Directory UMM :Data Elmu:jurnal:S:Stochastic Processes And Their Applications:Vol89.Issue2.2000:

Stochastic Processes and their Applications 89 (2000) 193–211
www.elsevier.com/locate/spa

Uniform iterated logarithm laws for martingales
and their application to functional estimation
in controlled Markov chains
R. Senoussi ∗
INRA, Laboratoire de Biometrie, Domaine St. Paul, Site Agroparc, 84 914.Avignon, Cedex 9, France
Received 27 March 1995; received in revised form 28 February 2000; accepted 28 February 2000

Abstract
In the rst part, we establish an upper bound of an iterated logarithm law for a sequence of
processes Mn (:) ∈ C(Rd ; Rp ) endowed with the uniform convergence on compacts, where Mn (x)
is a square integrable martingale for each x in Rd . In the second part we present an iterative
kernel estimator of the driving function f of the regression model:
Xn+1 = f(Xn ) + n+1 :
Strong convergences and CLT results are proved for this estimator and then extended to
controlled Markov models.
Resume:
La premiere partie e tablit une majoration de type loi du logarithme itere pour une suite de
processus stochastiques Mn (x) ∈ C(Rd ; Rp ) muni de la topologie de la convergence uniforme

sur les compacts, lorsque Mn (x) est une martingale de carre integrable pour tout x dans Rd . La
seconde traite par la methode des noyaux, le probleme de l’estimation iterative de la fonction f
du modele de regression:
Xn+1 = f(Xn ) + n+1 :
On prouve la consistance forte et e tablissons dierentes vitesses de convergence de l’estimateur.
On generalise ensuite ces resultats a d’autres exemples et en particulier au modele markovien
c 2000 Elsevier Science B.V. All rights reserved.
contrˆole.
MSC: primary 60F15; 62G05; secondary 60G42; 62M05
Keywords: Iterated logarithm law; Autoregressive model; Controlled model; Markov chain;
Kernel estimator

0. Introduction
Part I of the paper proves a lim sup version of an iterated logarithm law for a sequence of random processes (Mn (·))n¿1 with values in Rp and arguments or indices
in Rd . Processes (Mn (x))n¿1 are assumed to be square-integrable martingales for all
∗

Tel.: 19-33-90-31-61-33; fax: 19-33-90-31-62-44.

c 2000 Elsevier Science B.V. All rights reserved.

0304-4149/00/$ - see front matter
PII: S 0 3 0 4 - 4 1 4 9 ( 0 0 ) 0 0 0 2 5 - 9

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R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

x ∈ Rd and to have almost surely continuous paths for all n ∈ N. Strong laws on general Banach spaces have been established already (Mourier, 1953; Kuelbs, 1976), but
in our case the space C(Rd ; Rp ) endowed with the topology of uniform convergence
on compacts is not a Banach space. We give rst simple conditions that ensure the
strong uniform convergence and then strengthen this result by an iterated logarithm
law. Unlike Strassen’s law (Heyde and Scott, 1973) for sequences of real r.v. the result presented here is not an invariance principle but is only a strong law for variables
that take values in a function space.
Part II develops some aspects of function estimation in the context of autoregressive
models. Most studies of density or regression estimators generally use a Lp criterion,
but for the controlled models the a.s. convergence is crucial in order to adapt an optimal
control process. For this reason, we prove the a.s. uniform convergence (Devroy, 1988;
Hernandez-Lerma, 1991), an iterated logarithm law and the pointwise weak convergence of the regression function estimator f of the following controlled Markov model:
Xn+1 = f(Xn ) + C(Xn ; Un ) + n+1 :
Our results are quite similar to classical density and regression kernel estimators of

i.i.d. real sequences.
We now specify some notations that will be intensively used in this paper.
R) is the ball centered in x ∈ Rd with radius R in the Euclidean sense
Bd (x;p
P i 2
as for instance
(x ) . D generally denotes a dense countable set of Rd p
kxk =
S
D = m¿0 Dm where Dm = Zd =2m . The following function h(t) = 2LL(t) where
LL(t) = log(log(t)) is used throughout the paper. We recall that C(Rd ; Rp ) is the
metrisable space of continuous functions from Rp to Rd , endowed with the topology of uniform convergence on compacts. The modulus of continuity of function f on
[−N; +N ]d is denoted !(f; N; )=sup(kf(x)−f(y)k; kx−yk6; kxk6N; kyk6N ).
For the probability part, the existence of a stochastic basis (
; A; F = (Fn )n¿0 ; P)
satisfying the usual conditions, is always assumed, that is F is a P complete, increasing
and right continuous family of sigma elds.
The increasing process of a F-adapted, square-integrable vector martingale is the
predictable and increasing sequence of semi-denite positive matrices.
Pn

hM; M in = k=1 E(
Mk :t
Mk )|Fk−1 ), (or hM in ), where
Mn+1 = Mn+1 − Mn
stands for the martingale dierence. More generally, we denote hM (x)in for sequence
(Mn (·))n¿1 of random functions of C(Rd ; Rp ) such that, (Mn (x))n¿1 is a discrete,
F-adapted, square-integrable vector martingale for each x. Sometimes, with no loss of
rigor, it happens that we use a same notation for dierent constants to avoid their
profusion. Eventually, we simply refer to Du
o (1990) and Iosifescu and Grigorescu
(1990) each time we need to recall classical results on martingales.
Part A. Uniform strong laws
1. Strong laws in C(R d ; R p )
Rao (1963) established a strong law for stationary sequences Xn (·) of random functions in D([0; 1]; R), the metric Skorohod space of right continuous with left-hand

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

195

limit in each point under the assumption E(sup06s61 kXn (s)k) ¡ ∞. We give below
a comparable result for non-stationary sequences of random functions of martingale
type under Lipschitzian conditions. We rst dene a function on Rd to specify the
Lipschitzian conditions we need in the multivariate case, if x = (x1 ; : : : ; xd )

