getdoc115a. 288KB Jun 04 2011 12:04:06 AM

El e c t ro n ic

Jo ur

n a l o

f P

r o

b a b i_{l i t y} Vol. 15 (2010), Paper no. 37, pages 1161–1189.

Journal URL

http://www.math.washington.edu/~ejpecp/

The Green Functions

Of Two Dimensional Random Walks Killed On A Line

And Their Higher Dimensional Analogues

Kôhei UCHIYAMA

Department of Mathematics

Tokyo Institute of Technology

Oh-okayama, Meguro Tokyo, 152-8551, Japan

uchiyama@math.titech.ac.jp

Abstract

We obtain asymptotic estimates of the Green functions of random walks on the two-dimensional integer lattice that are killed on the horizontal axis. A basic asymptotic formula whose leading term is virtually the same as the explicit formula for the corresponding Green function of Brow-nian motion is established under the existence of second moments only. Some refinement of it is given under a slightly stronger moment condition. The extension of the results to random walks on the higher dimensional lattice is also given.

Key words:asymptotic formula, Green function, random walk of zero mean and finite variances, absorption on a line.

AMS 2000 Subject Classification:Primary 60G50; Secondary: 60J45. Submitted to EJP on October 5, 2009, final version accepted May 19, 2010.

1

Introduction and Results

LetS_nx = x+ξ₁+_{· · ·}+ξ_nbe a random walk onZ2 (the two dimensional integer lattice) starting at S₀x = x _∈Z2. Here ξ_j, j =1, 2, . . ., are independent and identically distributedZ2-valued random variables defined on a probability space(Ω,F,P). The walkS_n0 is supposed irreducible and having zero mean and finite variances. Let[x₁,x₂]stand for a point ofR2with components x₁and x₂and put

L=_{[s, 0]:s∈R} (the first coordinate axis).

In this paper we obtain asymptotic estimates of the Green functionG_Lof the walk killed on L: G_L(x,y) =

∞

X

n=0

pn_L(x,y)

wherepn_L(x,y) =P[S_nx = y,Sx_{j ∈}/ Lforj=1, . . . ,n₋1], then-step transition probability of the killed walk; in particular p0_L(x,y) =δ_x,y (Kronecker’s symbol). Note that pn_L(x,y)(with n≥ 1), hence G_L(x,y), may be positive even ifx,y _∈L. Our definition, thus different from usual one (but only if |x_{| · |}y_|=0), is convenient when duality relations are considered (see Remark 4 below). LetX and Y be the first and second component ofξ₁=S₁0, respectively, putσ2

1=EX2,σ12=EX Y,σ22=EY2,

σ_{j j} = σ2

j (j = 1, 2), Q = (σi j) (the covariance matrix of S10) and σ = |detQ|1

/4_{, and define the}

norm_kx_k, x _∈R2 by_kx_k2 =σ2Q−1(x), whereQ−1 andQ−1(x)stand for the inverse matrix ofQ and its quadratic form , respectively.

Fora,b_∈R,a_∨banda_∧bdenote, respectively, the maximum and the minimum ofa andb. The functiontlogt is understood continuously extended to t=0.

Theorem 1.1. Suppose that nk>0and let B_s,n,k be given via the equation

G_L([0,n],[s,k]) = 1

2πσ2log

1+ 4(σ/σ2) 2_nk

k[s,k₋n]_k2_∨1

+B_s,n,k. Then

B_s,n,k=o

1_∨log

nk k[s,k₋n]_k2_∨1

as nk_{→ ∞} (1.1)

uniformly in s; in particular Bs,n,k=o(logn)as n→ ∞uniformly in s,k. Moreover Bs,n,kis uniformly

bounded if and only if E[_|S₁0|2_log|S0

1|]<∞and if this is the case Bs,n,k→0ask[s,k−n]k → ∞.

Theorem 1.2. Suppose that nk<0. Then GL([0,n],[s,k])→0ask[s,k−n]k → ∞.

For each pair of k,n, GL([0,n],[s,k]) → 0 as |s| → ∞ and GL is everywhere positive. With this

taken into account it is inferred from the first half of Theorem 1.1 that under the conditionnk>0, G_Lis bounded away from zero if and only if so is nk/(_k[s,k₋n]_k2_∨1). Unless this is the case the estimate in Theorem 1.1 is crude. The next result is complementary in this respect (its proof is much more involved than that of Theorem 1.1).

Leta(n) (n∈Z2)denote the potential function of the one dimensional random walk of the second component ofS0_n: a(n) =P∞_k=0P[S_k0_∈Z_{× {}0_}]₋P[S_k0 _∈Z_{× {−}n_}]. (Cf. [7]:T29.1.) Also put a∗(n) =δ_0,n+a(n),

Theorem 1.3. If E[_X˜2_log

|_X˜_|]<_∞, then, as

k∨1)(n∨1)/(_k[s,k−n]_k2_∨1)_→0

GL([0,n],[s,k]) =

1 π

σ4₂a∗(n)a∗(₋k) +nk σ₂2_k[s,k₋n]_k2 +o

_(n

∨1)(_|k_{| ∨}1)

k[s,k−n]_k2

for n≥0,k,s∈Z.

REMARK1. Letλ=σ2/σ2₂ and observe that

k[s,m]_k2=λ−1(s₋µm)2+λm2=_k[s+2µm,₋m]_k2

and _k[s,k₋ n]_k2 +4λkn = _k[s+2µn,k+n]_k2. Then the formula of Theorem 1.1 is under-stood to be in accordance with the one for Brownian motion, say g◦(a,b), that is explicit ow-ing to reflection principle (see Appendix (B)). Donsker’s invariance principle implies that if two points a,b are taken from the upper half plane R_×(0,_∞), ǫ > 0 and α_x = _|a_|/_|x_| (x _∈ Z2), then α2

x

P

{y:|αxy−b|<ǫ}GL(x,y) converges to

R

|u−b|<ǫg

◦_{(a,u)du as |x| → ∞ in such a way that}

limα_xx=a, but not much more.

REMARK 2. For n< 0, the formula of Theorem 1.3 hold true asn[(−k)∨1]/k[s,k−n]k2 →0 by

duality. In the case when nk< 0 and_|k| ∧ |n| → ∞we haveσ4₂a(n)a(₋k) +nk= o(nk), so that any proper asymptotic form ofG_L is not given by it; the determination of it requires more detailed analysis than that carried out in this paper and will be made in a separate paper ([11]).

REMARK3. It follows from (1.1) that under the constraints2+(n−k)2> ǫnk>0 (for someǫ >0) the

convergence to zero ofB_s,n,k stated in Theorem 1.1 holds true without assumingE[_|S₁0_|2log_|S0_1|]< ∞, which however cannot be removed in general (see also Theorem 3.2). Similarly the moment condition E[X˜2log_|X˜_|] < _∞ in Theorem 1.3 is needed only in the case when _|k_|+_|n_| = o(_|s_|)

(Lemmas 4.2 and 4.3), while without it lim sup_|s|→∞s2G_L([0,n],[s,k]) can be infinite in view of Theorem 1.3 of[9].

REMARK 4. If ˜G_L denotes the Green function associated with the dual process (i.e. the process (₋S_n−x)), then ˜G_L([0,n],[s,k]) =G_L([0,k],[₋s,n]). Because of this duality we may suppose that |k_{| ≤}nfor the proof of Theorems above. In view of (1.4) (that also follows from the duality) the results onH[0,n](s)obtained in[9]provide the formula of Theorem 1.3 in the casenk=0. Thus we

may further suppose that|k|>0.

