NORMAL COhI"VERGENCESHOT NOISB AND RATES

Adv. Appl. Prob. 17,709-730 (19S5)Printed in N. Ireland

@ Applied Probability Trust 1985

OF MULTIDIMENSIONALOF THIS CON"VERGENCE

LOTHAR HEINRICH ANDVOLKER SCHMIDT, *Mining Academy of Freiberg

Abshact

Using a representation formula expressing the mixed cumulants of real-valued random variables by corresponding moments, sufficient conditions aregiven for the normal convergence of suitably standardized shot noise assumingthat the generating stationary point process is independently marked andBrillinger mixing and that its intensity tends to m. Furthermore, estimates forthe rate of this normal convergence are obtained by exploiting a generallemma on probabilities of large deviations and on the rate of normal con-vergence.

HIGH DENSITY; CUMUI-ANTS; MARKED POINT PROCESSES; MIXING CONDITION

L. Inhoduction

In this paper random fietds {u(r); t€Rd} of the form

( 1 . 1 ) u ( r ) : L f A - x x g i )

are considered, where {Ixu 9r]} is a stationary, independently marked pointprocess on Rd with some measurable mark space lK,%tl, and f :RdxK+R1some measurable real-valued function. Random fields of this form are calledmultidimensional (or generalized) shot noise processes (see [6], I29l). Here,

f(t-xuF) is interpreted as the effect at t€Rd caused by an event which ischaracterized by the random position xi at which this event occurs and by therandom mark p; giving additional information on the 'nature' of the eventconsidered, e.g. its magnitude. Thus, u(r) is the total effect observed at t, whichis caused by the sequence {[x,,0,]] of all events considered.

In connection with this, the main interest is concentrated on the limitingbehaviour of the standardized version

(r.2) (u (r) - Eo (r)Xvar u (r))-+

Received 12 October 1984.x Postal address: Bergakademie Freiberg, Sektion Mathematik, DDR-9200 Freiberg,

Bernhard-von-Cotta-Str. 2. GDR.

709

(1 3) [ , @)du* ' ( l ,@)a, ) ] [ " * l ,oau] -+

7ro LOTHAR HEINRICH AND VOLKER SCHMIDT

of o(r) when the intensity,l" of {x,} tends to m. In the literature, the normal

convergence of (1.2) is investigated in the case d: L and mostly under the

assumption that {x,} is a Poisson process [11], [t7], [18], [19] or a renewal

process [3] , [12] , [30] . Moreover, in [3] , [4] , [11] , [19] , [30] th is problem is

analysed only for special response functions f used for describing the number

of customers in queueing systems with infinitely many servers. Extending and

formalizing an idea sketched in [20], in Section 3 of our paper we give

sufficient conditions for the normal convergence of (1.2) assuming that {xr} is

mixing in the sense of Brillinger t5]. Moreover, in Section 4 we discuss

examples for which this mixing condition is satisfied. Referring to a result of

[19], in Section 5 we consider the integral |eu(u)du. In particular, we

investigate the normal convergence of the standardized version

of this integral when Ä lAl tends to @, where lAl denotes the Lebesgue

measure of the bounded Borel set A c Rd. In Section 6 estimates for the rates

of normal convergence of the random variables given by (1.2) and (L.3) are

obtained, and in Section 7 this is done for the special case that {xt} is Poisson.

Finally, in Section 8 we find conditions for the weak convergence in the

function space D([0, 1]d) of a suitably standardized version of the random field

{u(r)} to a Gaussian one.

In order to give rigorous proofs we use a representation formula (see [1a])expressing the mixed cumulants of real-valued random variables by corres-

ponding moments. This gives the possibility of extending a well-known formula

(see e.g. l23l,I24l) for the variance of (L.L) to higher-order cumulants (Lemma

1). Next, we use this formula for the higher-order cumulants of (1.L) given in

Lemma L, together with certain truncation techniques, to obtain sufficient

conditions for the normal convergence of (1.2) and (1-.3), respectively. Further-

more, for obtaining estimates for the rates of these normal convergences we

exploit a general lemma (see [21 ],l22l) on probabilities of large deviations and

on the rate of normal converqence.

2. Multidimensional shot noise

Let {[r,,p,]i be a simple stationary marked point process on Ro, de

{! ,2, . . . } , wi th some measurable mark space IK, l { l .As ment ioned in the

introduction, the x; cän be interpreted as random positions in Rd of event

occurrences, where the K-valued mark B, gives further information concerning

the event occurring at x,. For basic notions and results concerning stationary

marked point processes we refer to [7], [8], [13], [16].

Normal convergence of multidimensional shot noise

We assume with the exception of Example 3 in

7t t

0 < E [ca rd { i : 0 5 x ! )= l ; i : 1 ,

Section 4 that

, d ) l o < *

fo r every k : I ,2 , ' ' ' , x i : ( t5 t ' , ' ' ' , "5o) ) . Moreover , w i thout essent ia l res t r i c -

tion of the generality we assume that P(lim,,--card{i: lxrl 1u): *):1. Then,

for every measurable function f:RdxK->R1 we may formally define the

random f ie ld {r . : ( r ) ; f eRo} by (1.1).

We introduce the following notation:

l f ( t , F) ln B(dp)

andI

E u : L ,

, u f u ) d u f o r k : 1 , 2 , ' ' ' ,

where B denotes the Palm mark distribution of {[xt, ßt]]. From the Campbell

theorem for stationary marked point processes, it can easily be seen that Er < m

is sufficient for the convergence of the sum appearing in (1.1).