(x) =

d
Y
(|xi | + 5{|xi |=0} ):
i=1

Theorem 1.1. Let Mn (x) be a family of discrete martingales indexed by x ∈ Rd ; with
values in Rp and let us assume that for some continuous increasing function a(·) on
R+ and constants  ¿ 0;
¿ 0;
(a) E(kMn (0)k2 ) = O(n );
(b) for all integers N and x; y ∈ Bd (0; N );
E(kMn (x) − Mn (y)k2 )6a(N )n kx − yk
(x − y):
Then; for all  ¿ =2; the sequence n− Mn (·) converges a.s. to zero and uniformly
on compacts of Rd .
Proof. First, conditions (a) and (b) imply that Mn (·) has continuous paths a.s. and that
the strong law for square-integrable martingales applies (Neveu, 1964): limn n− Mn (x)=
0 a.s. for all x. Hence, a.s. n− Mn (x) converges to zero on every dense countable

set D.
Second, by Ascoli’s lemma we only have to prove that n− Mn (x) is a.s. an equicontinuous sequence. If we consider the partial oscillation, W (f; N; 2−m ) = sup(kf(x) −
f(y)k; x; y ∈ Bd (0; N ) ∩ Dm ; kx − yk62−m ) of f on Grid Dm = Zd =2m , we get
X
!(f; N; 2−m )6C
W (f; N; 2−r ):
(1)
r¿m

Next, for N ¿ 0;  ¿ 0; kxk; kyk6N , let us dene Events
A(n; x; y; ) = {sup(k − kMn (x) − Mn (y)k; 2n 6k62n+1 )¿}
and
B(n; m; N; ) =

[

(A(n; x; y; ); x; y ∈ Bd (0; N ) ∩ Dm ; kx − yk62−m ):

On the one hand, by Kolmogorov’s inequality for martingales, it follows
P(A(n; x; y; ))6a1 (N )kx − yk

(x − y)−2 (2n )−2 (2n+1 ) :
On the other hand, since the number of neighbors y ∈ Dm of x; ky − xk = 2−m is less
than C2md , we get
P(B(n; m; N; ))6a2 (N )−2 2−m
+n−2n :
S
If  ¡
=2 and C(n; N ) = m¿1 (B(n; m; N; 2−m ), then
X
2−
m+2m 6a4 (N )2−n(2−) :
P(C(n; N ))6a3 (N )2n−2n
m¿1

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R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

P
Since,

m¿1 P(C(n; N )) ¡ ∞, it follows by Borel–Cantelli Lemma that, from some
rank n∗ , we have for all m ∈ N; x; y ∈ B(0; N ) ∩ Dm ; kx − yk62−m .
n− kMn (x) − Mn (y)k6d−m :
S
Setting D = m Dm , by (1), we obtain for n¿n∗ ; x; y ∈ Bd (0; N ) ∩ D and kx − yk
P
62−k ; n− kMn (x) − Mn (y)k6C m¿k 2−m 6C2−k .
Since the paths are a.s. continuous, this inequality still holds on Bd (0; N ) and this
proves the a.s. equicontinuity of the sequence n− Mn ( ).
Corollary 1.1. Let (Zi )i¿1 be an i.i.d. sequence of square-integrable r.v. in Rs ;  a
positive continuous function on Rd × Rd and a mapping F from Rs × Rd to Rd which
meets the following conditions for some constants
; C1 ; : : : ; C4 :
(i) kF(z; 0)k6C1 kzk + C2 .

(ii) kF(z; x) − F(z; y)k2 6C3 kx − yk
(x − y)(C4 kzk2 + (x; y)).
Pn
Then; if n (x) = i=1 (F(Zi ; x) − E(F(Zi ; x))); the sequence n− n (·) converges a.s.
and uniformly on compacts to zero; for all  ¿ 21 .

Proof. The square-integrable martingale n (0) satises assumption (a) of Theorem 1.1
with  = 1. Moreover, we have


2 
n

X


E 
(F(Zi ; x) − F(Zi ; y)
 6Cnkx − yk
(x − y)(CEkZk2 + (x; y)):


i=1

2. Iterated logarithm law
Heyde and Scott (1973) generalized the invariance principle of Strassen’s log–log

law to discrete martingales and then to ergodic stationary sequences of r.v. by the
Skorokhod representation method. However, for our purpose, we follow in this paper
the classical approach by the exponential inequalities of Kolmogorov, adapted to randomly normed partial sums (Stout, 1970). Although we deal with the function space
C(Rd ; Rp ) (Ledoux and Talagrand, 1986), we recall that the result proved below is
not an invariance principle. Its proof relies on the following ILL for martingales. It
has been adapted from Stout (1970) and proved in Du
o (1990).
Theorem 2.1. (a) If Mn is a F-adapted real martingale and if sn2 is an adapted
sequence converging a.s. to +∞ that satisfy for some F0 -measurable r.v. C ¡ 1;
(i) |
Mn+1 | = |Mn+1 − Mn |6Csn2 =h(sn2 ) and

2
2
)61 + C=2 a:s:
; then limn |Mn |=h(sn−1
(ii) hM in 6sn−1

(b) This inequality continues to hold; if Condition |
Mn+1 |6Cn sn2 =h(sn2 ) is substituted
for (i); where Cn is an adapted sequence decreasing to C ¡ 1.

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

197

The interesting case is C = 0. This result can be extended to the topological space
C(Rd ; Rp ) in the following sense.
Theorem 2.2. Assume there exist an adapted sequence sn of r.v.; sn2 → +∞ and two
continuous functions a(·); b(·) from R+ to R+ such that
2
,
(i) k
Mn+1 (0)k6bsn2 =h(sn2 ) and trhM (0)in 6a2 sn−1

(ii) for some
¿  ¿ 0 and all x; y ∈ Rd

k
Mn+1 (x) −
Mn+1 (y)k6kx − yk b(sup(kxk; kyk))sn2 =h(sn2 );
2
:
tr(hM (x) − M (y)in )6kx − yk
a2 (sup(kxk; kyk))sn−1
2
) is a.s. relatively compact in
Then; the sequence of processes n (x) = Mn (x)=h(sn−1
d
p
C(R ; R ) and all cluster points (·) verify


√
3b(0)
3
=2

+ kxk a(kxk) + kxk b(kxk) :
(2)
k(x)k6 d a(0) +
2
2

Proof. The proofs of Theorems 1.1 and 2.2 are based on the maximal inequality for
positive supermartingales.
(1) It is enough to deal with the one dimension case, since if Mn = (Mn1 ; : : : ; Mnp ),
Pp
we have hM j (x)in 6trhM (x)in = j=1 hM j (x)in and |
Mnj (x) −
Mnj (y)|6k
Mn (x) −
Pp

Mn (y)k6 j=1 |
Mnj (x) −
Mnj (y)k:
(2) To show inequality (2), we put a(0) = a, b(0) = b, and consider the dierent
alternatives.
(i) If a ¿ b, we set sn∗2 = a2 sn2 and obtain hM (0)in+1 6sn∗2 and