This paper is in a sense a continuation of [9] in which the hitting distribution of L for the walk starting at[0,n]is evaluated. As in[9]put

ψ(t,l) =E[ei t X+il Y]

and fort6=0,

π_k(t) = 1

2π

Z π

−π

1 1₋ψ(t,l)e

−ikl_{d l and ρ(t) =} 1

π₀(t).

LetHx(s)denote the probability that the first entrance after time 0 of the walkSnx intoLtakes place

at[s, 0]: H_x(s) =P[for someτ_≥1,S_τx = [s, 0]andS_nx _∈/ Lfor 0<n< τ]. The evaluation ofH_x(s)

in[9]is based on the Fourier representation formula H[0,k](s) =

1 2π

Z π

−π

ρ(t)

h

π_−k(t)₋δ_0,k

i

while our starting point for evaluation ofG_Lis GL([0,n],[s,k]) =

1 2π

Z π

−π

h

π_−n+k(t)₋ρ(t)(π_−n(t)₋δ_0,n)π_k(t)ie−istd t, (1.3) which is derived in a standard way (see Appendix (A) for a proof). From (1.2) and (1.3) it follows that

G_L([0, 0],y) =H_−y(0). (1.4)

The asymptotic formulae presented above are extended to higher dimensional walks. Consider a d-dimensional random walk S_n0, moving on Zd, for d ≥3. Suppose that the walk is irreducible and has zero mean and the finite variances. LetX andY be the firstZd−1 component and thed-th component ofS0₁, respectively. Put σ2_d = E[Y2] and define the norm_{k · k} analogously to the two dimensional case. For k ∈ Z, x = (x1, . . . ,xd−1) ∈∈ Zd−1, let [x,k]stand for the d-dimensional

point(x1, . . . ,xd−1,k). Theorems 1.3 is extended as follows.

Theorem 1.4. Suppose the moment conditions E[_|X|d]<_∞and

E[Y2log_|Y|]<_∞ for d=3 and E[_|Y|d−1]<_∞for d≥4 (1.5) to hold. Then for n_≥0, k_∈Z, x _∈Zd−1, as([k_∨0]n+1)/(_k[x,k₋n]_k2_∨1)_→0

G_L([0,n],[x,k]) =Γ(d/2)

πd/2 ·

σ4_da∗(n)a∗(₋k) +nk σ2

dk[x,k−n]kd +o

_(n

∨1)(_|k| ∨1)

k[x,k−n]_kd

.

Letpm(x,y) =P[S_mx = y]andG(x,y):=P∞_m=0pm(x,y), the Green function of the walkS_nx. In the next theorem we give results corresponding to Theorems 1.1 and 1.2 under the following moment condition (to be a minimal one for the obtained estimates):

E[Y2log_|Y_|]<_∞ if d=4 and E[_|Y_|d−2]<_∞ if d_≥5. (1.6)

Theorem 1.5. Suppose (1.6) to hold. Then as_|n_|∨|k_|∨|x_{| → ∞}in such a way that_|nk_|/_k[x,k₋n]_k2

is bounded away from zero,

GL([0,n],[x,k]) (1.7)

=

(

G([0,n],[x,k])₋κ_dσ−2_{k[x+2nµ,k+n]k}−d+2_{(1+o(1)) if nk>0,}

o(_k[x,k₋n]_k−d+2) if nk<0,

whereµ=E[Y X] (_∈Rd−1)andκ_d= Γ(d/2)/(d−2)πd/2.

REMARK5. The Fourier representation (1.3) is a manifestation of the decomposition

G_L([0,n],[x,k]) =G([0,n],[x,k])₋X j∈Z

H[0,n](j)G([j, 0],[x,k]). (1.8)

As for the two-dimensional walk, ifA(x,y) =P∞_m=0[pm(0, 0)₋pm(x,y)], then as a two dimensional analogue of (1.8), with₋A(x,y)replacingG(x,y), we have

G_L([0,n],[s,k]) =₋A([0,n],[s,k]) +X j∈Z

If the walk satisfies a certain condition concerning symmetry and continuity in the vertical direction these identities reduce to the reflection principle. Otherwise, however, direct computation using them together with the estimates of H[0,n] and G (as found e.g. in [9], [8]) does not give any

correct asymptotic form ofG_L unless either the first term of the decomposition is dominant or we have nice estimates of the second order terms of H[0,n] andG; in any case the results obtained in

such a way are in general not sharp.

Two relevant matters on some closely related random walks are briefly discussed in the rest of this section.

Walks Killed on a half horizontal axis. Put L±=_{[s, 0]:±s≥0} ⊂Z2 (the negative and non-positive parts of the horizontal axis) and letG_L−(x,y)be the Green function of the two dimensional

walk killed onL−that is defined analogously toG_L(x,y). Then it follows that for y _∈/L−,

G_L−(x,y) =G_L(x,y) +

∞

X

s=1

H_x(s)G_L−([s, 0],y), (1.10)

and if y= [s, 0]_∈L−,

GL−(x,[s, 0]) =

∞

X

s1=1

Hx(s1)g(−∞,0](s1,s),

where g(−∞,0](s,s1) denotes the Green function of one dimensional walk which is the trace on

L+_{\ {}0_} of the walk (S_n[s,0]) killed on L−. Consider the last identity for the time-reversed walk. Indicating the dual objects by putting_∼over the notation as in Remark 4, we then find the identities G_L−(x,y) =G˜_L−(y,x), ˜H_y(s₁) =H_−y(−s₁) and ˜g_(−∞,0](s₁,s) = g_(−∞,0](s,s₁) and substitute these

into (1.10) to obtain that GL−(x,y) =G_L(x,y) +

∞

X

s=1 ∞

X

s1=1

Hx(s)H−y(−s1)g(−∞,0](s,s1) (y ∈/ L−); (1.11)

similarly

G_L−([s, 0],y) =G˜_L−(y,[s, 0]) =

  

H+_−y(₋s), s_≤0,

∞

X

s1=1

H_−y(₋s₁)g(−∞,0](s,s1) s>0,

whereH+_x(s)denotes the entrance distribution into L+(defined analogously toHx(s)).

Certain asymptotic estimates of H+_x(s) are obtained in [10] (under the present setting), [4] (for simple walk) and [1] (for a class of random walks with a finite range of jump by an algebraic method). Computation made in Section 4 of[10]would be helpful for evaluation of the double sum in (1.11).

Walks Killed on a half plane. Let D=_{[s,k]_∈Z2 :k≤0_}, the lower half plane. The evaluation of G_Lis intimately related to that ofG_D, the Green function of the walk killed onD. First it is pointed out that if the walk is either upwards or downwards continuous, namely either P[Y _≥ 2] =0 or P[Y ≤ −2] = 0, thenGL and GD agree onZ2\D. Let f+(n) (resp. f−(n)) (n=1, 2, . . .) be the

whose increment has the same law asY (resp. ₋Y): f_±(n) =E[f_±(n_±Y);n_±Y >0] (n_≥1)and lim_n→∞f_±(n)/n=1, which each exists uniquely ([7]:p.212). IfP[Y ≥2] =0, then forn≥1,

f+(n) = 1₂(σ22a(n) +n) and f−(n) =σ22a(−n) =n,

and the formula of Theorem 1.3 may be written as

G_D([0,n],[s,k]) = 2f+(n)f−(k)

πσ2

2k[s,k−n]k2

(1+o(1)) (1.12)

(n,k> 0, nk/_k[s,k₋n]_k2 _→0). In a separate paper [11] we prove (1.12) to be true in general underE[X˜2log_|X˜_|]<_∞, where the proof rests on Theorem 1.3 of the present paper.

