by applying Corollary 2.15. 4.3. Independent but not identically distributed sequences. Let fXi; i 2 N g be a sequence of independent random variables with values in R d and laws L(Xi) = i . Let k () = log E [exph; Xk i] are such that for all  2 R d the limit m X 1 (4.21) () = nlim !1 m i=1 i () exists and (
) is nite and dierentiable. Then fLn (1); n 2 N g satis es the LDP in M1(R d ) equipped with the weak topology with the convex rate function J ( ) de ned in (2.3) (see [11], where the LDP is proved in an even stronger topology). Moreover fLn (
); n 2 N g satis es the LDP in D([0; 1]; (M+(Rd ); )) equipped with

20

P. EICHELSBACHER

the sup norm topology with the convex good rate function de ned in (2.2) (see [11]). Consider the following additional assumption: assume that i   for all i 2 N and  2 M1(S ). Moreover we assume for fi := ddi that there exists a q > 1 such that sup kfi kq < 1; (4.22) i2N

where k
kq denotes the q-norm in L1 (S; S ; ).

Theorem 4.23. Consider a sequence fXi; i 2 N g as above. Assume that Condition 2.6 holds for h and . Then the measures fP  (Un (
))?1; n 2 N g satisfy a LDP on L1 ([0; 1]; (Rd ; k
k)) with a good rate function I1 (), de ned as in (2.11). Proof. The partial sums process fLn (
); n 2 N g satis es the LDP in (D([0; 1]; (M+(R d ))); d1) with rate function (2.2). We will check Condition 2.4 for Mn (t) = Lmn(t). For every ' 2 B (Rmd ) and the laws of Lmn(t) we get    Z ?
m ' dL (t)  E P exp(nB (Xi ; : : :; Xin )) E P exp n n m 1

S

via Hoeding's result in [17, Section 5], where B (X1; : : :; Xn) := k1 ['(X1; : : :; Xm) + '(Xm+1; : : :; X2m) +


+ '(Xkm?m+1; : : :; Xkm)]

and k := [ mn ]. Since B (
) is an average of independent random variables and E P (exp(m'(X1 ; : : :; Xm )))



Z

Sm

exp(pm'(X1; : : : ; Xm)) d m

1=p Z

Y m

S m i=1

fi (xi )

q

d m

1=q

by Holder's inequality, Condition 2.4 is ful lled by our assumption supi2N kfi kq < 1 for a q > 1 and Condition 2.6. As a Corollary we nally obtain Corollary 1 in [3], the Rd -valued case for independent but not identically distributed sequences.

Corollary 4.24. Suppose that the Xi are independent R d -valued and k () are such that for all  2 R d () (see (4.21)) exists and is nite and dierentiable. P ] Then the sequence fn ; n 2 N g of laws of Zn (t) = 1=n i[nt =1 Xi , t 2 [0; 1], satis es d the LDP in L1 ([0; 1]; (R ; k
k)) with the good rate function I () =

Z 1

0

 (_ (t)) dt if  2 AC ([0; 1]; (Rd ; jj
jj)); (0) = 0;

I () = 1 otherwise, where  is de ned as in Corollary (4.14).

LARGE DEVIATIONS FOR PARTIAL SUMS U -PROCESSES

Proof. We have to check

inf fJ ( (1));  (1) 2 \K1;1 and S

Z

S

21

xd (1) = (1)g =  ((1));

where K1;1 = L0 f% 2 M1(S ) : J (%)  Lg. Going along the line of Exercise 2.1.20 and 3.3.12 in [7], we only have to prove that there exists a lower semicontinuous function f : [0; 1) ! [0; 1] such that limx!1 f (x)=x = 1 and sup n1 log E exp h

n ?X

n2N

i=1


i

f (jjXijj) < 1

(see Proposition 1 in [4]). We choose f (x) = 1=2 sup2[0;1)fx ? log E [exp(x)]g. This function has the properties as desired using Lemma 2.2.20 in [5] and the inequality E [exp(f (jjXijj))]  4 < 1 (see inequality (5.1.6) in [5]): by the independence of the Xi we get h

E exp

n ?X i=1


i

f (jjXijj) 

n Y i=1

E [exp(f (jjXijj))]  4n :

Finally we want to transfer the LDP to the partial sums V -processes. Here we can get a result under weaker conditions: Assume that there exists at least one  > 0 such that Z

Sm

exp(kh   k) d < 1

(4.25)

for every map  : f1; : : :; mg ! f1; : : :; mg, where  : S m ! S m is de ned by  (s) = (s(1); : : :; s(m)) for every s = (s1; : : :; sm) 2 S m , and  as above. Consider the case t = 1. For every n  m we want to de ne a measurable function hn such that in the notation of the proof of Theorem 2.10 we get

F (L n m ) = Fn (Lmn );

(4.26)

