Large and moderate deviation principles for recursive kernel density estimators defined by stochastic approximation method

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by  Yousri Slaoui

Large and moderate deviation principles for recursive kernel
density estimators defined by stochastic approximation method
Yousri Slaoui
Abstract
Inthispaperweprovelargeandmoderatedeviationsprinciplesfortherecursivekernelestimators
of a probability density function defined by the stochastic approximation algorithm introduced by
3 Mokkadem et al. [2009. The stochastic approximation method for the estimation of a probability
1 density. J.Statist. Plann. Inference139, 2459-2478]. Weshowthattheestimator constructedusing
0 thestepsizewhichminimizethevarianceoftheclassoftherecursiveestimatorsdefinedinMokkadem
2 etal. (2009)givesthesamepointwiseLDPandMDPastheRosenblattkernelestimator. Weprovide
n results both for the pointwise and theuniform deviations.
a
2000 Mathematics Subject Classifiation: 62G07, 62L20 , 60F10.
J
Key Words: Density estimation; Stochastic approximation algorithm; Large and Moderate deviations
7
principles.
2

]
T 1 Introduction
S
. Let X ,...,X be independent, identically distributed Rd-valued random vectors, and let f denote the
h 1 n
t probability density of X1. To construct a stochastic algorithm, which approximates the function f at
a
a given point x, Mokkadem et al. (2009) defined an algorithm of search of the zero of the function
m
h :y 7→f(x)−y. They proceed as follows: (i) they set f (x)∈R; (ii) for all n≥1, they set
0
[

f (x)=f (x)+γ W (x)
1 n n−1 n n
v
whereW (x)isan“observation” ofthefunction hatthepointf (x)and(γ )isasequenceofpositive
2 n n−1 n
9 real numbers that goes to zero. To define Wn(x), they follow the approach of Révész (1973, 1977) and
3 of Tsybakov (1990), and introduced a kernel K (which is a function satisfying K(x)dx = 1) and a
Rd
6 bandwidth (h ) (which is a sequence of positive real numbers that goes to zero), and they set W (x)=
n n
1. h−dK(h−1[x−X ])−f (x). The stochastic approximationalgorithm introduRced in Mokkadem et al.
n n n n−1
0 (2009) which estimate recursively the density f at the point x is
3
1 x−X
v: fn(x)=(1−γn)fn−1(x)+γnh−ndK h n . (1)
(cid:18) n (cid:19)
i
X
Recently,largeandmoderatedeviationsresultshavebeenprovedforthe well-knownnonrecursivekernel
r densityestimatorintroducedbyRosenblatt(1956)(seealsoParzen,1962). Thelargedeviationsprinciple
a
has been studied by Louani (1998) and Worms (2001). Gao (2003) and Mokkadem et al. (2005) extend
these results and provide moderate deviations principles. The purpose of this paper is to establish
large and moderate deviations principles for the recursive density estimator defined by the stochastic
approximation algorithm (1).
Let us first recall that a Rm-valued sequence (Z ) satisfies a large deviations principle (LDP)
n n≥1
with speed (ν ) and good rate function I if :
n
1. (ν ) is a positive sequence such that lim ν =∞;
n n→∞ n
2. I :Rm →[0,∞] has compact level sets;
3. for every borel set B ⊂Rm,
− inf I(x) ≤ liminfν−1logP[Z ∈B]
◦ n→∞ n n
x∈B
≤ limsupν−1logP[Z ∈B]≤− inf I(x),
n n
n→∞ x∈B
1
◦
where B and B denote the interior and the closure of B respectively. Moreover, let (v ) be a
n
nonrandom sequence that goes to infinity; if (v Z ) satisfies a LDP, then (Z ) is said to satisfy a
n n n
moderate deviations principle (MDP).
The first aim of this paper is to establish pointwise LDP for the recursive kernel density estimators
defined by the stochastic approximation algorithm (1). It turns out that the rate function depend on
the choice of the stepsize (γ ); In the first part of this paper we focus on the following two special cases
n
: (1) (γ )= n−1 and (2) (γ )= hd n hd −1 , the first one belongs to the subclass of recursive
n n n k=1 k
kernel estima(cid:0)tors(cid:1)which have a min(cid:16)imu(cid:0)mPMSE o(cid:1)r M(cid:17)ISE and the second choice belongs to the subclass
of recursive kernel estimators which have a minimum variance (see Mokkadem et al., 2009).