The rest of the paper is organized as follows. In Section 2 some preliminary lemmas are established. Theorems 1.1 and 1.2 are proved in Section 3 and Theorem 1.3 in Section 4. In Section 5 their d-dimensional analogues, i.e., Theorems 1.4 and 1.5, are proved. Section 6, the last section, consists of Appendices (A), (B), (C) and (D) ; in (A) a proof of the Fourier representation formula (1.3) is given; we exhibit in (B) the formula for the Green function of Brownian motion corresponding to G_L for comparison with those given above and provide in (C) certain details for the caseσ₁₂₆=0 which are omitted in Section 2; in (D) two lemmas concerning Fourier type integrals are proved.

2

Preliminary Lemmas

Throuout the rest of the paper we suppose thatσ₁₂= 0 unless otherwise stated explicitly, so that the quadratic formQ(θ) =Q(θ₁,θ₂) =Pσ_{i j}θ_iθ_j (θ = (θ₁,θ₂)_∈R2) is of the form

Q(θ) =σ2₁θ₁2+σ2₂θ₂2.

The caseσ_{12 6}= 0 can be similarly treated and necessary modifications will be given in Appendix (C).

Letn>0. As an ideal substitute forπ_−n(t), we bring in

π◦_−n(t) = 1

2π

Z

R

2 Q(t,l)e

inl_{d l. (2.1)}

It follows that

π◦_−n(t) = e −nλ|t|

σ2_|t| , (2.2)

where, as in Remark 1,

λ=σ₁/σ₂=σ2/σ2₂.

On writingm=n₋kthe Fourier representation (1.3) is decomposed as

G_L([0,n],[s,k]) = 1

2π

Z π

−π

π_−m(t)₋π_−n(t)e−istd t+ 1

2π

Z π

−π

(π₀(t)₋π_k(t))π_−n(t)

π₀(t) e −ist_{d t.}

For the evaluation of the first integral above we observe that 1

2π

Z

R

(π◦_−m(t)₋π₋◦_n(t))e−istd t = 1

πσ2

Z ∞

0

e−|m|t/λ−e−|n|t/λ

t cosst d t

= 1

2πσ2log

_s2_+λ2_n2

s2+λ2_m2

. (2.3)

Lemma 2.1. Define J=J_s,n,mby

J= 1 (2π)2

Z π

−π

e−istd t

Z π

−π

2 Q(t,l)(e

iml

−einl)d l₋ 1 2πσ2log

s2+λ2n2 s2_+λ2_m2

. (2.4)

Then _|J_{| ≤} C

(s2_+1)(n2_∧m2₊₁₎ if s

2_{+ (m}2

∧n2)₆=0.

Proof.WriteJ =₋2π−2(I+I I), where

I:=

Z ∞

π

cosst d t

Z π

0

cosml₋cosnl

Q(t,l) d l and I I :=

Z ∞

0

cosst d t

Z ∞

π

cosml₋cosnl Q(t,l) d l.

ThenI≤C/(s2+1)(m2∧n2+1)since the functionR_π∞[Q(t,l)]−1_{cosst d tis a nice smooth function}

ofl bounded byC′/(s2+1); the same or a rather better bound is true for I I as readily seen from the identityI I=σ−2R_π∞cosml−_lcosnle−|s|l/λd l.

Proposition 2.1. For integers s,n,m such that s2+ (n2_∧m2)₆=0, define C_s,n,mvia the equation 1

2π

Z π

−π

h

π_−m(t)₋π_−n(t)ie−istd t= 1

πσ2log

k[s,n]_k

k[s,m]_k+Cs,n,m.

(a) Cs,n,mis bounded if and only if E0[|S01| 2_log

|S₁0|]<_∞, and if this is the case Cs,n,mtends to zero as

both|s|+_|n|and|s|+_|m|go to∞.

(b) Uniformly in s, C_s,n,m=o(_|n₋m_|/(_|n_{| ∧ |}m_|))as_|n_{| ∧ |}m_{| → ∞}. (c) Uniformly in s, C_s,n,m=o(1_∨log_{n2/(s2+m2)_})as n/(_|m_{| ∨}1)_{→ ∞}.

As an immediate corollary of Proposition 2.1 we obtain

Corollary 2.1. Suppose that n_{→ ∞}, s/n_→α_∈R_{∪ {±∞}}, m/n_→β_∈R. Then

Z π

−π

π_−m(t)₋π_−n(t)

e−istd t _−→ 1 σ2log

_(α/σ

1)2+ (1/σ2)2 (α/σ₁)2_{+ (β/σ}

2)2

,

where the limit value is understood to be0or_∞according as_|α_|=_∞orα=β=0. For each M>1 the convergence is uniform for s and m such that_|m_|<M n and_|m_{| ∨ |}s_|>n/M .

Proof of Proposition 2.1. LetJ =J_s,n,m be the function given by (2.4). Then

C_s,n,m= 1 (2π)2

Z π

−π

e−istd t

Z π

−π

₁

1−ψ(t,l)−

2 Q(t,l)

(eiml₋einl)d l+J (2.5)

According to Lemma 2.1J →0 as|s|+ (_|n| ∧ |m|)_{→ ∞}. The repeated integral on the right side is decomposed into the sum of the integral

Z

(−π,π]2

₁

1−ψ− 2 Q

(1₋e−ist+inl)d t d l

and a similar one (withmin place ofn). As_|s_|+_|n_{| → ∞}, the first integral converges to a finite limit (which equalsR₍

−π,π]2

h

1/(1₋ψ)₋2/Qid t d l) or diverges to+_∞according as E₀[_|S₁0_|2log_|S₁0_|]< ∞or=_∞(cf.[8]: proof of its Theorem 1), so that ifE0[|S01|2log|S10|]<∞,Cs,n,mremains bounded,

and otherwise it diverges to+_∞as_|n_{| → ∞}withs,mbeing fixed. This completes the proof of (a). For the proof of (b) we may suppose|m| ≤ n. Put k=n−m. By Lemma 2.1 J ≤C m−2 =o(k/n)

(as_|m_{| → ∞}). We must prove that the double integral in (2.5) iso(k/_|m_|). To this end we break it into three parts by dividing the inner integral atl =_±1/n. The part corresponding to the interval |l|<1/nis easy to evaluate to beo(k/n). For the evaluation of the remaining parts we put

D(t,l) = 1

1₋ψ(t,l)−

2

Q(t,l) and K ±₌

Z π

−π

e−istd t

Z π

1/n

D(t,±l)(e±iml−e±inl)d l.

It remains to proveK++K−=o(k/m). Since the factore−istdoes not come into play at all, lets=0 for simplicity. Writingeiml−einl= (e−ikl−1)einl we integrate by parts to have

K+ = 1

in

Z π

−π

d t

h

(eiml−einl)D

iπ

l=1/n+

k n

Z π

−π

d t

Z π

1/n

eimlDd l

− 1 in

Z π

−π

d t

Z π

1/n

(e−ikl₋1)einl∂_lDd l.

The first term on the right side is dominated bykn−2R₀π|D(t, 1/n)_|d t+r=o(k/n) +r, wherer is the boundary term corresponding toπ and cancels out with that arising fromK−. Integrating by parts once more and using the relations

Z π

−π

|∂_lνD(t,l)_|d t=o

l−ν−1

as l_→0 (ν=0, 1, 2), (2.6)

one can easily deduce that the second and third terms areo(k/m)ando(k/n), respectively.