1=2
sn∗2
sn∗2
b LL(sn∗2 )
6C
:
×
×
|
Mn+1 (0)|6
n
a LL(sn2 )
h(sn∗2 )
h(sn∗2 )
Since, limn Cn = b=a ¡ 1 a.s., by part (b) of Theorem 2.1, we get


b
b
Mn (0)
aMn (0)
=a+
a:s:
lim
= lim
=a 1+
n h(s2 )
n h(s∗2 )
2a
2
n−1
n−1
Now, if x 6= 0; we have |Mn (x)|6|Mn (0)| + |Mn (x) − Mn (0)|;
2
hM (x) − M (0)in 6kxk
a2 (kxk)sn−1
2
2
and |
Mn (x) −
Mn (0)|6kxk b(kxk)sn−1
=h(sn−1
).
2
, we get
Since, kxk−
=2 b(kxk)=a(kxk) ¡ 1 and sn∗ = kxk
a2 (kxk)sn−1

lim
n

|Mn (x) − Mn (0)|
1
6a(kxk):kxk
=2 + b(kxk)kxk ;
2 )
2
h(sn−1

which proves inequality (2).
(ii) If a6b and, or a(kxk)6kxk−
=2 b(kxk), put
a∗ = a∗ (0) = sup(a; b) + 

and

a∗ (s) = sup(a(s); b(s)s−
=2 ) + :

Then, substituting a∗ (·) for a(·), we see that the couple a∗ (·); b(·) meets the conditions of the theorem. a∗ is continuous, increasing and satises a∗ (·)¿a(·). With no

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R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

loss of generality, we may assume that a¿0, b¿0 and bound sup(a; b) by a + b.
Inequality (2) follows in this second case, since  is arbitrary.
Inequality (2) holds a.s. for each x. If the equicontinuity is established, each cluster
point (:) is continuous and then (2) holds a.s. on Rd .
(3) To prove the equicontinuity on compacts, it is enough to show that,
lim→0 supn !(n ; N; ) = 0 a.s.
(i) If x0 is the point from which the function g(t) = t=LL(t) increases, let us dene
the stopping time  = inf (n; sn2 ¿x0 ) and for  ¿ 1, the sequence tk = inf (n; sn2 ¿k ).
Fix N and for x; y in Bd (0; N ), put
Mn (x; y) = Mn (x) − Mn (y) and

Nn (x; y) = Minf (n; tk+1 ) (x; y):

Hn (x; y) = Nn+ (x; y) is family of martingales, adapted to ltration F = (Fn+ )n¿0 and
H0 (x; y) = M (x; y). For large k, tk ¿  and then,
|
Hn+1 (x; y)| = |
Mn++1 (x; y)|5{tk ¿n+}
2
2
6 kx − yk b(N )sn+
=h(sn+
)

6 kx − yk b(N ) k+1 =h( k ):
Let  ¿ 0 be a real so that,
 = b(N )=a(N ) ¡ 1 and  = h( k )=( k kx − yk a(N )):
If q ¡  and  = 12 h(k )kx − ykq a(N ); then  = kx − ykq− LL(k ), and the
F -martingale Hn has bounded increments,
|
Hn+1 (x; y)|6b(N )=a(N ) =  ¡ 1:
Its increasing process is also bounded: if +n ¡ tk+1 ; we get hH (x; y)in+1 6hM (x; y)in+1
6kx − yk
a2 (N ) k+1 . Whence,
 



P
sup Hn (x; y)¿ (1 + )kx − yk
a2 (N )k+1 +  F 6e− ;
2
+n¡t
k+1

P



sup

tk 6n6tk+1

 


Nn (x; y)
 F 6e−
¿A
∩
{t
¿}
k

2
h(sn−1 )

where A = (a(N )=2)((1 + )kx − yk
− + kx − ykq ).
If t = inf (q;
− ), t ∗ = sup(q;
− ) and


a(N )
−−t
q−t
((1 + )kx − yk
+ kx − yk ); x; y ∈ Bd (0; N ) ;
(N ) = sup
2
∗

we have (a(N )=2)6
(N )6a(N )=2(1 + 2a(N )=b(N )(2N )t −t for large N , and then
 



q−
k
Nn (x; y)
t
¿kx − yk
(N ) ∩ {tk ¿} F 6e−kx−yk LL( ) :
P
sup
2
tk 6n¡tk+1 h(sn−1 )

(ii) Since  ¡ ∞ a.s., we only have to prove the equicontinuity of (n+ )n . On the
set {tk ¿ }, we have


m(−q)
W (Mn ; N; 2−m )
−m
P
sup
;
¿2
(N ) 6C(N )2md (k log())−2
2
h(sn−1 )
tk 6n¡tk+1

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

199

where, function C(N ) depends only on N (but may vary below) and W (Mn ; N; 2−m ) =
sup{|Mn (x; y)|: x; y ∈ Dm (0; N ), y ∈ Vm (x)}.
We have, for all integers k; m and events

[
sup W (n ; N; 2−r ) ¿ 2−rt
(N ) ;
Bkm =
r¿m

tk 6n¡tk+1

P(Bkm )6C(N )

X

r(−q)

e{rd log(2)−2

LL(k )}

:

r¿m

For large m, k, say m¿m∗ , k¿k ∗ and r¿m, we have

 r(−q)
d log(2)
2
k
¿rLL(k ) = ruk :
−
r LL( )
r
LL(k)
P
If m∗ ¿ 1, P(Bkm )6C(N ) r¿m e−ruk 6C(N )e−muk ; and then
X
X
P(Bkm )6C(N )
(kLL())−m ¡ ∞:
k¿k ∗

k¿k ∗

By Borel–Cantelli Lemma, P(limk¿k ∗ Bkm )=0. We have thus proved that suptk 6n W (n ;
N; 2−r )6
(N )2−rt ; for large k, m and all r¿m.
Finally, part (2) of the proof implies that
X
2−rt 6C(N )2−tm a:s:
sup !(n ; N; 2−m )6
(N )
tk 6n

r¿m

This proves lim→0 supn !(n ; N; ) = 0 and the equicontinuity of the sequence n (·).
3. Examples
3.1. Usual rates

If the increments of the martingale family behave well, i.e., sn2 = n , the convergence
rate of Theorem 2.2 can be explicited.
Corollary 3.1. Let  ¿ 0 and
¿  ¿ 0; be constants such that
(i) k
Mn+1 (0)k6bn=2 =(LL(n))1=2 ; trhM (0)in 6an ;
(ii) for any; integer N and x; y ∈ Bd (0; N );

k
Mn (x) −
Mn (y)k6kx − yk b(N )n=2 =(LL(n))1=2 ;

trhM (x) − M (y)in 6kx − yk
a(N )n :
p
Then; for all  ¿ 0; supkxk6N kMn (x)k=( n=2 (LL(n))1+ ) → 0 a:s.