In what follows we assume that {[*u ßr]] is independently marked, i.e. we

assume the B, to be mutually independent and independent of the sequence

{x,}. Furthermore, we consider the following characteristics of the stationary

(non-marked) point process .P: {x,}; see [28]. By G we denote the generating

functional of V given by

Gl,nq: E ll 7((x,)

with 7(: Rd - [0, 1] a measurable function U"irrg equal to one outside of some

bounded subset of Rd. For a given bounded Borel set X - Rd, by tP(X) we

denote the number of points of V in X; V(X) : card {i: x, € X}.

By ar(Xr, ' ' ' , Xn) we denote the kth-order factorial moment of

v(Xr ) , . . . , v (&) ,

ar . (Xr , . . . , xn) : , , , , l iT. ro - : *c [ r+, t ruJx( . ) ] ,

where 1* denotes the indicator of the set Xr. Furthermore, by ?r(Xr, ' ' ' , Xu)

we denote the kth-order factorial cumulant of V(Xr), ' ' ' ,V(&);^ k r k l

^/r(Xr. ' ' ' ,&): - , .1.1T.,o *:-^ros clt*, I , u,1*,( ' )1.

Note that because of the stationarity of V, ]p cän be represented as follows:

( 2 . 1 ) ^ y t ( X r , . . . , X o ) : Ä [ T f " o ) ( & - u 1 . , ' ' ' , X u - u 1 ) d u 1 ,

where yf"o) is called the t,fr-ärd", reduced factorial cumulant of V; A :

EV( [0 , 1 ]d ) .With the exception of Section 4 we assume throughout this paper that tP is

Idr, fu):

)* t f {u, B))uB (dß), eo(u):t

7 L 2

Brillinger mixing [5],T[a"o'( . ) on (Rd)k-1 is

LOTHAR HEINRICH AND VOLKER SCHMIDT

i.e. that the total variation of the (signed) measuref i n i t e f o r k : 2 , 3 , . . . .

3. Normal convergence of high-density shot noise

In this section we consider shot noise processes generated by a fixedresponse function f, by a fixed mark distribution B and by a stationary pointprocess Vr : {xt} whose intensity Ä tends to @. We equip the correspondingrandom variable given by (1.1) with a subscript Ä. Furthermore we putu.r.:or(0) and, for brevity, the quantities ok, 1p and ?f"o) are mostly writtenwithout this subscript.

Under the additional condition that

(3.1) Ezl* ,

from the assumption that V^ is Brillinger mixing we get that the varianceVar o^ of u^ is finite. This can be seen from the formula Var u^ : Äofi, where(see, for example, t24l)

(3.2)

Now, assuming that 0 ( o^ ( oo, w€ seek for conditions ensuring that

I*. orrurd'Ju + w) duv|"o)(dr).

D(Ä) :.Yp l"(ffi.,)-a(a)l ̂**0,

o?: Ez* . | : ,

(3 .3)

where A(u):(2n)-iJi-exp (-t2lZ) dr is the standard normal distribution func-tion. In the special case when Vr. is a Poisson process and when the increase of^ is attained by compressing the scale (that is V^(X):V(ÄtrdX) for everybounded Borel set X-Rd, where V does not depend on Ä, and,EV([O, 1]o):L), this problem was attacked in [17] and [20] for d:t, respectively. In orderto prove (3.3) under weaker conditions we need the following lemma which isformulated in the case of an arbitrary (even non-stationary) generating pointprocess V^.

Lemma 1. Provided that it exists, the cumulant

cu'p {u^(r,), ' ' ' , ur(ru)}: i -:."Log

E "*o {, ,tru,u^(4)}lu1:.. :,k:o

can be represented in the form

curnp {u^(r . ) , . . . , ur( tk)}(3.4) k r .

: I I I n d s , 1 ( t t , - u i t k i e & ) y , ( d u r , - - . , d u r ) ,l : 1 ( t , k ) " ( R d ) ' i : 1

Normal conDergence of multidimensional shot noise 7 t3

where the sum

I : I( l , k ) K ' U " ' U K t : { 1 , " . ' k }

is taken over all partitions of the set {L, . . . , k} into I disjoint non-emptysubsets K,, lK,l denotes the number of elements of Kr and

d1K,1(r t , - u i )k i € Ki) : [ . [ ,

f ( tu,- u, , p)B(dp).

Proof. From [14] we have the formula

cumk {u^(rr), . . . , u,.(rr)} : , i , -1) '- ' ( ,- 1)!, ,L t*[ , o^(rr,) . . . EoF* r^,n,r.

Moreover, following the arguments used in [Zal for k :2, by means of the

Campbell theorem we find that

lKrl f , ,

E l l u^(ra) : I I I f l d ' * , ' ( r " " -u, )nne No)c, , (dur , . . . ,dw).k i € K i r i : 1 N r U . . . U N i : K i " ( R ' ) ' i p : 1

Thus, changing the order of summation and recording the summands we get

c u r r r p { u ^ ( t r ) , ' ' ' , u r ( r r ) }

k I lK i l

: I ( - 1 ) ' - ' ( t - 1 ) ! I I I I I' : t ,

r i

( l ' k ) i : 1 t : 1 N r u " ' U N " : K '

x I f l d,r, '(r no- ur; frp €No)c.,(dalr ' ' ' , du )J 1 p . 1 . , p : 1

: i I I t - 1 ) t - 1 ( t - 1 ) ! It:1 rlY'..:öi ':1 ':,"r,...1t''

f i l

x I f [ a ,n, ' { tno-ur inp€N")n as(duu. ;k ,eK") 'J ( R d ) i p : 1 r : L

Now, the well-known relationship (see [14])

yi(xr,' ' ' , xi) : ,i

(-1)'-'(, - 1)! .,Jr., ,ü

"Fg,p(Xk,i k, e K,): { r , . . . , i1 '

yields the desired formula (3.a).