We continue the argument made above for the proof of (c). ObviouslyK±=o(logn)(uniformly in s,k), hence one may suppose|m| ∧ |s| → ∞. First consider the case|m|>_|s|. We must prove that K± =o(log(n/_|m_|))as n/_|m_{| → ∞}. The contribution toK± of the part involvingeinl tends to zero asn_→0 uniformly ins, namely

sup

s

Z π

−π

e−istd t

Z π

1/n

D(t,_±l)e±inld l

→

With this in mind we see that as_|m_{| → ∞}

Z π

−π

e−istd t

Z π

1/n

D(t,l)eimld l

≤

C

Z 1/|m|

1/n

d l

Z π

0

|D_|d t+o(1) =o(1_∨log(n/_|m_|)),

in view of (2.6). Now lets_|s_{| ≥ |}m_|. In view of (2.7), with symmetric roles oft andl inD(t,l)taken into account, it suffices to prove that uniformly inm,

M+:=

Z 1/s

0

e−istd t

Z π

1/n

D(t,l)eimld l=o(1∨log(n/s)) (2.8)

ass_{→ ∞}. Changing the variable of integration we see that for anyǫ >0 andN>0 |M+_{| ≤} 1

s2

Z 1

0

d t

Z sπ

s/n

D(t/s,l/s)d l _≤

Z 1

0

d t

ZN

s/n

ǫ

t2+l2d l+

Z ∞

N

C l2d l

≤ 2ǫ[(log(n/s) +logN] + C

N for all sufficiently larges. Thus (2.8) is verified. Proof of Proposition 2.1 is complete.

Proposition 2.2. As_|k_{| → ∞}, uniformly for n_{≥ |}k_|and for s_∈Z,

Z π

−π

(π₀(t)₋π_k(t))π_−n(t)

π₀(t) e

−ist_{d t=} 1

σ2log

_s2_+λ2_(n+

|k_|)2

s2+λ2_n2

+o

_k

n

.

Proof. Writeρ◦(t)for 1/π₀◦(t) =σ2_|t|and make decomposition

ρ[π₀₋π_k]π_−n = ρ◦[π◦₀₋π◦_k]π◦_−n+nρ[π₀₋π_k]₋ρ◦[π◦₀₋π◦_k]oπ◦_−n

+ρ[π₀₋π_k](π_−n−π◦_−n). (2.9)

On using (2.2) the Fourier transform of the first term equals 2

σ2

Z ∞

0

e−λnt₋e−λ(n+|k|)t

t cosst d t= 1 σ2log

_s2_+λ2_(n+

|k_|)2

s2_+λ2_n2

, hence gives the principal term of the formula of the lemma.

For the remaining terms we use certain estimates of Fourier type integrals, which are collected in Appendix (D). From Lemma 6.1 (i) there it follows that

ρ(t)_|π₀(t)₋π_k(t)_{| ≤}C_|kt_|, which impliesRπ

−πρ[π0−πk](π−n−π

◦

−n)e−istd t=o(k/n)in view of Lemma 6.2 (ii).

By Lemma 6.1 (i)

lim

k→∞k

−1 _sup 0<|t|<1

|π₀(t)₋π_k(t)₋[π◦₀(t)₋π◦_k(t)]_|=0.

This, together with (2.9), shows that the middle term on the right side of (2.9) ise−λn|t|(_|k_|×o(_|t_|)+

|t| ×o(k)), hence its contribution is o(k/n)as above (but this time not only nbut_|k|must also be made large indefinitely: otherwiseo(k/n)must be replaced byO(k/n)).

REMARK6. Ifa{d=2}([s,n])denotes the potential function of the two-dimensional walkS_·x, namely

a{d=2}([s,n]) =P∞_j=0(P[S_j0=0]₋P[S[_js,n]=0]), then

a{d=2}([₋s,n])₋a{d=2}([₋s,n₋k]) = 1

2π

Z π

−π

h

π_−n(t)₋π_−n+k(t)

i

e−istd t.

The asymptotic estimates ofa{d=2}([s,n])given in[3]or[8]provide better estimates ofC_s,n,mthan in Proposition 2.1 but under certain stronger moment conditions.

3

Proof of Theorems 1.1 and 1.2

For simplicity we let 0<_|k| ≤nunless contrary is stated, which gives rise to no loss of generality as being pointed out in Remark 4. We continue to suppose thatQis diagonal.

Set for(α,δ)_∈R2with_|α_|+_|1₋δ_|>0,

Λ(α,δ) = 4δ+

α2_{+ (1−δ)}2 (δ+= (|δ|+δ)/2),

write

s_∗=s/σ₁, n_∗=n/σ₂ and k_∗=k/σ₂, and for_|s_|+_|n₋k_|>0, definer(s,k,n)by

GL([0,n],[s,k]) = (2πσ2)−1log

1+ Λ(s_∗/n_∗,k_∗/n_∗)+r(s,k,n). It follows thatk[s,n]_k2_=σ2_(s2

∗+n2∗). For simple random walk the reflection principle may apply

and it immediately follows from an asymptotic expansion of the potential functiona{d=2} as found in[3],[5]or[2]that

r(s,k,n) =O((s2+_|n₋k_|2+1)−1). We are to find a reasonable estimate ofr in general case.

We first consider the case whenn→ ∞under the constraint

Λ(s/n,k/n) = 4nk+

s2_{+ (n−k)}2 ≍ 1, (3.1)

namelyM−1_≤Λ(s/n,k/n)_≤M for someM>0. This condition is equivalent (or understood to be so) to the condition that there exists a constantǫ, 0< ǫ <1 such thatk> ǫn;_|s|<n/ǫ; and either k<(1−ǫ)nor|s|> ǫn.

Theorem 3.1. As n _{→ ∞} under (3.1), r(s,k,n)_→ 0 uniformly for s,k subject to the restriction specified above (for eachǫ); in particular, if k_∗/n_∗→δ, s_∗/n_∗→α, 0< δ_≤1and|1₋δ_|+_|α_|>0, then GL([0,n],[s,k])→(2πσ2)−1log

1+ Λ(α,δ).

Proof. The integrand of the integral on the right side of (1.3) (withµ=0,n>0) may be written in the form

and the assertion is inferred as an immediate consequence of Propositions 2.1 and 2.2. Next consider the case when

Λ(s/n,k/n)_{→ ∞}. (3.3)

Theorem 3.2. As n_{→ ∞}under (3.3), namely as k/n_→1and s/n_→0, the difference G_L([0,n],[s,k])₋ 1

πσ2 log

k[s,n+k]_k

k[s,n−k]_{k ∨}1

remains uniformly bounded if and only if E0[|S10|2log|S01|]<∞. If this is the case it converges to zero

as_|s_|+_|n₋k_{| → ∞}; if not it diverges to+_∞as n_{→ ∞}whenever_|s_|+_|n₋k_|is confined in any finite set.

Proof. Here again the expression on the right side of (3.2) is employed. From Proposition 2.2 it follows that (2π)−1Rπ

−π(πk−π0)ρπ−ne

−ist_{d t → (πσ}2₎−1_{log 2 in the limit under (3.3).}

Com-bined with this as well as with the equality lim_k[s,n+k]_k/_k[s,n]_k=2 (in the same limit) (a) of Proposition 2.1 shows the assertion of Theorem 3.2.

Proof of Theorem 1.1. On looking at (3.2) the first half of Theorem 1.1 follows from Propositions 2.1 and 2.2. The second half is included in Theorem 3.2.