3.2. Regression models

We now prove a useful result on the behavior of a special type of martingales that
will be used in the second part of this paper. These processes mainly occur in the
estimation theory of regression models.

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R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

By a zero mean noise (n )n¿1 with nite conditional moment of order ¿ 2, we mean
a F-adapted sequence of r.v. such that,
H1

t
E(n+1
n+1 |Fn ) = ;

(i) E(n+1 |Fn ) = 0 and

supn E(kn+1 k2+2 |Fn ) ¡ ∞:

(ii) for some  ¿ 0;

We also consider a sequence of processes Yn (x), F-adapted for all x; an increasing
continuous function b(·) from R+ to R+ . and assume there exist  ¿ 0 and an adapted
sequence of r.v. n such that, a.s.
H2

(i) |Yn (0)|6n ;

(ii) |Yn (x) − Yn (y)|6b(N )kx − yk n ; ∀x; y ∈ Bd (0; N ):

We study below the asymptotic behavior of the martingale family
Mn (x) =

n
X

Yk−1 (x)k :

k=1

Proposition 3.1. Under assumptions H1; H2 and if a.s.
sn2 =

n
X
k=1

k2 → ∞ and

∞
X
(2 )1+ (LL(s2 ))
k

k=1

k

(sk2 )1+

¡ ∞;

2
) is a.s. relatively compact in C(Rd ; Rp ).
the sequence Mn (·)=h(sn−1

P∞
Remark 3.1. If k=1 (k2 =sk2 )1+ is substituted for the above second series, we obtain
the pointwise convergence result of Du
o et al. (1990).

, we
Proof. (a) If kxk6N; and
get |Yn (x)|6a(N )n . Next, let us
 a(N ) = 1 + b(N )N
2
dene the events An+1 = a(N )n kn+1 k6sn =h(sn2 ) , their complements Acn+1 and put

n = n 5An − E(n 5An |Fn−1 );

n = n 5Acn − E(n 5Acn |Fn−1 ):

Since, E(n 5An |Fn−1 ) = −E(n 5Acn |Fn−1 ), the martingale Mn (·) splits into two martingales Mn (·) = M n (·) + M n (·); where
M n (x) =

n
X

Yk−1 (x)k

and

M n (x) =

n
X

Yk−1 (x)k :

k=1

k=1

2
2
) and n (x) = M n (x)=h(sn−1
).
Next, set n (x) = M n (x)=h(sn−1
(b) Theorem 2.2 applies to the family M n (·). Since, kn+1 k62sn2 =(a(N )n h(sn2 ))
and tr(E(tn n |Fn ))6tr( ), we get

(i) k
M n+1 (0)k6|Yn (0)| kn+1 k6b∗ sn2 =h(sn2 ),
tr(hM (0)in )6tr( )

n
X
k=1

2
2
Yk−1
(0)6a∗ sn−1
:

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

201

(ii) k
M n+1 (x) −
M n+1 (y)k6|Yn (x) − Yn (y)| kn+1 k6b∗ (N )kx − yk sn2 =h(sn2 ),
2

2

tr(hM (x) − M (y)in )6tr( )b (N )kx − yk

n
X

2
2
k−1
6a∗ (N )sn−1
:

k=1

Therefore, the sequence n (·) is a.s. relatively compact.
(c)(i) The increasing process (in the semi-denite sense) of Martingale Nn (x) =
Pn
2
k=1 Yk−1 (x)k =h(sk−1 ), satises
hN (x)in 6

n
2
X
Yk−1
(x)
E(kt k 5Ack |Fk−1 ):
2
2
h
(s
)
k−1
k=1

On the other hand, the moment assumption yields the Chebychev-type inequality
!2
2
k−1 h(sk−1
)
2 c
; whence the inequality
E(kk k 5Ak |Fk−1 )6C(N )
2
sk−1
tr(hN (x)in )6C(N )

n
X
k=1

2
(k−1
)1+
¡∞
2 )1+ (LL(s2 ))1−
(sk−1
k−1

a:s:

Thus, Nn (x) converges to a nite r.v. N∞ (x) a.s. for all x, and by the Kroneker lemma
we conclude that, n (·) → 0 a.s. on a dense countable set of Rd . It remains to establish
the equicontinuity of the sequence.
(c)(ii) For x; y ∈ Bd (0; N ), a rough approximation yields
n
X
|Yk−1 (x) − Yk−1 (y)| kk k
6b(N )kx − yk Rn ;
kn (x) − n (y)k6
2 )
h(s
k−1
k=1
Pn
2
where Rn = k=1 k−1 kk k=h(sk−1 ). Its compensation gives
Wn = Rn − R˜ n =

n
X
k−1 (kk k − E(kk k=Fk−1 ))
2 )
h(sk−1

k=1

:

The previous moment inequality, proves that
n
2
X
(k−1
)1+
¡ ∞ a:s:
trhW in 6C(N )
2 )1+ (LL(s2 ))1−
(sk−1
k−1
k=1

Thus, Wn → W∞ ¡ ∞ a.s. and once again by the Chebychev inequality, E(kk k|Fk−1 )
2
2
)=sk−1
)1+2 , it follows that
6C(N )(k−1 h(sk−1
R˜ n 6C(N )

n
2
X
(k−1 )1+ (LL(sk−1
))
k=1

2 )1+
(sk−1

¡∞

a:s:

Therefrom, R˜ n → R˜ ∞ and Rn converges a.s. to some nite r.v. R∞ . This implies,
supn kˆn (x) − ˆn (y)k6b(N )kx − yk R∞ , and proves the equicontinuity of n .
Corollary 3.2. If n2 = an
;
¿ 0; the sequence ((2n
+1 LL(n))−1=2 Mn (·))n¿1 is a.s.
relatively compact in C(Rd ; Rp ); for all  ¿ 0.
Pn
Proof. The asymptotic equivalence sn2 = a k=1 k
∼ C ten
+1 ,
P
P∞
∞


+1 1+
)
= k=1 (LL(k)) =k +1 ¡ ∞.
k=1 (LL(k)) (k =k

implies

that

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R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

