Limit theorems for conditional distributions with regard ...alzaig/limit.pdf · LIMIT THEOREMS FOR CONDITIONAL DISTRIBUTIONS WITH REGARD FOR LARGE DEVIATIONS 1189 In Sec. 2, we introduce

Ukrainian Mathematical Journal, Vol. 51, No. 8, 1999

L I M I T T H E O R E M S F O R C O N D I T I O N A L D I S T R I B U T I O N S

W I T H R E G A R D F O R L A R G E D E V I A T I O N S

A. Yu. Zaigraev UDC 519.2

Possible limit laws are studied for the multivariate conditional distribution of a subset of components of the sum of independent identically distributed random vectors under the condition that other components belong to the domain of large deviations. It is assumed that the considered distribution is absolutely continuous and belongs to the domain of attraction of the normal law but possesses "heavy tails." The approach suggested is based on the local theorem for large deviations.

Introduction

Let {, ~ l , ~2 . . . . be independent identically distributed random vectors with values in Ra.. Assume that E ~ =

0, E I ~ 12 < oo, and denote by B a covariance matrix of ~. We form the sums S,, = { l +--- + {n, n = 1,2 . . . . . Let

A c R d be a set of continuity of the Lebesgue measure. It follows from the central limit theorem [1, p. 78] that, for n ---) e~,

A

where ~PB is the density of the multidimensional normal distribution with zero mean value and the covariance

matrix B.

Assume that A depends on n and r n = n - 1 / 2 inf Ix[--~ oo as n ~ oo. In this case, the probability P (Sn~ xEA

A) is called the probability of large deviation of the sum S,, and the value r,~ is called the order of large deviation.

Analogously, the value x = x,, taken by the sum S n is called the large deviation if 17-1/21 x] ~ oo. It follows from

(I) that the probability of large deviation tends to zero. In the one-dimensional case, the asymptotics of the probability of large deviation is studied in detail. Among

the papers making a significant contribution to the part of the theory concerning the establishment of the so-called

exact asymptotic of P (S,~ e A), it is necessary to mention [2-8].

In the multidimensional case, the problem is more complicated and admits numerous different statements,

which is explained both by the variety of types of sets A worth considering (in the one-dimensional case, these are only finite or infinite intervals) and by the difference of the possible behavior of "tails" of the distribution. In this situation, in the author's opinion, the approach based on the proof of the local limit theorem is a unique real ap- proach to the investigation of limit laws with regard for large deviations.

The present paper is devoted to the study of possible limit laws for distribution of a subset of components of the sum of independent identically distributed random vectors under the condition that the other components lie in the domain of large deviations. We restrict ourselves to the case where the initial distribution is absolutely continuous and belongs to the domain of attraction of the normal law, but has more "heavy .... tails" than are admissible by the

so-called Cramer condition. In addition, assume that r,, > In n. Then, as in the one-dimensional case (see, e.g., [8]),

the large deviation of the sum is formed at the expense of one anomalously large term whereas the contribution of the other terms is negligibly small.

Institute of Mathematics, Ukrainian Academy of Sciences, Kiev; Kopemik Institute, Torun, Poland. Translated from Ukrainskii Matema-

ticheskii Zhurnal, Vol. 51, No. 8, pp. 1054-1064, August, 1999. Original article submitted January 29, 1997; revision submitted February 19, 1998.

1188 0041-5995/99/5108-1 188 $22.00 �9 2000 Kluwer Academic/Plenum Publishers

LIMIT THEOREMS FOR CONDITIONAL DISTRIBUTIONS WITH REGARD FOR LARGE DEVIATIONS 1189

In Sec. 2, we introduce the class of regularly varying densities, which is the subject of our investigation, and present the local limit theorem tbr large deviations.

In Sec. 3, the following problem is considered: Let f : Rd---~ R k be a nonlinear function such that E f ( ~ ) = O,

El f(~)]2 < o o . The subject of the investigation is the asymptotic of the distribution of the sum T,, = f (~ l ) + --- +

f(~n) under the condition that the sum S, has a large deviation. A particular example of a problem of such type is

the problem of investigating the limit distribution of the empirical coefficient of asymmetry lbr the large deviation of the empirical coefficient of excess, which was studied in the one-dimensional case (see [9]). The first fairly general investigation of such problems was carried out in [ 10].