We show that using the stepsize (γ ) = n−1 and (h ) ≡ (cn−a) with c > 0 and a ∈ ]0,1/d[, the
n n
sequence (f (x)−f(x)) satisfies a LDP with speed nhd and the rate function defined as follows:
n (cid:0) (cid:1) n
(cid:0) (cid:1)
if f(x)6=0, I :t→f(x)I 1+ t
a,x a f(x) (2)
( if f(x)=0, Ia,x(0)=0 and(cid:16)Ia,x(t)=(cid:17) +∞ for t6=0.
where
I (t)=sup{ut−ψ (u)}
a a
u∈R
ψ (u)= s−ad eusadK(z)−1 dsdz,
a
Z[0,1]×Rd (cid:16) (cid:17)
which is the same rate function for the LDP of the Wolverton and Wagner (1969) kernel estimator (see
Mokkadem et al., 2006).
Moreover, we show that using the stepsize (γ ) = hd n hd −1 and more general bandwiths
n n k=1 k
defined as hn = h(n) for all n, where h is a regularly v(cid:16)arin(cid:0)gPfunction(cid:1)wi(cid:17)th exponent (−a), a ∈ ]0,1/d[.
We prove that the sequence (f (x)−f(x)) satisfies a LDP with speed nhd and the rate function
n n
defined as follows:
(cid:0) (cid:1)
if f(x)6=0, I :t→f(x)I 1+ t
x f(x) (3)
( if f(x)=0, Ix(0)=0 and(cid:16) Ix(t)=(cid:17)+∞ for t6=0.
where
I(t)=sup{ut−ψ(u)}
u∈R
ψ(u)= euK(z)−1 dz,
ZRd(cid:16) (cid:17)
which is the same rate function for the LDP of the Rosenblatt kernel estimator (see Mokkadem et al.,
2005).
Our second aim is to provide pointwise MDP for the density estimator defined by the stochastic
approximation algorithm (1). In this case, we consider more general stepsizes defined as γ = γ(n) for
n
all n, where γ is a regularly function with exponent (−α), α ∈ ]1/2,1]. Throughout this paper we will
use the following notation:
ξ = lim (nγ )−1. (4)
n
n→+∞
For any positive sequence (v ) satisfying
n
γ v2
lim v =∞ and lim n n =0
n→∞ n n→∞ hd
n
and general bandwidths (h ), we prove that the sequence
n
v (f (x)−f(x))
n n
2
satisfies a LDP of speed hd/ γ v2 and rate function J (.) defined by
n n n a,α,x
if f(x(cid:0))6=0(cid:0), J (cid:1)(cid:1) :t→ t2(2−(α−ad)ξ)
a,α,x 2f(x)RRdK2(z)dz (5)
( if f(x)=0, Ja,α,x(0)=0 and Ja,α,x(t)=+∞ for t6=0.
−1
Let us point out that using the stepsize (γ )= hd d hd which minimize the variance of f ,
n n k=1 k n
weobtainthesameratefunctionforthepointwis(cid:18)eLD(cid:16)PPandMD(cid:17)Pa(cid:19)stheoneobtainedfortheRosenblatt
kernel estimator.
Finally, we give a uniform version of the previous results. More precisely, let U be a subset of Rd; we
establish large and moderate deviations principles for the sequence (sup |f (x)−f(x)|).
x∈U n
2 Assumptions and main results
We define the following class of regularly varying sequences.
Definition 1. Let γ ∈R and (v ) be a nonrandom positive sequence. We say that (v )∈GS(γ) if
n n≥1 n
v
n−1
lim n 1− =γ. (6)
n→+∞ (cid:20) vn (cid:21)
Condition (6) was introduced by Galambos and Seneta (1973) to define regularly varying sequences
(see also Bojanic and Seneta, 1973),and by Mokkadem and Pelletier (2007)in the context of stochastic
approximationalgorithms. Typicalsequences in GS(γ) are,for b∈R, nγ(logn)b, nγ(loglogn)b, andso
on.