Theorem 1. I f the condi t ion (3.1) is sat isf ied and i f for k: l ,z, . . . thereexist constants Cu not depending on tr such that for k:2,3, . . .

( 3 . 5 ) [ t r f " ü ( d u r , . . . , d u u ) l = c n < *Jlpa 1t t

71.4 LOTHAR HEINRICH AND VOLKER SCHMIDT

and

(3.6) crr. Z Cl ) 0,

then the convergence (3.3) holds.

Proof. First we replace f by the truncated response function

/t^'(r, F): f (u, F)1ro."vrr(lf(r, F)l)

for an arbitrary fixed e ) 0. With the abbreviations

uf)(r): I f(^)(r - xx ß) and u9): of)(0),

from (3.2) it follows immediately that

v", (55;{) : + | [ [ t,".,r,*, (lf (u,F)lxf(u, il\'n@il du\ V Ä o r / a ; L J R d J *

F r F f

+ | | | | 11,v1,-,( l f(ur, Fr) l)1r"r^.- i( l f(a, t ut,9)l) f (ur, ßr)f (u'* uz, ßz)Jpo Jeo JK JK

x B (dß r) B (dß r) t|"o' (d.u') O*']

/ f \ f t< o;2( r + | lyf"o'(au11 ) 1 | l ,.rt -,( lf(u, F)l)ff(a, PD,B@p) du.

\ Jpa / Jpa J6

Consequently, the conditions (3.1), (3.5) and (3.6) ensure that there exists a

sequence e: e(Ä)>0 tending to 0 as , \ . - ->oo such that Ä-t6r i tVar(r . r^-of))

converges to 0 as ,\, --+ m. Thus, it suffices to show that

(3.7) ,"p lP(ry.r)-ö(u)l--*-0.u ' | \ J Ä o ^ / ' l ^ +

For the sake of convenience we put fu{Y}:cumo{Y, " ' ,Y} for an arbitrary

random variable Y and

Ie[^) (u) :

. | . l f ' ^ ' ( u , P) l ' B(dP) , l : 1 ,2 , ' ' '

Then, Lemma 1. yields

trn{,S'}l=Ä i I t t efflq(uJli,fä],(u,*,,)l : 1 ( l , k ) J R d J l p a y t r

i : 2

x ly[ ' "o ' (duz, - . . , du)f duy

Now, taking into consideration Hölder's inequality and the inequality

s l ' " ' s f ' = 4 r s r * ' " * q r s r , w h e r e s ; , q i = 0 a n d Q t * " ' * q t : 1 , f r o m ( 3 . 5 )

Normal conuergence of multidimensional shot noise

(3.8)

fl [rf]'(u, * ur)1rtrrti : 2

x l y [ ' " o , ( d u 2 , . . . , d u t ) l d u ,

715

t Ä l= ̂ J", e[]'(u) ou

,)rJ,*,,, , lt','"o'(d,ur, ' ' ' , ar,)l ,I,

1

= sk/2(s(x))u-,E,(r.,!, t,L t)

Consequently, from (3.1) and (3.6) we see that

- f , l ' - r r$ ' lt.t= .ffJ-;*-o

we obtain the estimate

lru{rS)}l

= ̂ ,ä ,n, [, I.",,-, ['g'1u'1]'''''u

. r([" t

r,"-^.-rqf(u, p)l)ff(u, p))'B@il au[*^ly9"o,(d,r,ll, e,(w) dn)

implies

for k:3,4, . . . . Furthermore, f rom (3.2) and by using the Cauchy-Schwarz

inequality it is easy to see that

Ä-1 | Var u1-Var uf )l= t t 1r"vi,-r(l/(rz, F)l)ff( u, P))'B(dß) du

JRo JY

which

(3.e) Var u$ ' ,-------------;--

L.

Xai .\+

Thus, applying a well-known theorem of Fr6chet and Shohat (see [15], p. 187)

we obtain from (3.8) and (3.9) that (3.7) holds. This completes the proof.

4. Examples and remarks

Example L (Poisson process). Obviously, if V^ is Poisson for every Ä ) 0, the

condition (3.5) is satisfied because it suffices to note that in this case

T f ; " o ) ( X r , . . . , X u - r ) : 0 f o r k : 2 , 3 , . - . . F u r t h e r m o r e , f r o m ( 3 . 2 ) i t f o l l o w s

that (3.6) is satisfied if. Ez) 0, which is no essential restriction of the generality.

More general, assuming only that the increase of the intensity Ä of (a not

necessarily Poisson) V^ is attained by compressing the scale (i.e. tP^(X):

V(Äl/dx)), the condition (3.5) of Theorem f. is satisfied if the underlying point

process tP is Brillinger mixing in the usual sense.

Example 2 (cluster point process). If Vr is a cluster point process [16] with

7t6 LOTHAR HEINRICH AND VOLKER SCHMIDT

primary point process (parents) V(o) and secondary process (daughters) Vt"',the corresponding shot noise process given by (1.1) can in principle beconsidered as a shot noise process generated by the primary point process V(o)of parents. Namely, the mark of an atom x e V(o) is given by the correspondingindependently marked secondary process arising in x. On the other hand, inanalogy to relation (3.4), the factorial cumulant yp(Xr,. ..,Xu) of the clusterprocess V^ can be expressed by the formula

(4.1)I

ff off ls(X6 - ui; kie K;)yf")( d,tr,- . - , dur),i : 1

where af;) and y[o) denote the kth-order factorial moment of V(") and thekth-order factorial cumulant of V(o), respectively. Using this formula it ispossible to formulate a sufficient condition in terms of af;) and 1f;)'('"d) ensuringthat (3.5) is satisfied for Ve.