In applications, it is often necessary to know the limit behavior of the projection of the sum of independent identically distributed random vectors onto a certain subspace under the condition that its projection onto another subspace lies in the domain of large deviations. A problem of this type for the situation where the mentioned subspaces are determined by a fixed system of linearly independent vectors is considered in the last section. More

exactly, we study the asymptotic of the distribution of the vector ( ( S n, v l ) . . . . . ( S n, v s ) ) under the condition that

thevector ( ( S , , , u l ) . . . . . (Sn, uk) ) l iesinthedomain of large deviations if {ui, k s l)jJi=lj=l are linearly independent

vectors in the space R d, k + s < d. The result obtained actually generalizes Corollary 3 in [ 11] given without proof.

2. Regularly Varying Densities and the Local Limit Theorem for Large Deviations

We introduce the following class of absolutely continuous distributions with "heavy tails": Let nonnegative

functions r and h be defined on [0, ~ ) and the unit sphere R d-l, respectively.

Def in i t ion 1. We say that a func t ion p ~ P be longs to PTh i f it is u n i f o r m l y bounded in R d a n d

representable in the f o r m

p ( x ) = r(lxl)(h(e x) + w(Ixl)), e x = Ixl-'x, x ~ R d

w h e r e r ( t ) regularly varies as t ~ oo with the e x p o n e n t - 7, 7 > d, h ~ 0 is a cont inuous func t ion , and

w ( t ) - ~ O as t---~oo.

In what follows, we consider distributions with density p e P. As can easily be noted, in this case, the events

([ ~[ > t) and (eg cA) are asymptotically independent as t --~ oo.

The simplest example of the density of distribution of the class P is

p(x) = { C(Ixll +0 otherwise. +lXdlCr I x l > x ~ (2)

Here, r ( t ) = t -7, y= o ~ , h (e )= C(Ix, + ... + "[xdlcr -~ , and w( t )=0 for t > x o .

The following theorem proved in [11] is a basic assertion in studies of the asymptotic of conditional distributions.

Let Pn denote the density for Sn= ~l + ~2 + "'" + ~,~ and E be a support of the function h.

Theorem 1. I f p ~ P, 7 > d + 2, then, f o r any f i x e d e > 0 and n -+ o%

p , , (x ) = n p ( x ) ( 1 +o(1 ) )

1190 A. YU. ZAIGRAEV

uniformly with respect to x, I x [ > n I/2 In n, ex~ Ea = { x ~ R d- 1 : h (e) >_ ~ }, and

p . ( x ) = o ( n r ( I x l ) )

uniformly with respect to x, ix] > _ nll21nn, ex~ E.

3. Limit Conditional Distributions for Sums of Functionally Dependent Components of the Vector

Assume that f : Rd---> R k, k > d, is a linear function and there exists a (k • d)-matrix F x of lull rank, at least,

for all sufficiently large x such that

sup IFxul-'lf(x)- f ( x - . ) - Fxul : o ( 1 ) as I x l - - > = . (3) lul~,lul=o(Ixl)

Property (3) is true, e.g., in the case where f is differentiable with respect to every variable for all sufficiently

we take a matrix whose elements are the partial derivatives of the components of the large x. In this case, as F~

function f . For example,

2x i 0

0 2x 2

X2 X I

(a) F x =

(b) F x = (2x . . . . . ( k + l ) x k ) T

for f(xi, x2)= (x2, x~,xix2)r;

for f ( x ) = ( x 2 . . . . . xk+l) T.

Let Bf be the covariance matrix of the vector .f(~) and [Ill be the Frobenius norm in the set of all (k x d)-

matrices, i.e.,

k d ][FH2 = 2 Z F/2"

i=1 j= l

Denote by G x the covariance matrix of the vector f ( ~ ) - F ~ , i.e., G x = B f - F x BS - B o F f + F x a F f , where

B 0 = Ef(~) ~r. Further we assume that n -4 ~ and A is a set of continuity of Lebesgue measure. If A c R d and F is a (k •

d)-matrix, then by the set F A c R k, we mean { F a , a c A }.

Theorem 2. Assume that p ~ P, y > d + 2, and f is a function such that E f ( ~ ) = 0, E l f @ l 2 < ,,~. For

f ixed a > 0, uniformly with respect to x, Ix I-> n ' / 2 In n, exe Ee, the following statements are sati,~fied:

(i) 4f II Fx II ~ 0 as I x I -4 ~ , then

P(Tp - f(x) ~ ni/2A I Sn= x) = I ~Bm(u)du(1+~ A

for A c Rk;

LIMIT THEOREMS FOR CONDITIONAL DISTRIBUTIONS WITH REGARD FOR LARGE DEVIATIONS

(ii) i f IIF~II • 1 as Ix l - -~r162 then, for A c R k, w e h a v e

P ( T n - f ( x ) ~ nl/2al Sn=X ) = f ~pG x (u) du(1 +o(1)) A

and, moreover, in the case IIFx - gl[ --) O, f o r a certain

F B ff - B o FT+ F B F T instead o f G x in the limit law;

(iii) i f ][ F x [I --> ~ as Ix ] --> ~ , then

P ( T n - f ( x ) ~ nl/2FxAI Sn= x) =

f o r A c R k.