Part B. Functional estimation
This part deals with the kernel estimation of the unknown but smooth regression
function f from Rd to Rd that drives a controlled Markov model of type (Du
o,
1990)
Xn+1 = f(Xn ) + C(Xn ; Un ) + n+1 :

(3)

The control C is assumed to be known and the sequence (n ) to be a white noise
with respect to some ltration F = (Fn )n¿0 , i.e., all i have the same distribution and
n+1 is independent of Fn for all n. Model (3) extends the classical linear regression
model Xn+1 = AXn + n+1 and the nonlinear model of Hernandez-Lemma (1991),
Xn+1 = f(Xn ) + n+1 :

(4)

In the sequel, we content with a smooth kernel K and a bandwidth well adapted to
iterative computations and tracking (Masry and Gyorfy, 1987). We assume also that
K is Lipschitzian with order  and coecient k, and that
H3

d
K is a nonnegative
Z and compactly supported function on R ;
which satises
K(z) d z = 1 and |K(u) − K(v)|6kku − vk :

If the dynamic system (4) is stable and if the stationary distribution  has a density
h, a kernel estimator of h is dened for all  ¿ 0, as follows:
n
1 X d 
i K(i (Xi − x));
(5)
hˆ n (x) =
n
i=1

next, the function f of model (3) (or (4) if C ≡ 0) can be estimated by
Pn d 
i K(i (Xi − x))(Xi+1 − C(Xi ; Ui ))
ˆ
Pn d 
:
fn+1 (x) = i=1
i=1 i K(i (Xi − x))

(6)

We assume that fˆn+1 (x) = 0 whenever hˆ n (x) = 0.
The a.s. and weak convergence rates as well as the iterated logarithm laws of these
estimators are quite similar to those obtained in the i.i.d. case (Hall, 1981; Mack
and Silverman, 1982; Devroy and Penrod, 1984; Liero, 1989). The results rely on
stability criteria of Lyapounov type presented in Du
o (1990). Note that Iosifescu
and Grigorescu (1990) present a wide range of pointwise a.s., log–log laws and weak
convergence results and some invariance principles for dependent sequences (called
random systems with complete connections). However, our proofs seem to have no
counterparts in their framework.
We now brie
y recall the main denitions of Du
o (1990) used in the sequel. A
sequence (Xn ; Un ) of r.v. adapted to a ltration F with values in (E × U ; E ⊗ U), is
a controlled Markov chain if for some transition probability (x; u; dy) from E × U
in E, the distribution of Xn+1 conditionally to Fn is (Xn ; Un ; d x). E is said the
state space and U the control space. Any sequence  = (dn ) of measurable functions
dn from E n+1 to U is called a strategy. The strategy determines the control at any
time: Un = dn (X0 ; : : : ; Xn ). If, for a xed state x, the control dn (x0 ; : : : ; Xn−1 ; x) belongs
to a subset A(x) for all n, the strategy is said admissible. Next, let us note that

203

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

every sequence (Xn ) of r.v. gives rise to a sequence of empirical distributions
Pn
n (B)=1=(n+1) i=0 5{Xi ∈B} ; B ∈ B(Rd ). The sequence is said stable if, the sequence
n converges weakly to a stationary distribution  a.s. In the controlled case (3), a
class D of strategies stabilizes the sequence if a.s., for any admissible  ∈ D, any
initial distribution and ∀ ¿ 0, there exists a compact C such that limn n (C )¿1 − .
4. Non-controlled models
4.1. Strong convergence
Theorem 4.1. For the autoregressive model (4); assume that
1. f is continuous and limkxk→∞ kf(x)k=kxk ¡ 1.
2. n is a white noise with density p of class C 1 and p and its gradient are bounded.
3. Model (4) is stable.
Then;
(A) Stationary Rdistribution  has a bounded density h of class C 1 which satises h(x) = p(x − f(z))h(z) d z. Moreover; for all 0 ¡  ¡ 1=d and all initial
distribution; hˆ n (x) → h(x) a.s.; uniformly on compacts.
(B) For all x ∈ S = {x; h(x) ¿ 0}; fˆn (x) pointwise converges to f(x) a.s. If the
noise has a moment of order m ¿ 2 (assumption H1) and if 0 ¡  ¡ 1=2d; the
pointwise convergence strengthens to uniform convergence on compacts.
Finally; if f is of class C 1 ; then for all  ¿ 0; N ¡ ∞ and
=inf (; 21 −(d+));
sup kfˆ n (x) − f(x)k = O(n−
(LL(n))1=2 ) a:s:

kxk6N

We rst prove a lemma which enables us to convert the arithmetic mean to a type of
weighted means.
Let x be a sequence of real numbers x n , or in any normed spaces, and put Sn () =
Pn  n
Pn
i=1 i xi ; Sn =
i=1 xi .
Lemma 4.1. If Sn =n → s; then
(i) n−(1+) Sn () → s=(1 + ) if ¿0,
(ii) kSn ()k = O(n1+ ) if − 1 ¡  ¡ 0,
(iii) kSn ()k = O(log(n)) if  = −1,
(iv) kSn ()k = O(1) if  ¡ − 1.
Proof. If ai = i((i + 1) − i ) put n =
Sn () =

n
X
i=1

Pn

i=1

ai , then n = (n + 1)+1 −

i (Si − Si−1 ) = n+1 (Sn =n) −

n−1
X
i=1

ai (Si =i):

Pn+1
i=1

i , and

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R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

(i) if ¿0, the following inequalities
(n + 1)+1 =( + 1)6

n+1
X
i=1

(n + 1)

+1

1
1−
+1



i 6((n + 2)+1 − 1)=( + 1);
n+2
n+1

+1 !