Theorem 2. lt there exist constants a, b, ^y > 0 such that for every ̂ , >0

II l y f . " t 'G"d) (du1, . - . , duu- r ) l= ou(k ! ; t * " , k>2,

J l P a ; r - r

and

o f ; ) ( R ' , . . . , R o ) = b k ( k ! ) 1 * " , k = _ � r ,

then (3.5) holds with

I / 1 r ^ \ L l k

cr,:li##(k!)'*".g I \ I \ /

In order to save place we remark only that the proof is an immediateconsequence of (2.L) and (4.1) and of the fact that Ä : Ä(o)c!"(R'), whereÄ(p)denotes the intensity of V@). The reader interested in more details isreferred to the proof of Theorem 5, which is analogous.

Note that Theorem 2 can be used in order to obtain estimates for the rate ofthe normal convergence (3.3), see Section 6.

Example 3 (doubly stochastic Poisson process). Let V^ be a stationarydoubly stochastic Poisson process on Rd with intensity Ä, i.e. there exists somestationary random measure A^ with .E^Ä(0, 1]'): Ä such that

P(V^(Xr) : lr,. .. , V^(Xo) : I): Efi Sp exp (-A^(4))i : l L j .

\ . ( X y " ' , X n )k f

I I It : t ( t , k ) J ( R d ) r

for any disjoint bounded Borel sets X,, , Xu c Rd and for any non-negative

Normal conDergence of multidimensional shot noise 7 1 7

i n tegers 1 . , , . . . , lu ;k=1 [9 ] , [161. Our goa l i s to show how the asympto t icnormality of the 'continuous' shot noise Jn, dr(u)A ̂ (du) generated by A^ de-termines the normal convergence of o^:Lf(-xrF), where {[x,,F,]] is thecorresponding independently marked doubly stochastic Poisson process. First,we remark that because of

(4.2) ^yz(Xt, Xr) :Cov (A^ (Xr) , ̂ ^ (&)) : Ä [ C^(Xz- u\ duJX,

where C^ is called the second-order reduced cumulant of the random measureA^, we have yt*o)(X): Cr(X). Moreover, if ^r is a counting measure (pointprocess), we have

(4.3) c^(x) : ?f;:?o)(x) + öo(X),

where "/N:y denotes the second-order reduced factorial cumulant of A^ and6o(X) :1,.(0). Let us assume that

I(4.4\ l imsup t,

la^(du)l<*

and, for convenience, that the limit 7(2:limx_*7(7, Z(7:Je, Je*" dr(u) d1(u + w) du C^(dw), exists with

( 4 . 5 ) 0 < 7 ( 2 < * .

Theorem 3. If the conditions (3.1), (4.4), (4.5) and

| / 't / f \ \ I(4'6) ':o l"(frA (t, ,,(a)A^ (du)-tr^).")-<D(u)l-t=-o

are satisfied, then (3.3) holds.

Proof. Using the same truncation technique as in the proof of Theorem 1 wehave only to prove (3.7).We have (see [20])

f is )E exp {-+ (rr t ' -Err l^ ' l' l'/ x7(x

^ )

:exp {^J^, [ ["* (#ä f'^'(u, p')- t-ffi,r '^'(u,or]n us au]( r f l - / i s \ 1" r "*p | | | | "*p (+f,^,(u, p))- 1-* f,^,(u, AtlnUpl\J*a J* L \VÄUr / "/x7

x (Ar @.u)- L du) äl_^ J r '^ ' (u, ß)B(dp)(Ar(du) - Ä du)].

7 1 8

By the inequality

(4.-t)

LOTHAR HEINRICH AND VOLKER SCHMIDT

lexp(;x)-1- " . t l=# , x€R1,

and our assumptions the first factor on the right side converges toexp {-s2E rl27(').

In order to accomplish the proof we show that the term

r^(s):ul".o{[, L [".0(ä f '^ '(u,e))- t-ff i , f '^ '(u,F)]

.r Ix B(di l (A^(du) - L du) j - t

Itends to 0 for every f ixed seR'. By the inequali t ies lexp(x-y+iz)-I l=f l x -V l+ I r l l exp (x ) , where x ,y=0 and zeR1, and L -cos x=x2 l2 we ge t tha t

T^(s)=' l [ , t [ , - .o,w]"raBlrn^(du)-Xdu)l"-o(#)

*El t t fr i"&B)-'f '^,Ju'B)1"roBxA^(du) -�xdu'l /s'r ' \I JR, Jr( t .,/)tz(x .,/)tz(x ,

tx^^(du) - )t du) | exp

\rrr^)

Further on. it can be shown that

u",([,t [ ,-.o, 'wl"r,oln^(du))=w[, '*(du)|E,

and by lsin x - r l= l* l '16,

,,", ( [" | ["" W W]',ru,n^(au))=':: i l)) [ r.^ @u)t8,.:

367(1 J*o

The last two estimates together with the choice of the null sequence e(Ä) and

conditions (a.a) and (4.5) imply T^(s) l;+

0. Hence, from (4.3) and the

convergence theorem for characteristic functions we arrive at

E exp [ä('f '-t'r)]-r*+"*o

t- t?.HIfor every fixed s e Rt. This immediately implies (3.7).