Proof of Theorem 1.2. If|s| → ∞butnremains in a finite set, then the assertion of Theorem 1.2 is rather trivial since then the probability that the walkS_·[0,n] visits[s,k]earlier than Ltends to zero. If n → ∞, we have only to apply Propositions 2.1 and 2.2 as in the proof of Theorem 3.1 since m=n−k=n+_|k|in the casek<0.

4

Proof of Theorem 1.3

Theorem 1.3 pertains to the case whenΛ(s/n,k/n)_→0 and follows if we prove

Theorem 4.1. Suppose that E[_|X˜_|2_log

|_X˜_|]<_∞. Then, as(k∨1)/(n∨ |s|)_→0

GL([0,n],[s,k]) =

1 π

σ2₂a∗(n)a∗(₋k) +k_∗n_∗ k[s,n]_k2 +o

|nk_| s2+n2

for nk6=0.

Proof. As in the preceding section suppose that 0<_|k_{| ≤}nandQis diagonal. The proof is given in the following three subcases

(i)|s| ≤n; k=o(n), (ii)n≤ |s|; k=o(_|s|)and (iii)|s| ≤n; n≍(₋k)_∨1, separately. Put

e_n(t) =π_−n(t)₋π₀(t) +a(n) (4.1)

and

f(x) =_|x|−1(e−|x|−1) +1(= 1 2!|x| −

1 3!|x|

2_{+· · ·)}

as in [9] (see (5.12) of the next section for how f comes up in the next lemma). We need the following result from[9](Lemmas 4.2 and 5.5).

Lemma 4.1. (a) ρ(t) =σ2_|t_|+o(t), ρ′(t) =σ2t/_|t_|+o(1) and ρ′′(t) =o(1/t) as t_→0;

(b) e_n(t) =σ−_d2_|n_|f(λnt) +η_n(t)withsup_{n |}n_|−1_|η(_nj)(t)_|= o(_|t_|−j) for j =0, 1, 2; in particular, each ofe_n(t)/n, te′_n(t)/n and t2e′′_n(t)/n tends to zero as nt_→0and is uniformly bounded. (Hereη(j) denotes the derivative of the j-th order.)

Case (i)|s| ≤n; |k|=o(n). We make the decomposition

π_−n+k−ρ π_−nπ_k = (π_−n+k−π_−n) +a(₋k)ρ π_−n−ρ π_−ne_−k.

The contribution toGL of the first term on the right side is given in Proposition 2.1 (b) and that of

the second is given by Theorem 1.1 of[9](in view of (1.2)). The sum of these two may be written as

1 2πσ2log

k[s,n]_k

k[s,n₋k]_k+

a(₋k)n

π_k[s,n]_k2 +o(k/n) =

a(₋k)n+k_∗n_∗

π_k[s,n]_k2 +o(k/n) (4.2)

asn_{→ ∞}under (i). As for the third term we write

Z π

−π

(ρ π_−ne_−k)(t)e−istd t=

Z π

−π

(ρ π◦_−ne_−k)(t)e−istd t+

Z π

−π

(ρ[π_−n−π◦_−n]e_−k)(t)e−istd t.

Having the bound_|(ρ π◦_−ne_−k)(t)_{| ≤}C[_|kt_{| ∧}1]e−λn|t|in view of (2.2) and Lemma 4.1 (b), we infer that the first integral on the right side is o(k/n). That the second one admits the same estimate follows from Lemma 6.2 with the help of the bound_|e_−k(t)_{| ≤}C_|k_|. Thus we obtain the relation of Theorem 4.1.

Case (ii)n_{≤ |}s_|; _|k_|=o(_|s_|). Here we make the decomposition

π_−n+k−ρ π_−nπ_k = (π_−n+k−π_−n−π_k+π₀)₋(π_−n−π₀)(π_k−π₀)/π₀

= (π_−n+k−π_−n−π_k+π₀)₋ρe_ne_−k

+a(n)ρe_−k+a(₋k)ρe_n−a(n)a(₋k)ρ. (4.3) It suffices to compute the Fourier inversions for the first two terms of the right-most member, the other terms being dealt with in[9](see (4.17) and (4.18) given at the end of the present proof). The first and second of the two is dealt with in Lemmas 4.2 and 4.3 below, respectively.

Lemma 4.2. If E[X2log|X|]<_∞, then as|s| → ∞

Z π

−π

(π_−n+k−π_−n−π_k+π₀)(t)e−istd t= 1

σ2log

_(s2

∗+k2∗)(s∗2+n2∗)

s_∗2(s2_∗+ (n_∗−k_∗)2₎

+o

_nk

s2

.

Proof. Write

Z π

−π

(π_−n+k−π_−n−π_k+π₀)(t)e−istd t=I+r with

I=I(s,k,n) =

Z π

−π

e−istw(t)d t

Z π

−π

2(ei(n−k)l−einl−e−ikl+1)

r=r(s,k,n) =

Z π

−π

e−istd t

Z π

−π

₁

1₋ψ(t,l)−

2w(t)

Q(t,l)

(einl−1)(e−ikl−1)d l,

wherew(t)is a smooth cutoff function such thatw≥0;w=1 for_|t|<1/2;w=0 for_|t|>1. As before we have that for anyN

I= 1

σ2log

_(s2

∗+k∗2)(s∗2+n2∗)

s2

∗(s2∗+ (n∗−k∗)2)

+O(_|s_|−N)

as_|s_{| → ∞}. As for the error term r, perform integration by parts twice (to have the factors−2) and decompose(einl−1)(e−ikl−1) =sinnl sinkl+An,k(l), where

A_n,k(l) = (cosnl₋1)(coskl₋1)₋i(cosnl₋1)sinkl+i(coskl₋1)sinnl. Observe

F(t,l):=∂_t2

₁

1₋ψ(t,l)−

2w(t)

Q(t,l)

=o

₁

(t2+l2)2

(_|t| ∨ |l| →0), (4.5)

so thatR₍

−π,π]2|F(t,l)| ×d t l

3

→0 as l→0, and then deduce that sup

k,n

Z π

−π

e−istd t

Z π

−π

F(t,l)An,k(l)

nk d l

−→

0 (_|s_{| → ∞})

(cf. Lemma 11.1 of[9]). In order to deal with the contribution of sinnl sinklwe need the moment conditionE[X2log|X|]<_∞, with which we proceed as in[9]: Lemma 6.3 (estimation ofΘ_{I I} ). We decomposeF(t,l) =V(t,l) +R(t,l), where

V(t,l) = E[X

2_eiY l_(eiX t_−1)]

(1₋ψ(t,l))2 (4.6)

andRis the rest. On the one hand R admits once more differentiation with∂_tR=o((_|t|+_|l|)−3), and the integration by parts shows that its contribution iso(nk). On the other handV satisfies

Z π

−π

Z π

−π

(t2+l2)_|V(t,l)_|d t d l<_∞

ifE[X2log_|X_|]<_∞, so that the Riemann-Lebesgue lemma apply. These together show that

sup

k,n

Z π

−π

e−istd t

Z π

−π

F(t,l)sinnl sinkl

nk d l

−→

0 (_|s_{| → ∞}).

(Here the outer integral (for theRpart) must in general be understood improper and for the integral on_|t_|<1/_|s_|one should integrate by parts back to have the result above.) Consequentlyr(s,k,n) =

s−2n|k| ×o(1)witho(1)_→0 as_|s| → ∞uniformly inn,k. The next lemma is subtler than the preceding one.