6n 6


(n + 1)+1
+1

imply that limn n =(n + 1)+1 = =( + 1). Thus the lemma of Toepliz (on normed
spaces) applies and yields ( + 1)Sn ()=n+1 → s, since
!


n−1
X
Sn ()
Sn
n+1
+1
1
=
×
×
ai (Si =i) →  s

−
− s = s:
n−1
n−1
n
n−1

i=1

(ii) Since (Sn =n) converges, it is bounded by some constant M , and then
!!
!


n−1
n−1
X
X
i+1
+1
+1

+1
−1
6M n
+
kSn ()k6M n
+
i :
i
i
i=1

i=1

The last inequality follows from the Taylor formula of function x .
If −1 ¡  ¡ 0, we get kSn ()k6Cn1+ . Conditions (iii) and (iv) follow by similar
arguments.
Proof of Theorem 4.1. Since the noise has a density p and f is continuous, the
probability transition is strongly Fellerian and in this case, the stability is equivalent to
the positive recurrence. We recall also that assumption p ¿ 0 (or  ¡ 1) is sucient
to ensure the stability of the chain (Du
o, 1990).
A.1. Properties of the stationary distribution.
Thus R(Xn ) isR positive recurrent with
R
an invariant distribution  that satises g(x) d(x) = g(x)( p(x − f(z)) d(z)) d x,
for all Rbounded and measurable functions g. This is nothing but d(x) = h(x) d x where
h(x)= p(x−f(z))h(z) d z. Density h is bounded, continuous or dierentiable whenever
is p.
n
A.2. Study of hˆ (x): Let ¿d; ′ =  + 1 − d, be two constants and  a
non-negative function on RRd , with a compact support and Lipschitzian of order 
and coecient k. Put  = (z) d z and
Hn (x) = Hn (; x) =

n
X
i=1

i (i (Xi − x)):

A.2(a) For x ∈ RRd , let us consider the martingale Mn (x) = Hn (x) − H˜ n (x), where
Pn
˜
H n (x) = i=1 i−d (z)p(i− z + x − f(Xi−1 )) d z, and its increasing process
hM (x)in =

n
X
i=1

i

2

Z

2 (i (z − x))p(z − f(Xi−1 )) d z

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

−
6

Z

n
X
i=1

i

2



 (i (z − x))p(z − f(Xi−1 )) d z

2−d

Z

2 !

2 (z)p(i− z + x − f(Xi−1 )) d z6C
−′

′

205

n
X

i2−d :

i=1

Since 2 ¿ 2 − d + 1, we get n Mn (x) → 0 a.s. for each x ∈ Rd .
Next, we can nd constants a; b, such that
|
Mn (x)|6bn ; hM (x)in 6 an(1+2−d) ;

and easily prove that |(i (z − x) − (i (z − y)|6Cn kx − yk ; ∀i6n, for 0 ¡  ¡ 
for some constant C and thus,
|
Mn (x) −
Mn (y)| = n |((n (Xn − x)) − (n (Xn − y)))
Z
+ ((n (z − x)) − (n (z − y)))p(z − f(Xn−1 )) d z|
6 Cn+ kx − yk ;

hM (x) − M (y)in 6

n
X
i=1

i

2

Z

((i (z − x)) − (i (z − y)))2 p(z − f(Xi−1 )) d z
′

′

6 Ckx − yk n(1+2−d+ ) :

If 0 ¡ ′ = 2 ¡  where  is arbitrary and 0 ¡  ¡ 1=d, there exists ′ such that
′ = 1 +  − d ¿ =2 . Now, Theorem 2.2 applies with sn2 = n ,  = 1 + 2 − (d − ′ )
and asserts that the sequence ((n LL(n))−1=2 Mn (·))n is a.s. relatively compact.
A.2(b) Rewrite, Hn (x) = Mn (x) + (H˜ n (x) − Hn′ (x)) + Hn′ (x) where Hn′ (x) =
Pn −d

p(x − f(Xi−1 )).
 i=1 i
Since the chain is stable and p bounded and continuous, we rst get
Z
n
1X
p(x − f(Xi−1 )) → p(x − f(z))h(z) d z = h(x) a:s:
n
i=1

′

a.s. ∀x. Moreover,
By Lemma 4.1, ′ n− Hn′ (x) → h(x)
′
 −′
n− |Hn′ (x) − Hn′ (y)| 6 n

n
X
i=1

i−d |p(x − f(Xi−1 )) − p(y − f(Xi−1 ))|

6 Ctekx − ykn−
−′

Pn

′

n
X
i=1

i−d 6Ckx − yk:

−d

converges and this proves the a.s.
Lemma 4.1 again implies that, n
i=1 i
−′ ′
equicontinuity of sequence n Hn (·).
Finally, since grad p is bounded and  has a compact support, we have
Z
n
X
i−d (z)|p(x − f(Xi−1 )) − p(i− z + x − f(Xi−1 ))| d z
|H˜ n (x) − Hn′ (x)| 6
i=1

6C

Z

(z)kzk d z

n
X
i=1

i−d− :

206

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

Pn
′
Thus, supx∈Rd | H˜ n (x) − Hn′ (x)|6C i=1 i−d− = o(n− ) a.s., if  ¿ 0.
To resume, we have proved that, for all N ¡ ∞, 0 ¡  ¡ 1=d,
′

sup |′ n− Hn (x) − h(x)|
→0

a:s:

kxk6N

Taking  = d (′ = 1) and  = K ends the proof of statement A.
B. Study of fˆn (x): We decompose the bias into
fˆn+1 (x) − f(x) = (nhˆ n (x))−1 (Wn+1 (x) + Rn+1 (x));
where Ki (x) = K(i (Xi − x)), and
Wn+1 (x) =

n
X

id Ki (x)i+1 ;

Rn+1 (x) =

i=1

n
X
i=1

(7)

id Ki (x)(f(Xi ) − f(x)):

B.1. Uniform convergence of Rn (x): Let N ¡ ∞ and assume with no loss of generality, that supp(K) ⊂ Bd (0; R). Since f is continuous, then ∀ ¿ 0; ∃ = () ¿ 0;
such that kf(x) − f(y)k6, for all kxk6R + N , kyk6R + N and kx − yk6. Next,
observe that if x ∈ Bd (0; N ) we have either kXi − xk ¿ Ri− and then Ki (x) = 0,
or kXi − xk6Ri− and then kXi k6R + N . In this last case, the rst alternative
R6:i yields kf(Xi ) − f(x)k6 and the second alternative R ¿ i yields kf(Xi ) −
f(x)k62 supkzk6R+N kf(z)k = L.
We put n1 = inf (n; R6n ), and get the inequalities
kRn (x)k6C + :

n−1
X
i=n1

id K(x)6C + (n − 1)hˆ n−1 (x)

which prove the a.s. uniform convergence on compacts of Rn (·)=n, i.e., ∀ ¿ 0,
limn supkxk6N kRn (x)=nk6 supkxk6N h(x) a.s.
B.2. If f is of class C 1 and C = supx∈Bd (0; R+N ) kgrad fk, we have

Ki (x)kf(Xi ) − f(x)k6CRi− Ki (x);
Pn−1
and, kRn (x)k6C i=1 i(d−1) Ki (x). Then, Lemma 4.1 and the uniform convergence of
hˆ n (·) (part A) yield,
sup kRn (x)k = O(n1− )

a:s:

(8)

kxk6N

B.3. Pointwise convergence of fˆn (·) :
hW (x)in =

n−1
X

i2d Ki2 (x):

i=1

Since K is Lipschitzian, the kernel K 2 is also Lipschitzian with the same order . If
we take  = K 2 ,  = 2d, ′ =  − d + 1 = 1 + d, and proceed in the same way as
in part A, we get
Z
−(1+d)
hW (x)in − h(x) K 2 (z) d zk → 0 a:s:
(9)
sup k(1 + d)n
kxk6N

Since, trhW (x)in =O(n1+d ), n−
Wn (x) → 0 a.s. for all
¿ (1+d)=2; and the particular
case
= 1 proves the pointwise convergence of fˆn (·).

