Local limit theorems on some non unimodular groups

Local limit theorems on some non unimodular groups

Emile Le Page and Marc Peigne Abstract.

Let G_d be the semi-direct product of ^{R +} and ^Rd, d 1 and let us consider the product group Gd N = Gd ^RN, N 1. For a large class of probability measures on G_{d N}, one proves that there exists( )²]01] such that the sequence of nite measures

nn^(N+3)=2 ( )^{n n}

o

n1

converges weakly to a non-degenerate measure.

Resume.

Soit G_d le produit semi-direct de ^{R +} et de ^Rd et G_{d N} le groupe produit Gd^RN, N 0. Pour une large classe de mesures de probabilite sur G_{d N} nous montrons qu'il existe ( )²]01] tel que la suite de mesures nies

nn^(N+3)=2 ( )^{n n}

o

n1

converge vaguement vers une mesure non nulle.

1. Introduction.

Fix two integers d 1, N 0 and choose a norm ^kk on ^Rd and

RN (when N 1). Let Gd N be the connected group ^{R + Rd RN} 117

with the composition law

for all g= (aub) for all g⁰ = (a⁰u⁰b⁰)²G gg⁰ = (aa⁰au⁰+ub+b⁰):

We will note g = (a(g)u(g)b(g)) (or g = (aub) when there is no ambiguity). The group (Gd N) is a non unimodular solvable group with exponential growth and the right Haar measurem_D on G_{d N} is

mD(dadudb1dbN) = dadudb1dbN

a :

Note that Gd0 is the semi-direct product of ^{R +} and ^Rd in particular G1 0 is the ane group of the real line.

We consider a probability measure on G we denote by ⁿ its n^th power of convolution. Under quite general assumptions on we show that there exists ( )²]01 such that the sequence

nn^(N+3)=2 ( )^{n n}

o

n0

converges weakly to a non-degenerate measure. This problem has al- ready been tackled by Ph. Bougerol in 3] where were established local limit theorems on some solvable groups with exponential growth in particular, for a class R of probability measures on the ane group of the real line (that is d= 1 and N = 0) he showed that the sequence

n n³⁼² ( )^{n n}

o

n0

converges weakly to a non-degenerate measure. In 7] we extend this result to a quite large class of probability measures the new ingredient in our proof was the fact that there exists closed connections between this problem and the theory of the uctuations of a random walk on the real line. In the present paper, we extend this result to the case N 1 we rst obtain uniform upperbounds in the Local limit theorem for a random walk on ^Rd and, secondly, we use a generalisation of the Wiener-Hopf's factorisation due to Ch. Sunyach 9].

This study is also related with the work by N. T. Varopoulos 10], 11] where upperbounds and lowerbounds for the asymptotic behaviour

of the convolution powers ⁿ of a large class of probability measures are given.

From now on, we will suppose that N 1 and we set G =Gd N. We introduce the following conditions on :

Hypothesis G1.

There exists >0 such that

Z

G(e^jlogaj+^ku^k+^kb^k2) (dadudb)<+¹:

Hypothesis G2.

^Z_GLoga (dadudb) = 0 and

Z

Gb (dadudb) = 0:

Hypothesis G3.

The support of is included in ^{R +} (^R+)^{d RN}, the image of by the mapping (aub) ^7;! (Logab) is aperiodic in

RN+1 (see Denition 2:1) and there exists >0 such that

Z

G^ku^k; (dadudb)<+¹:

Hypothesis G'3.

The measure is absolutely continuous with respect to the Haar measure mD on G and its density satises

Z

]0 1]^RN

q s

Z

R

q(aub)du dadba <+¹: for some and q in ]1+¹:

We have the

Theorem 1.1.

Let be a probability measure on G satisfying hypothe- ses G1, G2 and G3 (or G'3). Then, the sequence of nite measures

fn^{(N+3)=2 ng}n0 converges weakly to a non-degenerate Radon mea- sure on G.

Note that the asymptotic behavior of the sequence^{f ng}_n1 does not depend on d.

When is not centered, that is

Z

GLoga (dadudb)⁶= 0

or ^Z

Gb (dadudb)⁶= 0

we bring back the study to the centered case as in 7]. We introduce the following conditions on :

Hypothesis G 1.

There exists > 0 such that

Z

G(a^t+^ku^k+ exp(t^kb^k)) (dadudb)<+¹ for any t^2R.