Note that, in distinction to Theorem 1, in Theorem 3 it is not assumed thatthe doubly stochastic Poisson process V^ is Brillinger mixing. However, we

Normal conuergence of multidimensional shot noise 71.9

remark that in extension of @.2) by using the representation of higher-order

moments of a doubly stochastic Poisson process (see t9]) it can be shown that

T " ( X r , ' ' ' , & ) : c u l n p { 4 " ( & ) , ' ' ' , A ^ ( & ) }

C r ( X r - u ) ' ' ' , X u - u ) d u

for any bounded Borel sets Xr, . . ' , Xt c Rd. Thus, V'^ is Brillinger mixing iff

I 1 * o ; n - , l C ^ ( d u 2 , " ' , d u u ) l < * f o r e v e r y k : 2 , 3 , " ' . O n t h e o t h e r h a n d , i f ^ Ä

is a counting measure, from (4.3) we get that 7(t^:

!o^d t ( r )du* . [p "Jn , d r (u )dJu+w)duy f : l 'U*1 . Thus , in th is case (4 .6 ) i s

anything but (3.3) for the shot noise generated by the (non-marked) A^ and by

the response function dr(u). Consequently, then sufficient conditions for the

validity of (a.6) are given in Theorem 1.

Example 4 (renewal process). Let V^ be a stationary recurrent point process

on R1 (then its restriction to the non-negative half-line is a stationary renewal

process) generated by the distribution F of the distance between two consecu-

tive points; Ä -1 : lfr uF(du). In this case, for pairwise different points

U L , " ' , U k w e h a v e

a y ( d u y ' ' ' , d r u ) : H ( d u - ( k ) - u n ( k - t l ) ' ' ' H ( d u - p 1 - u . , 1 1 - l ) t r d u n o ) ,

where Fr(X): Iä:rF *(X), Xc[0,oo),F * denotes the n-fold convolut ion ofF and the permutation zr reorders the integers l-, . . . , k in such a way thatU r (1 )

mixed cumulants (see t26l) we get

yu(dur

w h e r e u t l '

, d u u ) : (- r; '- ' I0,k)

Äü(K', , K,) S(duu; k i e Ki) ,

( uu and, for q, 1. . . 1q, ,

S(duo , , . . . , duo ) : [H(du^- u ,4 i , ) - Ä du^7) t duo,

Here, the integers .ltd(Kr, . . . , K,)20 depend only on the partition Kr,

and, i f Nr (Kr , . . . , K , ) )0 , then

L

: I,Xr

Tni : 1

k

II : 1

ni : 2

, K t

I r"ql {u, - u}= uu- t tr .o : 1

t ' i e r o

Thus, carrying out the reduction of Tr and taking into account the symmetryproperties of a stationary recurrent point process and its factorial moment

720

if

(4 .8 )

LOTHAR HEINRICH AND VOLKER SCHMIDT

measures it is easily seen that

t ( - f - f -

J*_ , l " r [ "o ' (d r t , " ' , duu ) l : 2u - ' ( k -1 ) ! Jo J , , " ' J ,__ , l t [ " " ' ( du r , " ' , dup ) l< rc

f -I u o l H ( d u ) - ) t d u l < a f o r P : 0 , 1 , ' ' ' , k - 2 .

J6

Exploiting some techniques developed in the papers of Statuleviöius (see e.g.

[25], l26D one can verify that under the condition that

(4.e)r -| .*p (6u) lH@u)- Ä dul<* for some ö > 0Js

we have the estimate

I

I lr ' ["o'(dur,- . . , duo)l= cok tJ P r - t

for k:2,3, . . . and for some constant C=0. Omit t ing the proof of th is resul t

we only remark that the following condition is sufficient for the validity of

@.9): F is 'spread-out', i .e. there exists some n>l such that F"* has a

non-trivial absolutely continuous component, and

f -| ""p (ötu)F(du) (- for some öt> 0 (see [27])'

J6

Note that the condition (3.5) in Theorem L is satisfied if (4.8) holds for

p:0, 1-, . . . and if the increase of Ä is attained by compressing the scale. In

[t2], the convergence (3.3) has been proved without the condition (4.8), but

under the assumption that the response function f is bounded and vanishes

outside of some bounded interval.

Under modified conditions and by using different techniques, asymptotic

normality including functional limit theorems was studied in [3] and [L].], see

also [4], for a special shot noise, namely, for the time-dependent number of

customers in queueing systems with infinitely many servers and exponentially

distributed service times. Moreover, in [30] the convergence (3.3) has been

obtained for the steadv-state number of customers in GIIMI*.

5. Normal convergence of integrated shot noise

In this section we again consider a general family of stationary point

processes V^ satisfying (3.5). Proceeding analogously to Section 3 we find

conditions ensuring that for a not necessarily fixed Ä and a family of bounded

Normal convergence of multidimensional shot noise 7 2l

Borel sets A c Rd the random variable

(5 .1) uA.^ : [ , ^ fu ) duJ4

is asymptotically normally distributed if the product Ä lAl increases to oo, where

lAl denotes the Irbesgue measure of A. First we note that uo.^ is of the form(1.1), too. Indeed, uA,Ä: Ir fo(-*r, ßr), where f o(u, F) : -fo f fu + w, ß) dw.

Thus, from (3.2) it follows that

'lü,T3Hfr'""i f (3.5) and

(s.z) t I f Vtu,ßt lau] 'n(dß)<*J 6 L J p o ' - J

are fulfilled. Furthermore, we assume that for some constant p > 0

(s.3) t1t,',1'^= o'I I A I

for sufficiently large Ä lAl.

Theorem 4. It (3.5), (5.2) and (5.3) are satisfied, then

(s.4) D(Ä, A): r"p lp(gglr'�^.1. r) -O(u)l --- --.----- O.u | \ V V a r O o . ^ / | r l A l +

Proof. Similar to the proof of Theorem 1 we replaee fe by the truncatedresponse function

fllor,(r, F): 1ro,"vrjÄu(lfo(u, p)l)f^ (u, g)

for an arbitrary fixed e ) 0 and we put

,}.f r,(r) : T fl'^l,(r - r,, g,), ,*.Fl) : ,*.f l,(o).