Lemma 4.3. If E[X2log_|X_|]<_∞, then as_|s_{| → ∞}with k=o(s)and n=O(s)

Z π

−π

ρ(t)e_n(t)e_−k(t)e−istd t=o

_nk

s2

Proof. Although one can proceed by extending the lines of the proof of Theorem 1.2 of[9](given in Sections 3 and 6), where evaluation of the integral Rρe_ne−istd t is carried out, we proceed somewhat differently in a way the proof works better in the higher dimensions.

From Lemma 4.1 we have the expression e_n(t) = f_n(t) +η_n(t)with the estimates

|f_n(j)(t)_{| ≤}C n(_|nt| ∧1)_|t|−j and η(_nj)(t) =n×o(t−j) for j=0, 1, 2, (4.7) where fn(t) =σ−_d2|n|f(λnt). In addition to the fact that bothρ and en do not necessarily admit

the differentiation of the third order, the difference of estimates between the derivatives of f_n and those ofη_n as given above causes the complication of arguments. To make the proof conceptually clear we replace(ρe_ne₋k)(t)byσ2(|t|fnf−k)(t)w(t)and compute the corresponding integral of the

latter and that of the difference between the two, separately. First we consider the difference, for which we need to find a way round the lack of differentiability. Writeρ◦(t)forσ2_|t_| (as in (2.9)) and put

g(t) = (ρe_ne_−k)(t)₋(ρ◦f_nf_−k)(t)w(t). (4.8) We may suppose that 0<k_≤n_≤s. After integrating by parts once we split the range of integration at t = _±1/_|s|. From Lemma 4.1 it follows that under the constraint of the variables s,n,s of the lemma

sup

|t|<1/s|

(ρe_ne_−k)′(t)_|/nk_→0 as s_{→ ∞},

which entails the same relation forg′in place of(ρe_ne_−k)′. We then deduce that the Fourier integral ofgon|t|<1/_|s|iso(kn/s2), and one more integration by parts gives

s2

Z π

−π

g(t)e−istd t=₋

Z

1/|s|<|t|<π

g′′(t)e−istd t+o(nk). (4.9)

For the latter we express g as the telescopic sum(ρ₋ρ◦)f_nf_k+ (e_n− f_n)ρf_k+ρe_n(e_−k−f_k)and deal with them separately. Among them we consider only the last term and verify that

Z

1/|s|<|t|<π

(ρe_nη_−k)′′(t)e−istd t=o(nk), (4.10)

the integrals for the other two being evaluated in the same way in view of (4.7). For the proof of (4.10) we claim that

Z

1/|s|<|t|<π

[ρe_n](t)η′′_−k(t)e−istd t=o(nk). (4.11)

Since _|ρe_{n| ≤} C_|nt_| and η′′_−k(t) = k_×o(t−2), the integrand is nk_×o(1/t). Let F and V be the functions given in (4.5) and (4.6), respectively. We may write

η′′_−k(t) = 1

2π

Z π

−π

F(t,l)(1₋e−ikl)d l+r_k(t)

withr_n(t), a nice function that is negligible for the present purpose, and thenη′′_−k(t) =v_k(t)+τ_k(t), where

v_k(t) = 1

2π

Z π

−π

andτ_kis the rest. Then, as in the preceding proof, we see that the functionτ_k(t)is differentiable for t6=0,τ′_k(t)/k=o(1/t3)and its contribution iso(nk)and that ifE[X2log_|X|]<_∞, the Riemann-Lebesgue lemma yields

Z

1/|s|<|t|<π

[ρe_n](t)v_k(t)e−istd t=o(nk). (4.12) That the last estimate is uniform in n and k requires proof. Since, by dominated convergence,

Rπ

−π|t vk(t)|d t/k → 0 as |k| → ∞, it suffices for the proof to show that n

−1 _{times the integral}

restricted on_|t_|> ǫtends to zero uniformly innfor eachǫ >0 andk. This follows from the fact that e_n=π_−n−π₀+a(n)and sup_ǫ<|t|<π|π_−n(t)₋π₀(t)_|=o(n). Thus the claim (4.11) has been verified. For the proof of (4.10) we must evaluate the integrals of other terms of

ρe_n[e_−k−f_k]

′′

(t), e.g., ρ′′(t)e_n(t)[e_−k− f_k](t), but their evaluations are quite similar, hence omitted.

It remains to prove

Z

R

(ρ◦fnfk)(t)w(t)e−istd t=o

_nk

s2

. (4.13)

It is not hard at all to verify this as in a similar way to the above, but we take up another way. We are concerned with the Fourier integral that has an explicit form ifw is removed and we shall seek out it. To this end the following decomposition ofρ◦f_nf_kis convenient:

ρ◦fnfk = ρ◦(π◦n−π◦0)(π◦k−π◦0) (4.14)

+σ₂−2kρ◦(π◦_n−π◦₀) +σ₂−2nρ◦(π◦_k−π◦₀) +σ−₂4nkρ◦.

Remember the identity π_n(t) = e−λ|nt|/σ2_|t| given in (2.2) and observe thatρ◦π◦_nπ◦_k = π◦_n+k, so that the first term on the right side of (4.14) equalsπ◦_n+k−π_n◦₋π◦_k+π◦₀, whose Fourier transform, already computed in the preceding lemma, equals

1 σ2log

_(s2

∗+k∗2)(s∗2+n2∗)

s_∗2(s2_∗+ (n_∗+k_∗)2₎

= −nk

σ₂2π_k[s,n]_k2 +o

_nk

s2

. (4.15)

The other terms are the transforms of Cauchy densities and by inverting them we have nk

σ2₂π_k[s,n]_k2 +

nk

σ2₂π_k[s,k]_k2 −

nk

σ2₂π_k[s, 0]_k2 (4.16)

for the Fourier integral of their sum (but the last term necessitates truncation by w or, otherwise, interpretation as a Schwartz distribution). Now summing (4.15) and (4.16) gives (4.13). This completes the proof of Lemma 4.3.

Now we can finish the proof of Theorem 4.1 in the case (ii). Proposition 6.1 of [9] says that if E[X2log|X|]<_∞, then as|s| → ∞withn≤ |s|

a∗(₋k)

2π

Z π

−π

ρ(t)e_n(t)e−istd t= a ∗_(−k)n

πσ2

₁

s_∗2+n2_∗ − 1 s2_∗

+o

_nk

s2

(4.17) and the corresponding Fourier coefficient of a∗(n)ρ(t)e₋k(t) isO(nk3/s4), hence o(nk/s2); from

[9]we also have

a∗(₋k)a∗(n)

2π

Z π

−π

(₋ρ(t))e−istd t= a

∗_(−k)a∗_(n)σ2 2

πσ2_s2 ∗

+o

nk s2

ζ₁(θ) =₋ρ 2 2π

Z π

−π

E[X_ω2eil Y(1₋w(_|X|θ))] (1₋ψ)2 d l,

and ζ₂(θ) = ρ

2 2π

Z π

−π

E[X_ω2eil Y(eiX·θ₋1)w(_|X_|θ)] (1₋ψ)2 d l. We may suppose that w(u) = 1 for |u| < 1

4 and 0 for |θ| > 1

2. Note that |∇ j

ωw(|X|θ)(|X||θ|)j is

bounded by a constant times the indicator function1(1₄<_|X_||θ_|< 1₂).