2.1 Pointwise LDP for the density estimator defined by the stochastic ap-
proximation algorithm (1)
2.1.1 Choices of (γ ) minimizing the MISE of f
n n
It was shown in Mokkadem et al. (2009) that to minimize the MISE of f , the stepsize (γ ) must be
n n
chosenin GS(−1) and must satisfy lim nγ =1. The most simple example of stepsize belonging to
n→∞ n
GS(−1) and such that lim nγ = 1 is (γ ) = n−1 . For this choice of stepsize, the estimator f
n→∞ n n n
defined by (1) equals the recursive kernel estimator introduced by Wolverton and Wagner (1969).
(cid:0) (cid:1)
To establish pointwise LDP for f in this case, we need the following assumptions.
n
(L1)K :Rd →Risaboundedandintegrablefunctionsatisfying K(z)dz =1,andlim K(z)=
Rd kzk→∞
0.
R
(L2) i) (h )=(cn−a) with a∈]0,1/d[ and c>0.
n
ii) (γ )= n−1 .
n
The following Th(cid:0)eore(cid:1)m gives the pointwise LDP for f in this case.
n
Theorem 1 (Pointwise LDP for Wolverton and Wagner estimator).
Let Assumptions (L1) and (L2) hold and assume that f is continuous at x. Then, the sequence
(f (x)−f(x)) satisfies a LDP with speed nhd and rate function defined by (2).
n n
(cid:0) (cid:1)
2.1.2 Choices of (γ ) minimizing the variance of f
n n
ItwasshowninMokkademetal. (2009)thattominimizetheasymptoticvarianceoff ,thestepsize(γ )
n n
mustbe chosenin GS(−1)andmustsatisfy lim nγ =1−ad. The mostsimple exampleofstepsize
n→∞ n
belonging to GS(−1) and such that lim nγ = 1−ad is (γ ) = (1−ad)n−1 , an other stepsize
n→∞ n n
satisfying this conditions is (γn)= hdn nk=1hdk −1 . For this last ch(cid:0)oice of stepsiz(cid:1)e, the estimator fn
defined by (1) produces the estimat(cid:16)or co(cid:0)Pnsidered(cid:1)by(cid:17)Deheuvels (1973) and Duflo (1997).
To establish pointwise LDP for f in this case, we assume that.
n
3
(L3) i) (h )∈GS(−a) with a∈]0,1/d[.
n
ii) (γ )= hd n hd −1 .
n n k=1 k
The following Th(cid:16)eore(cid:0)mPgives t(cid:1)he p(cid:17)ointwise LDP for f in this case.
n
Theorem 2 (Pointwise LDP for Deheuvels estimator).
Let Assumptions (L1) and (L3) hold and assume that f is continuous at x. Then, the sequence
(f (x)−f(x)) satisfies a LDP with speed nhd and rate function defined by (3).
n n
(cid:0) (cid:1)
2.2 Pointwise MDP for the density estimator defined by the stochastic ap-
proximation algorithm (1)
Let (v ) be a positive sequence; we assume that
n
(M1) K : Rd → R is a continuous, bounded function satisfying K(z)dz = 1, and, for all j ∈
Rd
{1,...d}, z K(z)dz =0 and z2|K(z)|dz <∞.
R j j Rd j R
(M2) i) (γ )∈RGS(−α) with α∈]1/2R,1].
n
ii) (h )∈GS(−a) with a∈]0,α/d[.
n
iii) lim (nγ )∈]min{2a,(α−ad)/2},∞].
n→∞ n
(M3) f is bounded, twice differentiable, and, for all i,j ∈{1,...d}, ∂2f/∂x ∂x is bounded.
i j
(M4) lim v =∞ and lim γ v2/hd =0.
n→∞ n n→∞ n n n
The following Theorem gives the pointwise MDP for f .
n
Theorem 3 (Pointwise MDP for the recursive estimators defined by (1)).
Let Assumptions (M1) − (M4) hold and assume that f is continuous at x. Then, the sequence
(f (x)−f(x)) satisfies a MDP with speed hd/ γ v2 and rate function J defined in (5).
n n n n a,α,x
(cid:0) (cid:0) (cid:1)(cid:1)
2.3 Uniform LDP and MDP for the density estimator defined by the stochas-
tic approximation algorithm (1)
To establish uniform large deviations principles for the density estimator defined by the stochastic ap-
proximation algorithm (1) on a bounded set, we need the following assumptions:
(U1) i) For all j ∈{1,...d}, z K(z)dz =0 and z2|K(z)|dz <∞.