Hypothesis G 2.

One has

Z

GLoga (dadudb)⁶= 0

with ^fg²G: a(g)<1^g>0 and ^fg²G: a(g)>1^g>0.

When satis es these two conditions, there exists a unique (s0t0)

2R RN such that

Z

Ga^s0e^{ht0 bi} (dadudb) = _{(s t)}inf

2RR N

Z

Ga^se^{ht bi} (dadudb): Furthermore,

( ) =^Z

Ga^s0e^{ht0 bi} (dadudb) belongs to ]01]. Note that the probability measure

0(dg) = 1( )a(g)^s0e^ht0b(g)i (dg)

satis es hypotheses G1 and G2. The following condition is the equiva- lent of Hypothesis G'3 in the non centered case:

Hypothesis G 3.

The measure is absolutely continuous with respect to the Haar measure mD on G and its density satises

Z

]0 1]^RN

q s

Z

R

q(aub)du dadba <+¹

for some q²]1+¹ and ²]1^;s0+¹.

Theorem 1.2.

Let be a probability measure on G satisfying condi- tions G 1,G 2 and G3 (or G 3) and let

( ) =_{(s t)}inf

2RR N

Z

Ga^se^{ht bi} (dadudb): Then, the sequence of nite measures

nn^(N+3)=2 ( )^{n n}

o

n1

weakly converges to a non-degenerate Radon measure on G.

The demonstration of Theorem 2.1 is closely related to the study of the uctuations of a random walk (X₁ⁿY₁ⁿ)n0 on^RN+1. In Section 2, we rst state the classical local limit theorem on ^RN+1 but we add in its statement uniform upperbounds relatively to the starting point of the random walk (X₁ⁿY₁ⁿ)n0. This result is thus very usefull to obtain a precise equivalent in Theorem 2.5 of the joint law of the random walk (X₁ⁿY₁ⁿ)n0 with its rst entrance time T+ in the half space^R+RN a local limit theorem for the process

(X₁ⁿmax^f0X₁₁:::X₁^ngY₁ⁿ)n0

is thus obtained (Theorem 2.6). In Section 3 we give the main steps of the proof of Theorem 1.1.

2. Fluctuations of a random walk on

^RN+1

.

Fix an integer N 1 and let (X1Y1)(X2Y2)::: be indepen- dent ^{R RN}-valued random variables with distribution p de ned on a probability space (^FP). Let (X₁ⁿY₁ⁿ)n0 be the associated random walk on^RRN starting from (00) and de ned byX₀₁ = 0Y₀₁ = 0 and X₁ⁿ =X1++XnY₁ⁿ =Y1++Ynforn 1 the distribution of the couple (X₁ⁿY₁ⁿ) is the n^th power of convolution p ⁿ of the measure p. Denote by^Fnthe -algebra generated by (X1Y1):::(XnYn)n 1.

Let us rst recall the

Denition 2.1.

Let p be a probability measure on ^Rk, k 1. The measure p is aperiodic on ^Rk if there is no closed and proper subgroup H of ^Rk and no ^2Rk such that p(+H) = 1.

Denote by ^p the characteristic function of p de ned by ^p(uv) =

Ee^{iuX1+ihv Y1i}] for any (uv) ^{2 R RN}. Recall that the probability measure p is aperiodic if and only if ^jp^(uv)^j<1 for (uv)⁶= (00).

For any ^{A R RN} let ^fT^A(k)gk0 be the the successive times of visit of the random walk (X₁ⁿY₁ⁿ)n1 to the set ^A one has T^A(0) = 0T^A(1) = inf^fn 1 : (X₁ⁿY₁ⁿ)^{2 Ag} and T^A(k+1) = inf^fn T^A(k)+ 1 : (X₁ⁿY₁ⁿ)^2Ag. Note that theT^A(k) are stopping times with respect to the ltration^fFn^gn1. We will associate to (p^A) the transition kernel P^A de ned by

P^A((xy)^B) =

Z

RR N

1

^Ac\B(x+x⁰y+y⁰)p(dx⁰dy⁰)

for any Borel set ^B in ^{R RN} note that for any k 1 one has P^Ak((00)^B) = ^ET^A > k](X₁^kY₁^k)^2B]. In order to simplify the no- tations we will set T^; = T^R;RNP^; = P^R;RN and T^;(k) = T^R(k;)RN similar notations will hold, with obvious modi cations, when ^A =

R

;

RN^{R+ RN} and ^{R + RN}.