On observing that_∇ωj ζ1(θ) =o(|θ|−j−1)for j=0, 1, 2 integrations by parts give

Z

T′

1/s

ζ₁(θ)e−i x·θdθ = i

s Z

∂T′

ζ₁(θ)e−i x·θω_·ndσ+o(1/s),

where ndenotes the outer normal vector to ∂T′. The boundary integral above vanishes since if

X 6=0,w(_|X|θ) =0 on∂T′so that the integrand is periodic. On integrating by parts and applying Fubini’s theorem

Z

T′

1/s

ζ₂(θ)e−i x·θdθ = E – Z

T′

1/s

e−i x·θdθ 2πs

Z π

−π

ρ2X_ω3eil YeiX·θw(_|X|θ) (1₋ψ)2 d l

™

+

Z

T′

1/s

e−i x·θdθ 2πsi

Z π

−π E

–

X_ω2eil Y(eiX·θ−1)_∇ω

_ρ2_w(

|X_|θ) (1₋ψ)2

™ d l.

The first term on the right side iso(1/s). Indeed if the expectation above is restricted to the event |X_|<K, it iso(1/s)for eachK in view of the Riemann-Lebesgue lemma. Changing the variables of the outer integral we see that the same expectation but on_|X_{| ≥}K is bounded in absolute value by a positive multiple of

E – Z

R2

u2w(u)du

2πs Z

R

|Xω|3

|X_|(u2_+l2₎2d l;|X| ≥K

™

≤ 1

sE[X

2_:

|X_|>K].

The second term is similarly dealt with. If the expectation involved in it is restricted to_|X_{| ≤}K, the corresponding part iso(1/s). The other part, after integrating by parts once more, is disposed of in the same way. These together verify

Z

T₁′_/s

ζ₂(θ)e−i x·θdθ =o(1/s).

We are left with ζ₀. Let 0< ǫ < 1/2. Split the range of integration into two parts according as |X₋x_{| ≥}ǫsor_|X₋x_|< ǫsand callJ₁andJ₂, respectively, their contributions toR_T′

1/s

ζ₀(θ)e−i x·θdθ. Since J₂ = (2π)2M_ǫ(x) if ǫ <1/2, it suffices to show thatJ₁ = o(1). We integrate by parts with respect toθ by factorizing the integrand asei(X−x)·θ×(the other) to deduce that for eachǫ >0,J1 equals

−

Z

T₁′_/s

dθ

Z π

−π E

–

X_ω2ei(X−x)·θ

2πi|X−x|e il Y

∇ω

₍₁

−w(_|X|θ))ρ2 (1₋ψ)2

;_|X₋x_{| ≥}ǫs ™

plus the boundary integral that iso(1/s)(the integral on∂T′vanish by the same reason as before). Repeat the same integration by parts once more and note that both _∇w(_|X|θ) and 1₋w(_|X|θ) vanishes if|X||θ_|< 1

4. We then observe that the double integral above is bounded in absolute value by a constant multiple of

E

_sX2

ω

|X−x|2;|X−x| ≥ǫs,|X| ≥

s

4

+

Z π

1/s

E

_X2

ω

|X−x|2;|X−x| ≥ǫs,|X| ≥ 1 4r

_{d r} r2 =o(1/s).

Finally, observing that the expectation involved in the second term is s−2 _×o(1) as r _→ 0, we conclude thatJ1=o(1/s)ass→ ∞. The proof of (5.18) is complete.

6

Appendices

(A) We prove the formula (1.3). The proof is based on the identity

pm_L([0,n],[s,k]) =pm([0,n],[s,k])₋ m

X

τ=1

X

j∈Z

F_n(τ,j)pm−τ([j, 0],[s,k]) (m_≥1), (6.1)

wherepm(x,y) =P[S_mx = y]and forτ=0, 1, 2, . . .

F_n(τ,s) =PhS[_τ0,n]= [s, 0],S[_j0,n]_∈/ Lforj=1, . . . ,τ₋1i

(the joint distribution of the timeτand positionsof the first entrance into Lof the walkS_·[0,n]). It is readily inferred (cf.[9], Appendix A) that for 0<_|t| ≤π,

∞ X

m=0

X

s∈Z

pm(0,[s,k])eist=π_k(t) and

∞ X

τ=0

X

j∈Z

F_n(τ,j)ei j t =ρ(t)π_−n(t) and with the help of these identities we derive from (6.1)

∞ X

m=0

X

s∈Z

pm_L([0,n],[s,k])eist=π_−n+k(t)₋ρ(t)π_−n(t)π_k(t).

The last double series is absolutely convergent: in factP∞_m=0P_s∈Zpm_L([0,n],[s,k])is nothing but the Green function of one-dimensional walk killed at the origin and hence equalsa(n) +a(₋k)₋

a(n−k)<_∞(cf.[7]). Now the Fourier inversion yields (1.3).

(B) Let B_t be the two dimensional standard Brownian motion and consider its linear transform

Xt=Q1/2Bt. If g◦(x,y)denotes the Green function ofXt killed on LandgB(x,y) =−_π1log|y−x| and g◦_B(x,y) = g_B(x,y)₋g_B(x′,y), where x′ stands for the mirror image of x relative to the line Q−1/2L, then, g◦(x,y) = g◦_B(Ax,Ay)detA(x ∈ R,y > 0) where A= Q−1/2. We may write (Ax)′=Ax˜, so that

g◦(x,y) = 1 2πσ2log

Q−1(y₋˜x)

The point ˜x is characterized by ˜x₂ = ₋x₂ and A(x ₋x˜)_·A[1, 0] = 0, which is solved to yield ˜

x1 = x1−2µx2 (µ= σ12/σ2). Thus if x = [0,n], then ˜x = [−2µn,−n], and the formula above may be written as

g◦([0,n],[s,k]) = 1 πσ2log

k[s+2µn,k+n]_k

k[s,k₋n]_k (nk>0). in accordance with (6.2) given below.

(C) Here we indicate a way the proofs given in Section 2 are modified in the caseσ₁₂₆=0. (For those in Sections 3 and 4 see Section 10 of[9].) It is noted that the formula of Theorem 1.1 may be written as

G_L([0,n],[s,k]) = 1 πσ2log

k[s+2µ(n+k₋),n+_|k|]_k

k[s,k−n]_{k ∨}1 +Bs,n,k, (6.2) wherek₋= (_|k| −k)/2. If we define

˜

π_k(t) = 1 2π

Z π+µt −π+µt

e−ikl

1−ψ(t,l−µt)d l, then

G_L([0,n],[s,k]) = 1 2π

Z π

−π h

˜

π_−n+k(t)₋ρ(t)π˜_−n(t)π˜_k(t)ie−i(s+µn−µk)td t. The principal part of 1₋ψ(t,l−µt) =1₋E[ei tX˜+il Y]is given by half the quadratic form

˜

Q(θ) =Q(θ₁,θ₂+µθ₁) =σ˜2₁θ₁2+σ₂2θ₂2 with σ˜2₁=σ2₁₋µσ₁₂=σ4/σ2₂.

For the quadratic form of the inverse matrix ˜Q−1 we have ˜Q−1(α₋µβ,_±β) =Q−1(α,β); we shall be interested in the following version of it

˜

Q−1(α+µ(1₋δ), 1+_|δ_|) =Q−1(α+2µ(1+δ₋), 1+_|δ_|) =

¨

Q−1(α+2µ, 1+δ) δ >0,

Q−1(α,δ₋1) δ <0. (6.3) By changing the variables according tou=l+µt,

Z π

−π

π_−m(t)₋π_−n(t)e−istd t= 1 2π

Z π

−π d t

Z π+µt −π+µt

A+B

1₋ψ(t,u₋µt)du, where

A=e−i(s+µm)t(eimu₋einu), B=

e−i(s+µm)t₋e−i(s+µn)t

einu.