R j j Rd j
ii) K is Hölder continuous.
R R
(U2) f is bounded, twice differentiable, and, supx∈RdkD2f(x)k<∞.
(U3) lim γnvn2log(1/hn) =0 and lim γnvn2logvn =0.
n→∞ hd n→∞ hd
n n
SetU ⊆Rd;inordertostateinacompactformtheuniformlargeandmoderatedeviationsprinciples
for the density estimator defined by the stochastic approximationalgorithm (1) on U, we set:
kfk I 1+ δ when v ≡1 , (L1) and (L2) hold
U,∞ a kfkU,∞ n
g (δ) =  kfk I (cid:16)1+ δ (cid:17) when v ≡1 , (L1) and (L3) hold
U  δ2U(,2∞−(α(cid:16)−ad)ξ)kfkU,∞(cid:17) when vn →∞ , (M1)−(M4) hold
2kfkU,∞RRdK2(z)dz n
g˜U(δ) = min{gU(δ),gU(−δ)}
where kfk =sup |f(x)|.
U,∞ x∈U
Remark 1. The functions g (.) and g˜ (.) are non-negative, continuous, increasing on ]0,+∞[ and
U U
decreasing on ]−∞,0[, with a unique global minimum in 0 (g˜ (0) = g (0) = 0). They are thus good
U U
rate functions (and g (.) is strictly convex).
U
Theorem 4 below states uniform LDP on U in the case U is bounded, and Theorem 5 in the case U
is unbounded.
4
Theorem 4 (Uniformdeviationsonaboundedsetfortherecursiveestimatordefinedby(1)). Let(U1)−
(U3) hold. Then for any bounded subset U of Rd and for all δ >0,
γ v2
lim n n logP supv |f (x)−f(x)|≥δ =−g˜ (δ) (7)
n→∞ hdn (cid:20)x∈U n n (cid:21) U
To establish uniform large deviations principles for the density estimator defined by the stochastic
approximation algorithm (1) on an unbounded set, we need the following additionnal assumptions:
(U4) i) There exists β >0 such that kxkβf(x)dx<∞.
Rd
ii) f is uniformly continuous.
R
(U5) There exists τ >0 such that z 7→kzkτK(z) is a bounded function.
(U6) i) There exists ζ >0 such that kzkζ|K(z)|dz <∞
Rd
ii) There exists η >0 such that z 7→kzkηf(z) is a bounded function.
R
Theorem 5 (Uniform deviations on an unbounded set for the recursive estimator defined by (1)). Let
(U1)−(U6) hold. Then for any subset U of Rd and for all δ >0,
γ v2
−g˜ (δ) ≤ liminf n n logP supv |f (x)−f(x)|≥δ
U n→∞ hdn (cid:20)x∈U n n (cid:21)
γ v2 β
≤ limsup n n logP supv |f (x)−f(x)|≥δ ≤− g˜ (δ)
hd n n β+d U
n→∞ n (cid:20)x∈U (cid:21)
The following corollary is a straightforwardconsequence of Theorem 5.
Corollary 1. Under the assumptions of Theorem 5, if kxkξf(x)dx < ∞ for all ξ in R, then for any
Rd
subset U of Rd,
R
γ v2
lim n n logP supv |f (x)−f(x)|≥δ =−g˜ (δ) (8)
n→∞ hdn (cid:20)x∈U n n (cid:21) U
Comment. Since the sequence (sup |f (x)−f(x)|) is positive and since g˜ is continuous on
x∈U n U
[0,+∞[, increasing and goes to infinity as δ → ∞, the application of Lemma 5 in Worms (2001) al-
lowstodeduce from(7)or(8)thatsup |f (x)−f(x)| satisfiesaLDPwith speed γ−1hd andgood
x∈U n n n
rate function g˜ on R+.