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

207

B.4. Uniform convergence of fˆn (·): Clearly, if the noise has a moment of order
m ¿ 2, Wn meets assumption H1 of Proposition 1 and then, it is enough to verify
assumption H2.
Indeed, Inequality |Yn (0)|6Cnd Ki (0) and arguments as in part A.2 show that,
|Yn (x) − Yn (y)|6Cn(d+) kx − yk ,  ¿ 0. Therefore, Corollary 3 applies and says
that the sequence Wn (·)=(2n1+2(d+) LL(n))1=2 is a.s. relatively compact.
Since, for all
¿ (1 + 2d)=2, there exists  ¿ 0 such that
¿ (1 + 2(d + ))=2,
we get supkxk6N n−
Wn (x) → 0 a.s.
In particular, the value
= 1 is possible if  ¡ 1=2d.
Summing up, we have proved that,
if  ¿ 0 and 12 (d +  + 1)6 ¡ 12 (d + ), then
sup kfˆn (x) − f(x)k = O((n(2(d+)−1) LL(n))1=2 ):
kxk6N

Note that the case ¿ 21 (d + ) is useless, since the uniform convergence of the estimator is not ensured and that the other case 0 ¡ 6 21 (d +  + 1) yields
sup kfˆn (x) − f(x)k = O(n− (LL(n))1=2 ):
kxk6N

We do not know yet if the value  = 21 (d + 1) which gives the best rate is attainable.
4.2. Pointwise CLT and ILL
Theorem 4.2. If the assumptions of Theorem 4:1 hold; if f is of class C 1 and if
1=(d + 2) ¡  ¡ 1=d; then for x1 ; : : : ; xq ∈ S:
(A) (Zn (x1 ); : : : ; Zn (xq )) where Zn (xj )=n(1−d)=2 (fˆn (xj )−f(xj )); converges weakly to a
Gaussian distribution
in Rd×q which has q independent components Nd (0; ( =(1+
R 2
d)h(xj )) K (z) d z); j = 1; : : : ; q:
(B) Moreover; if the noise has a nite conditional moment of order m ¿ 2; a pointwise iterated logarithm law holds on S;
 1−d 
Z
tr
n
2
ˆ
kfn (x) − f(x)k 6
K 2 (z) d z a:s:
lim
n
2 LL(n)
(1 + d)h(x)
Proof. Considering bias (7), we have already proved that hˆ n (x) → h(x) ¿ 0 a.s. on
S, if  ¡ 1=d. Now, if f is C 1 and  ¿ 1=(d + 2); the upper bound (8) improves
into, supkxk6N kRn (x)k = o(n(1+d)=2 ) a.s., and it remains only to study the asymptotic
behavior (CLT and ILL) of Wn (x).
(A) We start checking the CLT assumptions for martingales
(Du
o, 1990).
R 2
−(1+d)
hW (x)in → h(x) K (z) d z a.s.
By (9), we get (1 + d)n
For the Lindeberg’s condition, we note that V (t) = E(kk2 5{kk¿t} ) → 0 if t → ∞,
and that for  ¿ 0 and kKk = sup(|K(u): u ∈ Rd ), we have
n
X
E(k
Wi (x)k2 5{k
Wi (x)k¿:n(1+d)=2 } )
i=1

6

n
X
i=1

i2d Ki2 (x)V



n(1−d)=2
kKk



= o(n1+d ) a:s:

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R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

This is enough to prove the weak convergence of each Z(xi ).
For the independence of components, it is enough to prove that, a.s.,
lim hW (x); W (y)in = lim
n

n

n
X
i=1

i(2d) Ki (x)Ki (y) ¡ ∞;

if x 6= y:

(10)

Considering the events Ai = {Xi ∈ Bd (x; Ri− ) ∩ Bd (y; Ri− )}, we dene the martingale
Pn
Mn = i=1 i2d (5Ai − P(Ai |Fi−1 )) and its increasing process hM in . Since the density
is bounded, it follows by integration on Rd that,
P(Ai | Fi−1 ) = P(i ∈ Bd (x − f(Xi−1 ); Ri− ) ∩ Bd (y − f(Xi−1 ); Ri− )| Fi−1 )
6 5{kx−yk62Ri− } P(i ∈ Bd (x − f(Xi−1 ); 2Ri− )| Fi−1 )
6 Ci−d 5{kx−yk62Ri− } :

Put N (x; y) = inf (i: kx − yk ¿ 2Ri− ) and observe that,
hM in 6

n
X
i=1

N (x;y)

i4d P(Ai | Fi−1 )6C

X
i=1

i3d ¡ ∞:

Thus, Mn converges a.s. to a nite r.v. M∞ . Moreover, since
n
X
i=1

i2d P(Ai | Fi−1 )6C

n
X
i=1

id 5{kx−yk62Ri− } 6CN (x; y) ¡ ∞ a:s:;

2

the bound Ki (x)Ki (y)6kKk 5Ai implies that (10) holds.
(B) The second part of theorem is a simple consequence of Proposition 3.1 and RePn
P∞
mark 3.1 if we take m=2+2, i =id Ki (x), sn2 = i=1 i2 and prove that i=1 (i2 =si2 )1+
converges a.s.
R
Since, limn (1+d)n−(1+d) sn2 =h(x) K 2 (z) d z, it is enough to show the convergence
P∞
of i=1 i−
Zi where Zi = id Ki2(1+) (x) and
= (1 − d)(1 + ) + d:
Pn
First, observe that
i=1 Zi =n converges if  ¡ 1=d, and then apply Lemma 4.1
P∞ −
i
Z
;
since

¡
1: Next, note that limn (2sn2 LL(sn2 ))−1 kWn (x)k2 6tr a.s. to
to
i
i=1
complete the proof.
4.3. Noise density estimator
If, in addition to functions f and h of the non-controlled model (4), we need to
estimate the noise density p, we consider on R2 × R2 the autoregressive model, Zn+1 =
∗
, where
F(Zn ) + n+1






Xn+1
n+1
f(x)
∗
; F(x; y) =
:
and n+1 =
Zn+1 =
Xn
n
f(y)
Put K ∗ =K ⊗K, choose a point x0 , h(x0 ) ¿ 0, and dene the following kernel estimate:
∗

pˆ n (y) = (hˆ n (x0 ))−1 hˆ n (x0 ; y + fˆn (x0 ));
Pn
∗
where hˆ n (x; y) = n−1 i=1 i2d K ∗ (i− (Xi−1 − x; Xi − y)).