Troughout this paragraph, for any k 1, we denote by _k the Lebesgue measure on ^Rk. Furthermore, for any > 0, ^H(^Rk) is the space of ^C-valued functions 'on ^Rk such that

xsup^2Rk(1 +^kx^k)^kj'(x)^j<+¹:

2.1. Preliminaries.

The local limit theorem gives the asymptotic behaviour of the se- quence ^fp ⁿ(')^gn1 for any continuous function ' with compact sup- port on^RN+1 we state it here and we precise some uniform upperbound for the sequence^fp ⁿ(')^gn1 when'belongs to^H(^RN+1) with > 4.

Theorem 2.2.

Assume that:

i) the common distribution p of the variables (X_nY_n), n 1, is aperiodic on ^RN+1,

ii) ^EjX1^j2+^kY1^k2]<+¹ and ^EX1] = 0, ^EY1] = 0. Then:

i) for any continuous function ' with compact support on ^RN+1 one has

n^!lim+¹n^(N+1)=2E'(X₁ⁿY₁ⁿ)]

= 1

(2)^(N+1)=2pjC^j

Z

R N+1

'(xy)1(dx)_N(dy) where^jC^jdenotes the determinant of the positive denite quadratic form

C(uv) =^E(uX1 +^hvY1ⁱ)²]:

ii) For any function ' in ^H(^RN+1) with > 4, the sequence

fn^(N+1)=2E'(x+X₁ⁿy+Y₁ⁿ)]^gn1 is bounded uniformly in (xy) ²

R RN.

Proof. The rst assumption is the classical local limit theorem. To obtain the second claim, x a non negative function whose Fourier transform has a compact support K(^). Recall that

p^(uv) = 1^; 1

2C(uv)(1 +"(uv))

with lim(u v)^!(0 0)"(uv) = 0 so there exists > 0 such that for ^ju^j+

kv^k< one has

jp^(uv)^j1^; 1

4 C(uv)e^{;C(u v)=4}:

On the other hand, by the aperiodicity ofp there exists=(pK(^)) such that^jp^(uv)^j as soon as (uv) belongs toK(^) and^ju^j+^kv^k . It follows that

(2 n)^(N+1)=2E(X₁ⁿY₁ⁿ)]

n^(N+1)=2Z

ju^j+^kv^k<^j^(uv)^jjp^(uv)^jn1(du)N(dv) +n^(N+1)=2Z

ju^j+^kv^kj^(uv)^jjp^(uv)^jn1(du)N(dv)

n^(N+1)=2Z

ju^j+^kv^k<n^(N+1)=2 ^ u

pn v

pn

e^{;(n=4)C(u=pn v=pn)}

1(du)N(dv) +n^{(N+1)=2nk^k}1

k^^k1Z

RR N

e^{;C(u v)=4}1(du)N(dv) +n^{(N+1)=2nk^k}1 : Now set_{x y}(x⁰y⁰) =(x+x⁰y+y⁰) for any (xy)^2RRN and note that ^_{x y}(uv) = e^{iux+ihv yi^}(uv) the functions ^_{x y} and ^ thus have the same compact support and satis es the equalities ^k^x y^k1 = ^k^^k1

and ^k^x y^k1 =^k^^k1. For any (xy)^{2R RN} one thus has

j(2 n)^(N+1)=2Ex y(X₁ⁿY₁ⁿ)]^j

k^^k1Z

RR N

e^{;C(u v)=4}1(du)N(dv) +n^{(N+1)=2nk^k}1 : The assertion ii) thus holds for any function whose Fourier transform has a compact support. To achieve the proof of ii) it suces to show that for any function'in ^H(^RN+1) with >4 there exists a function whose Fourier transform has a compact support and ^j'^j . It is an immediate consequence of the following result we thank here J. P.

Conze for helpfull discussions about this fact.

Lemma 2.3.

Set

h"(x) = 1 1 +^jx^j4+"

for any x^{2 R}. If " >0 there exists a function h" greater than h" and whose Fourier transform has a compact support in ^R.