Because of

t l| | 1r.J^1o-1,-,(lfo@, F)l)ffo(u, P))'B(dß) du

J R d J 1 q

= lAl [ l,..u.",.-,( [ V@, ß)lr,)f I Vo, pll auf'n@B),J K \ J p a / L J p a J

i t follows from (3.5), (5.2) and (5.3) that there exists a sequence of e: e(Ä lAl)

722 LoTHAR HEINRICH AND VoLKER ScHMIDT

tending to 0 as ̂ lAl- * such that

(Var oo.J-t Var (oa^- otf D) -^LaF O.

Thus, as in the proof of Theorem L we can estimate the cumulants of thetruncated random variable ,1.f1) by using Lemma 1. Arguing as in Section 3we arrive at the estimate

which imp l ies f6 { (Var u , . . ^ ) - t / tu t , f l ' } ^ rÄr - - -+0 fo r k :3 ,4 , . . . . Wi thout

difficulty we can also show that Var ulFl)/Var ua..r. ^rAG;+ 1. Consequently,

in order to complete the proof it again suffices to apply the theorem of Fr6chetand Shohat.

Remark. By using Theorem 4, a result given in [19] concerning the normalconvergence of a special integrated shot noise process on the real line Rt canbe derived under weaker assumptions. Namely, let d: I and let V^ be a fixedstationary point process on R 1 which is Brillinger mixing. Then, for theresponse function f,

lro{ril.f r,}l= (r lAl)n,'("(Ä lA1;;,u-',,'J I f" ftr, pSl auf'n6p1

* (t * ,f,r,,L t)

f ( u , F ) : 1 1 o = , , = E , 1 i I : ( N ; € r , t z , ' ' ' ) ,

considered in [L9] we have

and, consequently, the condition (5.2) is satisfied if EN2(m and I3 u2p(du)<oo, where for each mark p the ti are i.i.d. non-negative random variables withthe distribution ,8, independent of the random integer N.

Note that for A : [0, T] and for fixed ,\ > 0 the condition (5.3) is satisfied iftor v : Jfr cum, {rr^ (0), u^ (u)} du we have

(s.s)

(s.6)

N

Ii : 1

t I J , r,,, B1t auf'nsp1 :T !* u'r{ao + -ErN(N- 1)r[l , F@üf

v ) 0

because limr-*T-1Varolo,rt..r,:2v. Furthermore, we remark that for f givenby (5.5), the random variable u^(f) can be interpreted as derived from a specialcluster point process (see Example 2 in Section 4). Moreover, if additionally V^

Normal conuergence of multidimensional shot noise 723

is Poisson, it can be easily shown (see [23]) that

t E N f - - _ . - _ f - f - . l, : x l ; l u ' r @ u ) + E I N ( N - 1 ) l | | ( 1 - F ( w ) X l - F ( u + w ) ) d w d u J .

t Z J 6 J g J g

Consequently, for (5.6) to be satisfied it suffices in this case that EN>O and

l; u'F (du)> 0. Thus, if VÄ is Poisson and if f is given by (5.5), for the normal

convergence (5.4) the finiteness of the third moments EN3 and Jä utF(du) as

assumed in [19] is not necessary.

6. Rates ol normal convergence

The starting point of this section is a general lemma on probabilities of large

deviations (see tzID and on the rate of normal convergence (see [22D.

Lemma 2 lztl, 1221. Let Y: Y(A) be a real-valued random variable

depending on a real parameter A>0 and sat isfy ing EY:0, VarY:1, and

( 6 . 1 ) l f n {Y } l =H*+# , k : 3 , 4 , . - . ,

for some real numbers 1>0 and H>-I . Then, in the interval OSu=LrlH

with A" :+(öL1611l(1+2r) the relations

P(Y=u) : (1- o(u)) exp (L,,(u))f 1+ Hf ,,(u)+.l,L A - J

6.2)

P(Y < -u):<D(-u) exp (L"(-rl l [r + Hfr(u, #]

and

(6.3) sup lP(Y <u)-o(u)l=1gg' a "

are valid.

Here, the function fr(a) and fzfu) are uniformly bounded in the interval

[0, a,,/H],

( * i f ? : 0 ,

L r ( u ) : I c k u k , w h e r e p : { ( 1 ' r

3<k<p I min lZ+: ,#(L.)r l i t y > 0,L l . 1 ' - '

r ' )

724 LOTHAR HEINRICH AND VOLKER SCHMIDT

and (see t10l)

.n :G+)k'r',-r)'-'(k .'r-t) .,- ;,:--,ü m,-

k , = 1

e.B. c: : äf3{Y}.In the case ? :0 the above lemma was derived in [25] (see also [26D. This

case is of special interest in applications because then the magnitude of theerror term in (6.2) and (6.3) is mostly the best possible.

First we apply Lemma 2 to the situation of Theorem 1.

Theorem 5. Let the stationary point processes V^ satisfy the conditions (3.5)

a n d ( 3 . 6 ) w i t h C o : a n ( k ! ) t * " f o r k : 2 , 3 , . . . , a n d l e t

(6.4) *r ]p"

ro(u) < bulk ! ; t*" and Eu€bk(k ! ; t*"

fo r k :1 ,2 , " 'and fo r some f in i te cons tan ts a ,b ,^y >0 no t depend ing on Ä.