U
(cid:0) (cid:1)
3 Proofs
Throught this section we use the following notation:
n
Π = (1−γ ),
n j
j=1
Y
Z (x)=h−dY ,
n n n
x−X
n
Y =K (9)
n
h
(cid:18) n (cid:19)
Throughout the proofs, we repeatedly apply Lemma 2 in Mokkadem et al. (2009). For the convenience
of the reader, we state it now.
Lemma 1. Let (v ) ∈ GS(v∗), (γ ) ∈ GS(−α), and m > 0 such that m−v∗ξ > 0 where ξ is defined
n n
in (4). We have
n
γ 1
lim v Πm Π−m k = .
n→+∞ n n k vk m−v∗ξ
k=1
X
Moreover,for all positive sequence (α ) such that lim α =0, and for all δ ∈R,
n n→+∞ n
n
γ
lim v Πm Π−m kα +δ =0.
n→+∞ n n " k vk k #
k=1
X
5
Noting that, in view of (1), we have
f (x)−f(x) = (1−γ )(f (x)−f(x))+γ (Z (x)−f(x))
n n n−1 n n
n−1 n n
= (1−γ ) γ (Z (x)−f(x))+γ (Z (x)−f(x))+ (1−γ ) (f (x)−f(x))
j k k n n j 0
   
k=1 j=k+1 j=1
X Y Y
n   
= Π Π−1γ (Z (x)−f(x))+Π (f (x)−f(x)).
n k k k n 0
k=1
X
It follows that
n
E[f (x)]−f(x)=Π Π−1γ (E[Z (x)]−f(x))+Π (f (x)−f(x)).
n n k k k n 0
k=1
X
Then, we can write that
n
f (x)−E[f (x)] = Π Π−1γ (Z (x)−E[Z (x)])
n n n k k k k
k=1
X
n
= Π Π−1γ h−d(Y −E[Y ])
n k k k k k
k=1
X
Let (Ψ ) and (B ) be the sequences defined as
n n
n
Ψ (x) = Π Π−1γ h−d(Y −E[Y ])
n n k k k k k
k=1
X
B (x) = E[f (x)]−f(x)
n n
We have:
f (x)−f(x)=Ψ (x)+B (x) (10)
n n n
Theorems 1, 2, 3, 4 and 5 are consequences of (10) and the following propositions.
Proposition 1 (Pointwise LDP and MDP for (Ψ )).
n
1. Undertheassumptions(L1)and(L2),thesequence(f (x)−E(f (x)))satisfiesaLDPwithspeed
n n
nhd and rate function I .
n a,x
2. (cid:0)Unde(cid:1)rtheassumptions(L1)and(L3),thesequence(f (x)−E(f (x)))satisfiesaLDPwithspeed
n n
nhd and rate function I .
n x
3. (cid:0)Unde(cid:1)rtheassumptions(M1)−(M4),thesequence(v Ψ (x))satisfiesaLDPwithspeed hd/ γ v2
n n n n n
and rate function J .
a,α,x
(cid:0) (cid:0) (cid:1)(cid:1)
Proposition 2 (Uniform LDP and MDP for (Ψ )).
n
1. Let (U1)−(U3) hold. Then for any bounded subset U of Rd and for all δ >0,
γ v2
lim n n logP supv |Ψ (x)|≥δ =−g˜ (δ)
n→∞ hdn (cid:20)x∈U n n (cid:21) U
2. Let (U1)−(U6) hold. Then for any subset U of Rd and for all δ >0,
γ v2
−g˜ (δ) ≤ liminf n n logP supv |Ψ (x)|≥δ
U n→∞ hdn (cid:20)x∈U n n (cid:21)
γ v2 ξ
≤ limsup n n logP supv |Ψ (x)|≥δ ≤− g˜ (δ)
hd n n ξ+d U
n→∞ n (cid:20)x∈U (cid:21)
The proof of the following proposition is given in Mokkadem et al. (2009).
6
Proposition 3 (Pointwise and uniform convergence rate of (B )).
n
Let Assumptions (M1)−(M3) hold.