Proof. Set

h"(x) =C^sin2x

x^{2 + sin2}x x²

for some and C in ^{R +} which will depend on ". Assume =^{2 Q}, the function h_" is strictly positive on ^R it thus suces to show that there exists =^2Q such that

x^!lim+¹x^2+"(sin²x+ sin²(x)) = +¹:

If such a real did not exist, then for any =^{2 Q} there should exist a sequence^fxn^gn1 which tends to +¹and a constantC" >0 such that for all n 1,

sin²xn+ sin²(xn) C x²⁺n ^" :

So there should exist two strictly increasing sequences of integers

fkn^gn1 and ^fln^gn1 such that

jxn^;kn^j C⁰

x¹⁺n ^{"=2 j}xn^;ln^j C⁰ x¹⁺n ^"=2

which implies

^; ln

kn C⁰⁰ k²⁺n ^"=2

for some positive constants C⁰ and C⁰⁰. This leads to a contradiction because for almost all^2R (with respect with the Lebesgue measure), this last inequality has at most a nite number of solutions in ^N2 2].

The lemma is proved.

2.2. A local limit theorem for a killed random walk on a half space.

In 7], we proved the following

Theorem 2.4.

Let the hypotheses of Theorem 2:2 hold. Then for any continuous function with compact support ' on ^R; we have

n^!lim+¹n^3=2ET+ > n]'(X₁ⁿ)] = 1 (X1)^p2

Z 0

;1

'(x)^;₁ U ^;(dx) where ^;₁ denotes the restriction of the Lebesgue measure on ^R; and U ^; is the -nite measure on ^R; dened by

U ^;(^B) =^X+1

k=1

E

1

^B(X₁^{T;(k )})]

for any Borel set ^B. In the same way, one has

n^!lim+¹n^3=2ET + > n]'(X₁ⁿ)] = 1 (X1)^p2

Z 0

;1

'(x)^;₁ U^;(dx)

where U^; is the -nite measure on ^R; dened by U^;(^B) =^+X1

k=1

E

1

^B(X₁^{T;(k )})]

for any Borel set ^B.

Recall that the random walks^fX₁^{T;(k )g}_k1and^fX₁^{T;(k )g}_k1are tran- sient on ^R; it follows that the series^P+_k=0^1ET+ > k]'(x+X₁^k)] and

P+¹

k=0^ET + > k]'(x+X₁^k)] do converge. Furthermore one has

+¹

X

k=0

ET+ > k]'(x+X₁^k)] =

Z 0

;1

'(x)U ^;(dx)

and ₊

1

X

k=0

ET + > k]'(x+X₁^k)] =^{Z 0}

;1

'(x)U^;(dx): Let us now state the following

Theorem 2.5.

Let the hypotheses of Theorem 2:2 hold. Then:

i)For any continuous function'with compact support on^R;RN one has

n^!lim+¹n^(N+3)=2ET+ > n]'(X₁ⁿY₁ⁿ))]

= 1

(2)^(N+1)=2pjC^j

Z

R

;

R N

'(xy)^;₁ U ^;(dx)N(dy): ii) For any continuous function f with compact support on ^R and any g in ^H(^RN) with >4, the sequence

fn^(N+3)=2ET+ > n]f(X₁ⁿ)g(y+Y₁ⁿ)]^g_n1

is bounded, uniformly in y^2RN. In the same way, one has

n^!lim+¹n^(N+3)=2ET + > n]'(X₁ⁿY₁ⁿ)]

= 1

(2)^(N+1)=2pjC^j

Z

R

;

R N

'(xy)^;₁ U^;(dx)N(dy)

and the sequence

fn^(N+3)=2ET+ > n]f(X₁ⁿ)g(y+Y₁ⁿ)]^g_n1

is bounded, uniformly in y^2RN.

Proof. We prove this theorem by induction overN. Theorem 2.2 deals with the caseN = 0 we will suppose that this result hold for someN 0 and we consider a sequence (XnYnZn)n1 of independent identically distributed random variables on ^{R RN R}. By a classical argument in probability theory, it suces to show the above convergence hold for '(xyz) = e^ax

1

^R;(x)(y)(z) where a ^{2 R +} and are ^C- valued functions whose Fourier transform are continuous with compact supports. By the inverse Fourier transform one has

I_n=^ET+ > n]e^aX1n(Y₁ⁿ)(Z₁ⁿ)]

= 1

(2)^(N+1)=2

Z

R N

R

^(v) ^(w)n(avw)N(dv)1(dw) with n(avw) = ^ET+ > n]e^{aX1n+ihv Y1ni+iwZ1n}].