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

209

Corollary 4.1. Under the assumptions of Theorem 4:1; pˆ n (·) converges a.s. to p(·)
uniformly on S ∩ C ; for all compact C for all  ¡ 1=2d.
If f is known at some point x0 ; h(x0 ) ¿ 0, we had to substitute advantageously the
value fˆn (x0 ) for f(x0 ). Another method is to vary x0 and take x0 = y for example.
Sketch of the proof. (1) Note rst, that if K is of order  and meets assumption (H3),
so is K ∗ on Rd+d . Note also that, the noise ∗ is not white anymore and the rst part
of the proof of Theorem 4.1 demands a slight modication. However, the stability of
the chain (Xn ) implies the stability of (Zn ), which has a stationary distribution ∗ with
density h∗ (x; y) = h(x)p(y − f(x)).
Pn
(2) Split up Hn∗ (x; y) = i=1 i K ∗ (i (Xi−1 − x; Xi − y)), into
∗

∗

∗

∗

∗

∗

Hn∗ = (Hn∗ − H˜ n ) + (H˜ n − Hˆ n ) + (Hˆ n − H n ) + H n ;

∗
where, H n (x; y) = p(y − f(x))
∗
H˜ n (x; y) =

n
X

i−d Ki−1 (x)

i=1

∗
Hˆ n (x; y) =

n
X

Pn

i



i=1

Z

i=1

i−2d p(x − f(Xi−2 )),

Z

K(v)p(i− v + y − f(Xi−1 )) dv;

K ∗ (i (u; v))p(u + x − f(Xi−2 ))p(v + y − f(u + x)) du dv:
′

We rst prove that, limn→∞ ′ n− Hn∗ (x; y) = h(x)p(x − f(x)) for all ¿2d and
 =  − 2d + 1.
∗
∗
Note that, H˜ n is the F-compensator of Hn∗ and that Hˆ n is the F ∗ -compensator of
∗
H˜ n where F ∗ = (Fn−1 )n¿1 .
(i) We follow the proof of Theorem 4.1 and readily obtain the a.s. uniform conver′
∗
gence on compacts of ′ n− H n to h(x)p(x − f(x)), if we take  = .
(ii) Put en = sup(!n ; 2Rn− ); where !n = !(f; R + N; n− ) is the continuity modulus
of f. Since p and its gradient are bounded and continuous, K has its support included
in Bd (0; R), and !n → 0, we get for all x; y; kxk6N; kyk6N;
′

K ∗ (u; v)|p(i− u + x − f(Xi−2 ))p(i− v + y − f(i− u + x))
−p(x − f(Xi−2 ))p(y − f(x))|

6C(i− (kuk + kvk) + kf(i− u + x) − f(x)k)5{kuk6R; kvk6R} 6Cei :

For example, applying Lemma 4.1 to the sequence ei , we obtain
sup
kxk6N;kyk6N

∗
∗
n− |Hˆ n − H n |(x; y)6Cn−
′

′

n
X
i=1

i−2d ei → 0:

The iterated logarithm laws for martingales are proved as in Theorem 4.1 for the other
two components and complete the proof.

210

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

5. Controlled model
The controlled models (3) have no stationary distribution in general, and the statistic
hˆ n (x) is not intended to estimate anything actual. However fˆn (x) continue to estimate
the regression function f as shown below.
Theorem 5.1. Assume that
(1) C is known and f unknown but continuous;
(2) Noise (n ) is white with a strictly positive and C 1 -dierentiable density p and
if p and its gradient are bounded.
(3) u(x) = supu∈A(x) kC(x; u)k is bounded on compacts and
supu∈A(x) kf(x) + C(x; u)k
¡ 1:
kxk
kxk→+∞
lim

Then; for all initial distributions and admissible strategies; statement B of Theorem
4:1 continues to hold on Rd .
Proof. Only minor modications of the proof of Theorem 4.1 are needed. First, there
is nothing to change in studying Mn = Hn − H˜ n , since if F(x; u) = f(x) + C(x; u), the
process Hn (x) = nhˆ n (x) has compensator
n Z
X
˜
K(z)p(i− z + x − F(Xi−1 ; Ui−1 )) d z:
Hn =
i=1

Next, we note that we must only bound hˆ n (:) on compacts (and not to deal with its
convergence), the Lyapounov condition (3) enables us to stabilize the chain (Du
o,
1990) and then, to get a constant M so that,
n
1X
lim
kXi k2 6M and for r ¿ M; lim n (Bd (0; r))¿1 − (M=r)2 :
n n
n
i=1

Set, d(r)=sup(kF(x; u)k; u ∈ A(x); kxk6r): It follows that m(r)=inf (p(z): kzk6R+
N + d(r)) ¿ 0; since p ¿ 0 and continuous.
For large r, so that d(r)6r where  ¡ 1, we obtain
Z
kpk ¿ K(z)p(i− z + x − F(Xi−1 ; Ui−1 )) d z
Z
¿ m(r)5{kxk6N; kXi−1 k6r} K(z) d z = m(r)5{kxk6N; kXi−1 k6r} ;
˜
i.e., kpk¿n−1
√H n (x)¿m(r)n (Bd (0; r)); for all kxk6N .
Thus, if r ¿ 2M , by the Lyapounov condition we get the bounds
m(r)
6 lim inf h˜n (x)6 lim sup h˜n (x)6kpk =
¡ ∞:
0 ¡ 6
n kxk6N
2
n kxk6N

Since all terms in Lemma 4.1 are positive, we get
!
n−1
X
−(1+)
−1
1+
ai (Si =i) 6Sn =n:
Sn () = n Sn − n
n
i=1

Thus, bounds (8) and (9) remain true, and this ends the proof.

R. Senoussi / Stochastic Processes and their Applications 89 (2000) 193–211

211

6. For further reading
The following reference is also of interest to the reader: Stute, 1982.
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