1. If for all i,j ∈{1,...d}, ∂2f/∂x ∂x is continuous at x. We have
i j
If a≤α/(d+4), then
B (x)=O h2 .
n n
(cid:0) (cid:1)
If a>α/(d+4), then
B (x)=o γ h−d .
n n n
(cid:18)q (cid:19)
2. If (U2) holds, then:
If a≤α/(d+4), then
sup |B (x)|=O h2 .
n n
x∈Rd
(cid:0) (cid:1)
If a>α/(d+4), then
sup |B (x)|=o γ h−d .
n n n
x∈Rd (cid:18)q (cid:19)
Setx∈Rd;sincetheassumptionsofTheorems1and2guaranteethatlim B (x)=0,Theorem1
n→∞ n
(respectively Theorem 2) is a straightforward consequence of the application of Part 1 (respectively of
Part 2) of Proposition 1. Moreover, under the assumptions of Theorem 3, we have by application of
Propostion3,lim v B (x)=0;Theorem3thusstraightfullyfollowsfromtheapplicationofPart3of
n→∞ n n
Proposition1. Finaly,Theorem4and5followsfromProposition2andthe secondpartofProposition3.
We now state a preliminary lemma, which will be used in the proof of Proposition 1.
For any u∈R, Set
γ v2 hd
Λ (u) = n n logE exp u n Ψ (x)
n,x hd γ v n
n (cid:20) (cid:18) n n (cid:19)(cid:21)
ΛL,1(u) = f(x)(ψ (u)−u),
x a
ΛL,2(u) = f(x)(ψ(u)−u),
x
u2
ΛM(u) = f(x) K2(z)dz
x 2(2−(α−ad)ξ) ZRd
Lemma 2. [Convergence of Λ ]
n,x
1. (Pointwise convergence)
If f is continuous at x, then for all u∈R
lim Λ (u)=Λ (u) (11)
n,x x
n→∞
where
ΛL,1(u) when v ≡1 , (L1) and (L2) hold
x n
Λ (u)= ΛL,2(u) when v ≡1 , (L1) and (L3) hold
x  x n
ΛM(u) when v →∞ , (M1)−(M4) hold
 x n

2. (Uniform convergence)
If f is uniformly continuous, then the convergence (11) holds uniformly in x∈U.
Ourproofsarenoworganizedasfollows: Lemma2isprovedinSection3.1,Proposition1inSection3.4
and Proposition 2 in Section 3.3.
7
3.1 Proof of Lemma 2.
Set u∈R, u =u/v and a =hdγ−1. We have:
n n n n n
v2
Λ (u) = n logE[exp(u a Ψ (x))]
n,x n n n
a
n
v2 n
= n logE exp u a Π Π−1a−1(Y −E[Y ])
a n n n k k k k
n " !#
k=1
X
v2 n a Π n
= n logE exp u n nY −uv Π Π−1a−1E[Y ]
a na Π k n n k k k
n k=1 (cid:20) (cid:18) k k (cid:19)(cid:21) k=1
X X
By Taylor expansion, there exists c between 1 and E exp u anΠnY such that
k,n nakΠk k
h (cid:16) (cid:17)i
a Π a Π 1 a Π 2
logE exp u n nY =E exp u n nY −1 − E exp u n nY −1
na Π k na Π k 2c2 na Π k
(cid:20) (cid:18) k k (cid:19)(cid:21) (cid:20) (cid:18) k k (cid:19) (cid:21) k,n (cid:18) (cid:20) (cid:18) k k (cid:19) (cid:21)(cid:19)
and Λ can be rewriten as
n,x
v2 n a Π v2 n 1 a Π 2
Λ (u) = n E exp u n nY −1 − n E exp u n nY −1
n,x a na Π k 2a c2 na Π k