The Spitzer's factorisation for random walks on ^R gives for all a >0, for alls ²01

+¹

X

n=0s^nET+ > n]e^aX1n] = exp^X+1

n=1

sⁿ

n ^EX₁ⁿ <0]e^aX1n]: Using the fact that ^{R+ RN+1} and ^{R ; RN+1} are semi-groups in

RN+2, Ch. Sunyach extended this factorisation to the multidimension- nal case (9, Corollary 3, p. 553 and Theorem 5, p. 556]) for anya > 0, v^2RN, w^2R and s²01 one thus has

+¹

X

n=0s^nET+ > n]e^{aX1n+ihv Y1ni+iwZ1n}]

= exp^+X1

n=1

sⁿ

n ^EX₁ⁿ <0]e^{aX1n+ihv Y1ni+iwZ1n}] that is

(n+ 1)n+1(avw) =^Xn

k=0n+1^;k(avw)k(avw)

with n(avw) = ^EX₁ⁿ <0]e^{aX1n+ihv Y1ni+iwZ1n}]. Finally In= 1n+ 1

n

X

k=0In k

with

In k = 1 (2)^(N+1)=2

Z

R N

R

n+1^;k(avw)k(avw)

^(v) ^(w)N(dv)1(dw): Set

I = 1

(2)^(N+2)=2pjC^j

Z

R N

R

+¹

X

k=0

E

hT+ > k]e^aX1k a

i

(y)(z)N(dy)1(dz) since

I =^;₁ U ^;(e^a)_N()1()

it suces to show that ^fn^(N+4)=2I_n^g_n1 converges to I, that is 1) for allk >0, lim_n

!+¹n^(N+2)=2In k =I k, 2) ^+X1

k=0

jI k^j<+¹ and ^X+1

k=0I k=I, 3) limsup_l

!+¹ limsup_n

!+¹ n^(N+2)=2Xn

k=l

jIn k^j= 0.

To prove the assertion 1), note that

I_{n k} =^ET+ > k]^\X_kⁿ₊₁⁺¹ >0]e^aX1n+1(Y₁ⁿ⁺¹)(Z₁ⁿ⁺¹)]

by the local limit theorem on^RN+2 the assertion 1) follows with I k = 1

2^(N+2)=2pjC^j

ET+ > k]e^aX1k] a

Z

N

(y)N(dy)

Z

(z)1(dz):

The fact that the series ⁺_k=0^jI k^j converges is a direct consequence of Theorem 2.4. To prove the assertion 3), note that

jIn k^jET+ > k]^\X_kⁿ₊₁⁺¹ <0]e^aX1n+1j(Y₁ⁿ⁺¹)^jj(Z₁ⁿ⁺¹)^j]

E

hT+ > k]e^aX1kZ

R

;

R N

R

e^axj(y+Y₁^k)^j

j(z+Z₁^k)^jp ^(n+1;k)(dxdy dz)ⁱ

C(a)

(n+ 1^;k)^{(N+2)=2 E}T+ > k]e^aX1k] by Theorem 2.2.ii)

C1

(n+ 1^;k)^(N+2)=2k³⁼² by Theorem 2.4.

On the other hand

jIn k^jkk1Z

R

;

R N

R

ET+ > k]e^aXk1j(y+Y₁^k)^j

e^axp ^(n+1;k)(dxdy dz)]

k^k1C(a) k^(N+3)=2

EX_kⁿ₊₁⁺¹ <0]e^{aXk +1n+1}] by hypothesis of induction

C2

k^(N+3)=2pn+ 1^;k :

The assertion 3) follows since for any " >0 one has n^(N+2)=2Xn

k=l

jIn k^jC1

n(1X^;")]

k=l

n^(N+2)=2

k³⁼²(n+ 1^;k)^(N+2)=2 +C2 ^Xn

n(1^;")+1]

n^(N+2)=2 k^(N+3)=2pn+ 1^;k

C1

"^(N+2)=2

n(1X^;")]

k=l

k³1⁼²

+ C2

pn(1^;")^(N+3)=2

n

X

n(1^;")+1]

1

pn+ 1^;k

C"^(N+2)1=2pl ⁺

p"

(1^;")^(N+3)=2

: Since " is arbitrarily small, the assertion 3) follows.