Then, Lemma 2 holds for the random variable Y:(u^ _�Eu)lJio^ with

(6 .S) A : JÄ - -9 ' - and Fr : max f [ ( t + a )b f ' " ' � \(r+ a)b

an., ra : max tL c, I ' t J'

Proof. Starting from (3.4) we obtain

l " f u r :Eu^J l= 1 f tf ' u[ JÄ o^ J |

:rt .e-2\t2ck,?,,,ft.,

" [ [ , , * ,1(ur) f i , , * , { r . *u; ) ly t ' "o)(dur , . - - ,du) ldu,J R d J l p a ; t , j : U

=f f i l t . ,ä o , * . -4u , :uo ' (k , ! ' ' ' k , ! t ! ) " ]k , > 1

= (f,!):l l

f (1+_a)blk.

= S(k-2)t2 L C, I

'

Here, we have used our assumptions, the identities

r t : ( 1 - l ) s ' - 1 r , ; k t ;

k r + . . . + k r : k . r - r , ' , f t * r ' - I ! u r * . . L u 4 : t k r ! " ' k r !k i > 1 k i = 1

and the inequa l i t y k r ! . . - k r t l ! s (k r+ . . .+kr ) ! fo r a rb i t ra ry in tegers

I ,k r , - . . , k r> l .Thus , compar ing the above es t imate o f f r {u^ /JÄo^} w i th(6.1) we recognize the validity of (6.5).

Normal conuergence of multidimensionql shot noise

Next, we apply Lemma 2 to the standardized(s .1))

(6.6)

725

integrated shot noise (see

Y _JVar %"

Theorem 6. Let the conditions (5.3), (3.5) with Cu: ak(k!;r*" for k-

2 , 3 , - . . a n d

(6.7) F. : [ | f vtu, ß)l aulu n@p)=bk(k !;r*"- ( J r L J p '

' ' J

fo r k :L ,z , . . . be sa t is f ied fo r some abso lu te cons tan ts a ,b ,y=-0 . Then,

Lemma 2 holds for the random variable Y given by (6.6) with

(6.8) a : +ry and Fr: max {[(t + a)al', ,].

( l + a ) b l . L p J ' )

Proof . Taking into account the stationarity of {oo,^r,r}, for Y given by (6.6) we

have

Iu{ Y} : (Var t)o.^)-u't fJ4 J4

x cumk {oo,^(0), u e.x(tz- fr), . . ., oA.Ä(tu - rr)} dtu . . - dt, dtr.

Thus, making use of Lemma L we obtain the estimate

l f o {Y } t=+ f r f f t tVVar ,-^ ,?t ,rto, Jo J,*,,n-, Jlpayr-r JRo

x 1d16,1(- t t r , tk,- ut)k, e Kr\{1}) l l [ ld1a1(rr . , - u i ;k i e Ki) lj : 2

x ly[ ' "o ' (du2,. . . , du) l du, dtu . . . dtrdtr .

Now, proceeding analogously to the proof of Theorem 5 we obtain (6.8).

It should be mentioned that when ? :0, some results on large deviations for

integrated stochastic processes on the real line Rl without any special structurewere derived in [26]. The above results show how the general conditions mustbe specified in the case of a shot noise process. On the other hand we have

treated the more general situation of random fields and y > 0.

7. Rates of nomal convergence in the Poisson case

The aim of this section is to weaken the moment condition (6.4) in order toobtain a Berry-Esseen estimate of the discrepancy D(Ä) (see (3.3)) in the casethat V^ is a stationary Poisson process.

726 LOTHAR HEINRICH AND VOLKER SCHMIDT

Obviously, then we have (see e.g. [20], t23l)

( f t lE u ^ : Ä l I f ( r , ß ) B ( d ß ) d u , o l : 6 2 : l I f ' ( u , ß ) B ( d ß ) d u

J R , J 6 J R d J g

and, with the abbreviation f^(s) : E exp {is(r.r^ - Eu^ )lJ) o\,

r^(s) : exp {^[" [ [".0 (# rtu, ß,) - t -# ru, il]Bras a,]

Thus, as an immediate consequence of @.7) we get, for E. q o (see Section 2),

l - ^ , \ , s ' l - l r l ' E ,l lo* f^(s)+t1=i l#.

Since l"' - ll=lzl er'r we may continue with the estimate

l /^(s)-exp eJrz) l=#?".0 ( -1O- ' ) )

i f lsl< 3al) o'lEr: L for some a e (0, 1). Together with Esseen's fundamentalinequality (see e.g. [15], p.297) this gives

D(Ä) =? [ ' l f^(s)-exP (-s ' lz) l

, , * - 4-T T J o I s I r r " l Z n L

- E , | . g _ 1 I: JAo 'Lon 3 (1 - a f J '

Hence, by taking in the last expression the infimum over 0 < a ( 1 we get thefollowing result.

Theorem 7 . I t 0<L : L( f )4m, then

( 7 . 1 ) D ( I ) =2 .2 r+. V.l. o'

We remark that in [18] the inequality (7.L) is given with the larger constant(4BJzr.

Analogous considerations lead to a corresponding result for integrated shotnoise generated by a stationary Poisson process V^.

Theorem 8. If (5.3) is satisfied and if 0< Fzlm, then

(7.2) D(Ä, A) = 2 .2I - :+-' - - J , t l A l p ' '

The estimate (7.2) establishes a rate of the normal convergence which has beenproved in [19], see also Section 5.

Normal conoergence of multidimensional shot noise

8. Asymptotic Gaussianity of {t^(r), teTo:[0, 1]d]

727

In this section sufficient conditions will be given to ensure that the randomfield {Z^(r), r e Ta} with

(8 .1 ) Z^(t): # (u^ (r) - Er,,^ (r))

having the covariance function K^(r) : EZ^(0)Z^(I) weakly converges (as

Ä+*) to a stationary D(Td)-valued Gaussian field {Z(t), re Ta} with mean 0,and the covariance function

(8.2) K(r): EZ(Q)Z(1): l im K^(r),.1,".-"-D

provided the latter limit exists for all t e To.For the general theory of stochastic processes whose sample functions belong

to D(T) and for the weak convergence of multiparameter D(Td)-valuedstochastic processes we refer to [2] for d: l" and [1] for d=2. To describe thehypotheses under which we shall derive the result of this section we make useof the following notation and terminology.