n k=1 (cid:20) (cid:18) k k (cid:19) (cid:21) n k=1 k,n (cid:18) (cid:20) (cid:18) k k (cid:19) (cid:21)(cid:19)
X X
n
−uv Π Π−1a−1E[Y ] (12)
n n k k k
k=1
X
First case: v →∞. ATaylor’sexpansionimpliestheexistenceofc′ between0andu anΠnY such
n k,n nakΠk k
that
2 3
E exp unaanΠΠnYk −1 =unaanΠΠnE[Yk]+ 21 unaanΠΠn E Yk2 + 61 unaanΠΠn E Yk3ec′k,n
(cid:20) (cid:18) k k (cid:19) (cid:21) k k (cid:18) k k(cid:19) (cid:2) (cid:3) (cid:18) k k(cid:19) h i
Therefore,
n n
Λn,x(u) = 12u2anΠ2n Π−k2a−k2E Yk2 + 61u2una2nΠ3n Π−k3a−k3E Yk3ec′k,n
Xk=1 (cid:2) (cid:3) Xk=1 h i
v2 n 1 a Π 2
− n E exp u n nY −1
2a c2 na Π k
n k=1 k,n (cid:18) (cid:20) (cid:18) k k (cid:19) (cid:21)(cid:19)
X
n
1
= f(x)u2a Π2 Π−2a−1γ K2(z)dz+R(1) (u)+R(2) (u) (13)
2 n nk=1 k k kZRd n,x n,x
X
with
n
1
R(1) (u) = u2a Π2 Π−2a−1γ K2(z)[f(x−zh )−f(x)]dz
n,x 2 n nk=1 k k kZRd k
X
Rn(2,)x(u) = 61uv3a2nΠ3n n Π−k3a−k3E Yk3ec′k,n − 2van2 n c21 E exp unaanΠΠnYk −1 2
n kX=1 h i n kX=1 k,n (cid:18) (cid:20) (cid:18) k k (cid:19) (cid:21)(cid:19)
Sincef iscontinuous,wehavelim |f(x−zh )−f(x)|=0,andthus,bythedominatedconvergence
k→∞ k
theorem, (M1) implies that
lim K2(z)|f(x−zh )−f(x)|dz =0.
k
k→∞ZRd
Since (a )∈GS(α−ad), and lim (nγ )>(α−ad)/2. Lemma 1 then ensures that
n n→∞ n
n
1
a Π2 Π−2a−1γ = +o(1), (14)
n n k k k (2−(α−ad)ξ)
k=1
X
8
it follows that lim R(1) (u) =0.
n→∞ n,x
Moreover,in view of (9(cid:12)), we hav(cid:12)e |Y |≤kKk , then
(cid:12) (cid:12) k ∞
(cid:12) (cid:12)
a Π
c′ ≤ u n nY
k,n na Π k
(cid:12) k k (cid:12)
≤ (cid:12)|u |kKk (cid:12) (15)
(cid:12) n ∞ (cid:12)
(cid:12) (cid:12)
Noting that E|Y |3 ≤ hdkfk K3(z) dz. Hence, using Lemma 1 and (15), there exists a positive
k k ∞ Rd
constant c such that, for n large enough,
1 R (cid:12) (cid:12)
(cid:12) (cid:12)
which goes to 0(cid:12)(cid:12)(cid:12)(cid:12)uvan3san2nΠ→3nkX∞=n1sΠin−kc3eav−k3E→hY∞k3.ec′k,ni(cid:12)(cid:12)(cid:12)(cid:12) ≤ c1e|un|kKk∞vun3 kfk∞ZRd(cid:12)(cid:12)K3(z)(cid:12)(cid:12)dz (16)
(cid:12) n (cid:12)
Moreover,Lemma 1 ensures that
v2 n 1 a Π 2
n E exp u n nY −1
(cid:12)(cid:12)2an Xk=1c2k,n (cid:18) (cid:20) (cid:18) nakΠk k(cid:19) (cid:21)(cid:19) (cid:12)(cid:12)
(cid:12)(cid:12)(cid:12) ≤ vn2 n E exp u anΠnY −1 (cid:12)(cid:12)(cid:12)2
n k
2a a Π
n k=1(cid:18) (cid:20) (cid:18) k k (cid:19) (cid:21)(cid:19)
X
u2 n n
≤ kfk2 a Π2 Π−2a−1γ hd+o a Π2 Π−2a−1γ hd
2 ∞ n n k k k k n n k k k k
!