R

m where Fn (Lm n ) := hn dLn . One possible choice is given in [15], i.e. for all x = (x1; : : :; xm) 2 S m we set

hn (x) =

m n X X (j )

(4.27) nm  2Tj h(x (1) ; : : :; x (m)); where Tj denotes the set of all surjective maps  : f1; : : :; mg ! f1; : : :; j g with  (1) = 1 and  (k)  1+maxf (1); : : :;  (k ? 1)g for all k 2 f2; : : :; mg (see proof of Theorem 1.11(b) in [15]). De ne e = (mm?1 + m2 )?1 with  as in Condition 4.25. j =1

22

P. EICHELSBACHER

By using the exponential Chebychev-Markov inequality, independence, Hoeding's formula [17, Section 5] and Holder's inequality we get P(kUn (1) ? Vn (1)k  ")

m = P(kF (Lm n (1)) ? F (Ln (1))k  ")  e?e"n E [exp(en2 kF (Lmn(1)) ? Fn (Lmn(1))k)] (4.28)  e?e"n E [exp(pemnkh(x) ? hn (x)k)][n=m]1=p 2

2

 Y q [n=m]1=q m

m E fi (xi ) d :

i=1 m 2 Note that 1 ? n(m) =n  m =n. Using (4.27), it follows that mX ?1 X 2 nkh(x) ? hn (x)k  m kh(x)k + kh   (x)k j =1  2Tj S for all x 2 S m. Using this estimate, j jm=1?1 Tj j  mm?1 , Holder's inequality and Condition 4.25 as well as the assumption supi2N kfikq < 1 for a q > 1, it follows

that the product of the integrals in (4.28) is bounded by a constant which does not depend on n. Hence, for any sequence fnk gk2N satisfying km  nk for all k 2 N , m lim sup n1 log P(kF (Lm n (1)) ? Fn (Ln (1))k  ") = ?1: k!1 For t 2 [0; 1) the calculations are identical. Thus we have proved that fUn (
); n 2 N g and fVn (
); n 2 N g are exponentially equivalent under the weaker moment Condition 4.25. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

References M. A. Arcones and E. Gine, Limit theorems for U -processes, Ann. Probab. 21 (1991), 1494{ 1542. M. A. Arcones and B. Yu, Central limit theorems for empirical and U -processes of stationary mixing sequences, J. Theoret. Probab. 7 (1994), no. 1, 47{71. A. Dembo and T. Zajic, Large deviations: From emprical mean and measure to partial sums process, Stochastic Process. Appl. 57 (1995), 191{224. , Uniform large and moderate deviations for functional empirical processes, preprint, 1995. A. Dembo and O. Zeitouni, Large Deviations Techniques and Applications, Jones and Bartlett, Boston, MA, 1993. M. Denker and G. Keller, On U -statistics and v. Mises' statistics for weakly dependent processes, Z. Wahrsch. Verw. Gebiete 64 (1983), 505{522. J.-D. Deuschel and D. W. Stroock, Large Deviations, Academic Press, San Diego, 1989. M. D. Donsker and S. R. S. Varadhan, Asymptotic evaluation of certain Markov process expectations for large time III, Comm. Pure Appl. Math. 29 (1976), 389{461. R. M. Dudley, Probabilities and metrics, Aarhus Universitet, Lecture notes series No. 45, Aarhus Universitet, Mathematisk Institut, 1976. P. Eichelsbacher, Large deviations for products of empirical probability measures in the  topology, J. Theoret. Probab., (to appear).

LARGE DEVIATIONS FOR PARTIAL SUMS U -PROCESSES

23

11. , Partial sums large deviations for independent random variables, in preparation, 1997. 12. P. Eichelsbacher and M. Lowe, Large deviation principle for m-variate von Mises-statistics and U -statistics, J. Theoret. Probab. 8 (1995), 807{824. , Large deviations for partial sums U -processes, SFB-preprint 95-115, 1995. 13. 14. P. Eichelsbacher and U. Schmock, Exponential approximations in completely regular topological spaces and extensions of Sanov's theorem, SFB-preprint 96-109, 1996. , Large deviations of products of empirical measures and U -empirical measures in 15. strong topologies, SFB-preprint 96-027, 1996. , Large deviations for products of empirical measure of dependent random sequences 16. in strong topologies, preprint, 1997. 17. W. Hoeding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13{30. 18. V. Perez-Abreu and C. Tudor, Large deviations for a class of chaos expansions, J. Theoret. Probab. 7 (1994), no. 4, 757{765. 19. T. Zajic, Large exceedances for uniformly recurrent Markov-additive processes and strong mixing processes, J. Appl. Probab. 32 (1995), no. 3, 679{691. (Peter Eichelsbacher) Fakultat fur Mathematik, Universitat Bielefeld, Universitats-

strae 1, D-33501 Bielefeld, Germany

E-mail address, Peter Eichelsbacher:

[email protected]