Denote by I_AandI_B the integrals corresponding toAandB, respectively. Then, the same proofs of Propositions 2.1 and 2.2 yield that in the limit indicated in Corollary 2.1

IA → 1 σ2log

_Q˜−1_{(α+µβ, 1)}

˜

Q−1_(α+µβ,β)

and IB → 1 σ2log

_Q˜−1_{(α+µ, 1)}

˜

Q−1_{(α+µβ, 1)}

respectively. This shows Proposition 2.1 that is so modified that the limit in Corollary 2.1 becomes σ−2log[Q˜−1(α+µ, 1)/Q˜−1(α+µβ,β)]. For Proposition 2.2 we can proceed similarly. We write down

Z π

−π

(π₀₋π_k)π_−n

π₀ e

−ist_{d t=} 1 (2π)2

Z π

−π d t

Z π+µt −π+µt

(A+B)d l

1₋ψ(t,l₋µt)

Z π+µt −π+µt

ρ(t)einudu

1₋ψ(t,u₋µt), where

A=e−i(s+µn)t(1₋e−ikl), B=

e−i(s+µn))t₋e−i(s+µ(n−k)t

e−ikl, and denote byI_AandI_Bthe corresponding integrals. Then, ass/n_→αandk/n_→δ,

I_{A →} 1

σ2log

_Q˜−1_{(α+µ, 1+}

|δ_|) ˜

Q−1_{(α+µ, 1)}

, I_{B →} 1

σ2log

_Q˜−1_(α+µ(1

−δ), 1+_|δ_|) ˜

Q−1_{(α+µ, 1+|δ|)}

, which shows Proposition 2.2 modified in an obvious way.

(D) Let θ denote the (d−1)-dimensional variable as in Section 5 (but here the case d = 2 is included). It is supposed thatX andY are uncorrelated.

Lemma 6.1. (i) There exists a constant C such that|π₀(θ)₋π_k(θ)_{| ≤}C|k|(0<_|θ_|<1).

(ii) lim k→∞k

−1 _sup 0<|θ|<1|

π₀(θ)₋π_k(θ)₋[π◦₀(θ)₋π◦_k(θ)]_|=0.

Proof. We decompose 2π[π₀(θ)₋π_k(θ)] =H_c(θ) +H_s(θ), where

H_c(θ) =

Z π

−π

1−coskl

1−ψ(θ,l)d l and Hs(θ) =−i

Z π

−π

sinkl

1−ψ(θ,l)d l. We claim that

1

k0<sup|θ|<1|

Hs(θ)| →0 as |k| → ∞. (6.4) Noticing that only the odd part of(1₋ψ)−1is relevant, for the proof we writeψ(θ,l)₋ψ(θ,₋l) = 2i E[(eiX·θ −1)sinY l] +2i E[sinY l]so that

H_s(θ)

k =

Z π

−π

E[(eiX·θ₋1)sinY l] (1₋ψ(θ,l))(1₋ψ(θ,₋l))

sinkl k d l+

Z π

−π

E[sinY l₋Y l] (1₋ψ(θ,l))(1₋ψ(θ,₋l))

sinkl k d l.

The first integral iso(1)uniformly in_|θ_|>0 since its integrand iso(_|θ_|/[θ2+l2])as_|θ_|+_|l_{| →}0 (due toE[Y X] =0). The integrand of the second one is dominated in absolute value byC E_|sinY l₋ Y l|/_|l|3, which is integrable so that the dominated convergence theorem may be applied to conclude that it also iso(1). This verifies the claim (6.4).

SinceRπ

−π(1−coskl)l

−2_{d l < kπ, we have|Hc| ≤C k. Combined with (6.4), this shows (i). (ii) is} obtained in a similar way.

Lemma 6.2. Let h(θ)be a periodic bounded Borel function on the(d₋1)-dimensional torus T′and put

I_n=

Z

T′ dθ

Z π

−π h(θ)

₁

1−ψ(θ,l)− 2w(l)

Q(θ,l)

einld l.

(i) Let_|h(θ)_{| ≤}1. Then I_n=o(n−d+2), provided that

E[Y2log_|Y_|]<_∞ for d=2, 4 and E[_|Y_|d−2]<_∞ for d_≥5. (ii) Let_|h(θ)_{| ≤ |}θ_|. Then I_n=o(n−d+1), provided that

E[Y2log_|Y_|]<_∞ for d=3 and E[_|Y_|d−2]<_∞ for d_≥4.

In both(i)and(ii)the estimates of I_n hold uniformly for h(θ)that satisfy the assumed bounds.

Proof.Write r=pθ2_+l2_,B

n={r≤1/n}andT=T′×(−π,π]. Put

Dw(θ,l) = 1 1₋ψ(θ,l)−

2w(l)

Q(θ,l).

We prove (ii) first. If d =2, we decompose the range T intoB_n and its complement. Clearly the integral on B_n iso(1/n) under the assumption of (ii). We integrate by parts (w.r.t. the variablel) the integral onT\Bn, which is then transformed into

the boundary integral

in −

1

in Z

T\Bn

h(θ)∂_lDw(θ,l)einldθd l. (6.5) Since_|∂Bn|=O(1/n)andh∂_lDw =o(n)on∂Bn, the boundary integral iso(1). Integrating the last integral by parts once more we find it to be at most a constant multiple of n−1R₁∞_/nr−2d r =o(1). Thus I_n=o(n−1)as required.

Ford=3 we first perform integration by parts to obtainnIn=i

R

Th(θ)∂lD

w_(θ,l)einl_{dθd l and then} apply the same procedure as above to the right side to see that

n2In=−

Z

T\Bn

h(θ)∂_l2Dw(θ,l)einldθd l+o(1) (6.6) We make decompositionh∂_l2Dw=K+τ, where

K(θ,l) = h(θ)E[e

iX·θ_(eiY l_−1)Y2_] (1₋ψ(θ,l))2 ,

andτ=τ(θ,l)is the rest. Since thenτadmits one more differentiation w.r.t. land∂_lτ=o(1/r3), its contribution to the integral in (6.6) is shown to beo(1). Now use the supposed moment condition

E[Y2log_|Y|]<_∞to see that K is integrable onT (cf. Lemma 6.1 of[9]), which fact shows that this integral tends to zero asn→ ∞in view of the Riemann-Lebesgue lemma.

Ifd≥4, underE[_|Y|d−1]<_∞we can decompose∂_ld−1Dw(θ,l) =v(θ,l) +τ(θ,l)with_|v| ≤C r−4

nd−1I_n =id−1R

T′_\Bnh(θ)[v+τ](θ,l)e−

i x·θ+inl_{dθd l+o(1). The contribution of v is evaluated by} the Riemann-Lebesgue lemma and that ofτby the integration by parts, both resulting ino(1). The proof of (i) is carried out similarly. Ford =4 we apply integration by parts two times, which results in o(r−4)for the integrand, and, the further integration by parts being not allowed under the condition EY2 <_∞, we impose the logarithmic moment condition on Y to guarantee the inte-grability. In the cased=2 we also need to suppose the same moment condition ofY to guarantee the integrability of Dw (so thatI_n =o(1)), but the reason is slightly different: ifd =2 we cannot dispose of the integral about the origin in advance,Dw itself being possibly non-integrable.

We must ensure the uniformity of convergence with respect to functions hfor the integral that is disposed of by means of the Riemann-Lebesgue lemma. This is readily done by approximating the function ofl that results in from integration over|θ_|> ǫ by a smooth function for each positiveǫ, the other integral approaching zero uniformly asǫ_→0.

Acknowledgments. The author wishes to thank the anonymous referees for their valuable sug-gestions to the original manuscript, which lead the author to revise the paper in several significant ways.

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