k=1 k=1
X X
=o(1) (17)
The combination of (16) and (17) ensures that lim R(2) (u) = 0. Then, we obtain from (13) and
n→∞ n,x
(14), lim Λ (u)=ΛM(u). (cid:12) (cid:12)
n→∞ n,x x (cid:12) (cid:12)
(cid:12) (cid:12)
Second case: (v )≡1. It follows from (12) that
n
n n 2
1 a Π 1 1 a Π
Λ (u) = E exp u n nY −1 − E exp u n nY −1
n,x a a Π k 2a c2 a Π k
n k=1 (cid:20) (cid:18) k k (cid:19) (cid:21) n k=1 k,n (cid:18) (cid:20) (cid:18) k k (cid:19) (cid:21)(cid:19)
X X
n
−uΠ Π−1a−1E[Y ]
n k k k
k=1
X
n n
1 a Π
= hd exp u n nK(z) −1 f(x)dz−uΠ Π−1γ K(z)f(x)dz
an k=1 kZRd(cid:20) (cid:18) akΠk (cid:19) (cid:21) nk=1 k kZRd
X X
−R(3) (u)+R(4) (u)
n,x n,x
n
1
= f(x) hd (exp(uV K(z))−1)−uV K(z) dz
an k=1 k(cid:20)ZRd n,k n,k (cid:21)
X
−R(3) (u)+R(4) (u) (18)
n,x n,x
with
a Π
n n
V =
n,k
a Π
k k
n 2
1 1 a Π
R(3) (u) = E exp u n nY −1
n,x 2a c2 a Π k
n k=1 k,n (cid:18) (cid:20) (cid:18) k k (cid:19) (cid:21)(cid:19)
X
n
1 a Π
R(4) (u) = hd exp u n nK(z) −1 [f(x−zh )−f(x)]dz
n,x an k=1 kZRd(cid:20) (cid:18) akΠk (cid:19) (cid:21) k
X
n
−uΠ Π−1γ K(z)[f(x−zh )−f(x)]dz.
n k k k
k=1 ZRd
X
9
It follows from (17), that lim R(3) (u) =0.
n→∞ n,x
Since |et−1|≤|t|e|t|, we have (cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
n
1 a Π
R(4) (u) ≤ hd exp u n nK(z) −1 [f(x−zh )−f(x)] dz
(cid:12) n,x (cid:12) an Xk=1 kZRd(cid:12)(cid:12)(cid:20) (cid:18) akΠk (cid:19) (cid:21) k (cid:12)(cid:12)
(cid:12) (cid:12) n (cid:12) (cid:12)
(cid:12) (cid:12) +|u|Π Π(cid:12)−1γ |K(z)||f(x−zh )−f(x)|dz (cid:12)
n k k k
k=1 ZRd
X
n
≤ |u|e|u|kKk∞Π Π−1γ |K(z)||f(x−zh )−f(x)|dz
n k k k
k=1 ZRd
X
n
+|u|Π Π−1γ |K(z)||f(x−zh )−f(x)|dz
n k k k
k=1 ZRd
X
n
≤ |u| e|u|kKk∞ +1 Π Π−1γ |K(z)||f(x−zh )−f(x)|dz
n k k k
(cid:16) (cid:17) Xk=1 ZRd
In view of Lemma 1 the sequence Π n Π−1γ is bounded, then, the dominated convergence the-
n k=1 k k
orem ensures that lim R(4) (u)=0.
n→∞ n,x (cid:0) P (cid:1)
Inthecasef isuniformlycontinuous,setε>0andletM >0suchthat2kfk |K(z)|dz ≤ε/2.
∞ kzk≤M
We need to prove that for n sufficiently large
R
sup |K(z)||f(x−zh )−f(x)|dz ≤ε/2
k
x∈RdZkzk≤M
which is a straightforwardconsequence of the uniform continuity of f.
Then, it follows from (18), that
n
γ
lim Λ (u) = lim f(x) n hd [(exp(uV K(z))−1)−uV K(z)]dz (19)
n→∞ n,x n→∞ hdn k=1 kZRd n,k n,k
X
In the case when (v )≡1, (L1) and (L2) hold
n
We have
n
Π
n
= (1−γ )
j
Π
k
j=k+1
Y
k
= ,
n
then,
a Π
n n
V =
n,k
a Π
k k
ad
k
= .
n
(cid:18) (cid:19)
Consequently, it follows from (19) and from some analysis considerations that
1
lim Λ (u) = f(x) s−ad exp usadK(z) −1−usadK(z) ds dz
n,x
n→∞ ZRd(cid:20)Z0 (cid:21)
= ΛL,1(u) (cid:0) (cid:0) (cid:1) (cid:1)
x
10