The proof of ii) is also made by induction overN. Ifg^2H(^RN+1) there exist^2H(^RN) and ^2H(^R1) such that^jg^j. We set

In(yz) =^ET+ > n]e^aX1n(y+Y₁ⁿ)(z+Z₁ⁿ)]

and we have

I_n(yz) = 1n+ 1

n

X

k=0I_{n k}(yz) with

In k(yz) =^ET+ > k]^\X_kⁿ₊₁⁺¹ <0]e^aX1n+1(y+Y₁ⁿ⁺¹)(z+Z₁ⁿ⁺¹)]: As above, one has

jI_{n k}(yz)^jinfⁿ C1

(n+ 1^;k)^(N+2)=2k³⁼² C2

k^(N+3)=2pn+ 1^;k

o

which proves that the sequence

nn^(N+2)=2Xn

k=0

jIn k(yz)^jo_n

1

is uniformly bounded in yz. This achieves the proof of ii).

The convergence of the sequence

fn^(N+3)=2ET + > n]'(X₁ⁿY₁ⁿ)]^gn1

is obtained with similar arguments.

2.3. Behaviour of the process

((X₁ⁿmax^f0X₁₁:::X₁^ngY₁ⁿ))n0

.

For any n 0 set ^X₁ⁿ = max^f0X₁₁:::X₁^ng and let T_n be the random variable de ned on (^FP) by Tn = inf^f0 k n : ^X₁ⁿ =

X₁^kg for any continuous function 'with compact support on^RN+1 we have

E'(^X₁^nX₁^n;X₁ⁿY₁ⁿ)]

= ^Xn

k=0

ETn =k]'(X₁^k;X_nk+1Y₁ⁿ)]

= ^Xn

k=0

E0 < X₁^kX₁₁ < X₁^k:::X₁^k;1 < X₁^k

X₁^k+1 X₁^k:::X₁ⁿX₁^k]'(X₁^k;X_nk+1Y₁ⁿ)]

= ^Xn

k=0

EX₁₁ >0:::X₁^k >0]^\X_k^k₊₁⁺¹ 0:::X_nk+1 0]

'(X₁^k;X_nk+1Y₁ⁿ)]: One obtains the following factorisation

E'(^X₁^nX₁^n;X₁ⁿY₁ⁿ)] =^Xn

k=0Jn k(') with

Jn k(')

=

Z

R N+1

'(x^;x⁰y+y⁰)P^;k((00)dxdy)Pⁿ₊^;k((00)dx⁰dy⁰): The behaviour of the process (^X₁^nX₁^n;X₁ⁿY₁ⁿ) is thus closely related to the one of the iterates of the transition kernels P^; and P +. Using this factorisation one proves the

Theorem 2.6.

Suppose that the hypotheses of Theorem 2:2 hold.

Then, for any continuous function with compact support on ^R+

R+ RN the sequence

fn^(N+3)=2E'(^X₁^nX₁^n;X₁ⁿY₁ⁿ)]^gn1

converges to (2)^(N+1)1^=2pjC^j

Z

R +

R +

R N

'(s^;ty)U ⁺(ds)^;₁ U^;(dt)_N(dy)

+ 1

(2)^(N+1)=2pjC^j

Z

R +

R +

R N

'(s^;ty)+U ⁺(ds)U^;(dt)N(dy): Furthermore, for any continuous function f with compact support on

R+ R+ and any g in ^H(^RN), the sequence

fn^(N+3)=2Ef(^X₁^nX₁^n;X₁ⁿ)g(y+Y₁ⁿ)]^gn1

is bounded, uniformly in y^2RN.

Proof. We only proof the rst assertion the second one may ob- tained with obvious modi cations as in Theorem 2.5. Set '(xty) = '1(x)'2(t)'3(y) where '1'2 and '3 are continuous with compact support. Fixk 0 by Theorem 2.5, the sequence

nn^(N+3)=2Z

R

;

R N

'2(x⁰)'3(y+y⁰)Pⁿ₊^;k((00)dx⁰dy⁰)^o_n

1

is bounded uniformly in y^2RN and converges to (2)^(N+1)1^=2pjC^j

Z 0

;1

'2(^;t)^;₁ U^;(dt)N('3):

By the dominated convergence theorem, one thus obtains, for any xed i 1

n^!lim+¹n^(N+3)=2Xi

k=0Jn k(')

= 1

(2)^(N+1)=2pjC^j

i

X

k=0

ET^; > k]'1(X₁^k)]

Z 0

'2(^;t)^;₁ U^;(dt)_N('3):