A block B in To is a subset of Td of the form (s, t]: Xf:, (s(t), t(t)] withr : (s( t ) , . . , s(al ; and t : ( t ( t ) , . . , t ( ' ) ) ; the pth face of B: (s, t ] isXi=rr(s(i), 1ti)1. Disjoint blocks B and C are neighbours if they have the same

p t h f a c e f o r s o m e p € { 1 , . . . , d } .For each block B:(s, r] and an arbitrary function g:To+Rl, let

g (B) : I ( -1 ;o - " ' - " ' -ea*1r ( r )+ e r ( f ( t ) -s ( t ) ) , . . , s (a ) + ea( t@- s (o) ) )e: e{O,1}i : 1 , " ' d

be the increment of g around B. Further, let 9obe the collection of subsets of

Ta of. the form S: Xf:r $, where each $ contains 0 and l" and has countablecomplement. To formulate our convergence theorem we next state the follow-ing slightly changed and simpified variant of the main result in [1.], cf. Theorem2 and 3. To that end let X". n2I, and X be elements of D(Ta).

Lemma 3 [1]. Suppose that the following conditions are satisfied:

( 8 . 3 ) ( X " ( s r ) , . . . , X , " ( s o ) ) c o n v e r g e s w e a k l y t o ( X ( s r ) , . . . , X ( s o ) ) a s n - + afor all finite subsets {sr, . . . , so} of some member S of {a,

( 9 . 4 ) t i m p ( l X ( t , t , , . . . , t ( r ) , . . , t ( a r ; - X ( r t r l , . . - , ! , . . . , t ( o ) ) l . e ) : gt ( e ) 1 1

for every e )0 and al l p €{1, ' ' ' , d} ; there exists an integer n6 such that

728 LOTHAR HEINRICH AND VOLKER SCHMIDT

(8.s) E IX,'(B)I" lx"(c)|" : (p(B))'(p(C))'

for every pair (B,C) of neighbouring blocks in Ta and k2ho, where

T>0, a>) and p is a finite non-negative measure on Q.

Then X, converges weakly to X as n-+Q.In comparison with Theorem 3 in t1] we have omitted the additional

assumption on the continuity of the marginals of p and have used the fact

that the almost sure continuity of X at the upper boundary of Td is alreadygiven by (8.3)-(8.5) (see [2], Theorem 15.4 and 1,5.7 for d: l).

Theorem 8. Let the shot noise process {u,,.(t), te Ta} in (8.1) satisfy the

conditions of Theorem L and let the limit (8.2) exist. Further, it is assumed for

some real number Äo>0 that

(8.6)

for all Ä

(8.7)

and

(8 .8 )

K^(h) I a2x and K(h) + K(0)

? Äo, as h tends to 0 from above,

t f

J*, )*f ' {a + u, g)f ' (c + u, i lB(di l du= (p(B)p(c)) '

t f| | f '@ + u, p)B(dil du = (p(B))'

J R ' J y

for every pair (I|, C) of neighbouring blocks in Ta, where a>) and p is a finitenon-negative measure on Ta. Then the random fields Z^, ,1, 2 Äo, and theGaussian field Z arc D(T)-valued and Z^ converges weakly to Z as Ä+m.

Proof. This proof consists in verifying the conditions of Lemma 3, where X^

and X are replaced by Z^ and Z, respectively. In view of (8.6) and Öebyshev's

inequality it is seen that (8.4) holds and together with (8.7), (8.8) and Theorem4 in [L] it turns out that the sample paths of Z^, Ä 2 Äo, and Z belong to D(?i)with probability L. The convergence of the finite-dimensional distribution of. Z^can be easily proved by the Cram6r-Wold method (see [2]). To this end it isenough to show that

converges weakly to a normally distributed random variable with mean 0 and thevariance lr='j=o w,qK(s, - s,) for all real p-tuples (wr, . . ., wo) f (0,. . ., 0)and all finite subsets {sr, . . . , so} of some member S of. {a. In fact, applying

(8.e) fr*,2^1r,):;|fr[i:::';;*")(x,,F,),where

(8.10) F[ir:.'";*") (x, ß): ,ä ,,[ftr, - x, p)- Ä J , I* orru, ouf ,

Normal conergence of multidimensional shot noise 729

Theorem L to the special response function (8.10) we obtain this desired resultand so (8.3) holds.

Last but not least we have to prove the tightness condition (8.5) for y:2and c > j. For any two random variables Yr, Y, with EY?(m, i :1,,2, wehave

E(Yr- EY)2(Y2- EYr)': curr4 {Yr, yr, yr, yjJ'+ f2{y1}f ,{yr}(8.11) * 2(cum2 {Yr, Yr})'.

Using the latter identity for Yr: Z^(B) and Y2: Zx(C), formula (3.4) and thecondition (8.7) and (8.8) we find after a lengthy computation involving liberaluse of Hölder's inequality an estimate similar to (8.5), where only the right sideof (8.5) is enlarged by some multiplicative constant greater than 1. Thiscompletes the proof of Theorem 8.

Remark. If in Theorem 8 the condition (8.8) holds for some a) | and ppossesses continuous marginals then the condition (8.6) and (8.7) can beomitted and the Gaussian field Z has continuous sample paths with probabilityL. This can be derived by a multidimensional extension of the correspondingconvergence criteria for the case d: | (see [2], Theorem I2.3 and 15.5).

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