( k x d ) -matr ix

f q)B(U) du(1 + o(1)) A

Proof . Let P = P ( T n - f ( x ) ~ nl/2AI S n = x) . Obviously,

P = (pn(x)) -1 limlA1-1 P(Tn - f ( x ) ~ nl/2A, x -< S n -< x + A), I~Xl--*0

where -< ( 5 ) means the coordinatewise ordering. Consider the events

a 0 = {<gj, ex><clx I, j = 1 . . . . . n},

A1 = ex><clxl, j= l . . . . . <~,,ex>>_clxl},

A 2 = LJ {<~k, ex>>-clxl, <~j, ex>>-clxl}, k,j=l,k;ej

where 0 < c < ( ~ / - 1 ) - l ( ~ / - d - 2 ) is fixed. It is clear that the representation

p n ( x ) = PnO(X) + P n l ( X ) + Pn2(X)

holds, where

p . j ( x ) -- lim A - 1 P ( x - < S. -< x + A, Aj ) Vx ~o - _

and (see the proof of Theorem 1 in [11])

p,,o(X) = o ( n p ( x ) ) , phi (x) = p ( x ) ( l + o ( 1 ) ) , p,,2(x) = o ( n p ( x ) ) .

It follows from (4)-(6) that

P - (pn(x)) -1 l imIA1-1P(T n - f ( x ) ~ na/2A, x -< S n -< x + A, al) . I~F~o

1191

F, we have G = B f -

(4)

(5)

(6)

(7)

1192 A. YU. ZAIGRAEV

Let ~, ~,, ~2 . . . . be independent identically distributed random vectors with values in { u ~ Re: (u, ex) <

We prove that

IEf (~) I = o(n -v2),

for Ix I > nl/21nn. Indeed, since

I f (u )p (u)du = -

~(u) = p(u) P((~,ex) < clxl)"

IEf i (~) f j (~) - Ef~(~)fj(~)l = o ( i ) ,

I f (u) p(u) du,

Moreover, since IE,~(~)II.6(~)I < o~, we get

] e f i ( ~ ) f j ( ~ ) - E.f~(~).fj(~)l - ( (P( (~ ,ex )<ClXl ) ) - ' - I )

+

(.,ex) < ,'lxl (u,,'x) -> ,'lxl

lbr sufficiently large n, in view of IEf(~)II~I < ~ , we have

I f f ( ~ ) ' = O( I I . f ( u ) l p (u )du )= O(Ixl-' [I.f(u)llulp(u)du) I.l>-clxl [ul->clxl

~lf,(u)llfj(u)lp(u)du (,,e~)<,q~l

~l.f~(u)llfj(u)lp(u)du = o(1) . (.,ex)>-clxl

Relations (8) are proved. The validity of (8) implies that (1) holds for the sum

n = 1, 2 . . . . . Similar reasoning yields that (1) also holds for the sum Note further that

limlAl-' P(T,, - f(x) ~ nl/2A, x -< S n -.< x + A, AI) IZxl~0

= (P(<~,ex)<C[X]))"-' IP(T,,_, + f ( x - u ) - f ( x ) (.,,.x)<_(,-,)lxl

By virtue of the Chebyshev inequality lor Ix] _> n 1/2 In n, we have

whence

Sn = ~1 + . . - + ~,,, n = 1,2 . . . . .

n'/ZA I Sn_,=u)~,,_,(u)p(x-u)du.

p((~,ex) > clx[ ) = O(ix 1-2) = o(n - I ) ,

P( (~ ,ex) < clxl) ~ = 1 + o 0 ) . (lO)

i . j = 1, 2 . . . . . k. (8)

i = 1 ,2 . . . . ,k,

= o(n-1/2).

.f(~l) + -'- + f(~, , ) ,

(9)

c [x ] } and density

LIMIT THEOREMS FOR CONDITIONAL DISTRIBUTIONS WITH REGARD FOR LARGE DEVIATIONS 1193

Let X > 0 be an arbitrarily large fixed number. Relations (7), (9), and (10) yield

P - (p(x))- ' IP(T,_,+f(x--u)--f(x)~nI/2AI Sn_,=bl)[3n_l(U)p(x-lA)du (.,~.)<o-,:)lxl

- .fP(~._~ + f(x-u)-f(x)~nl/ZAI ~,,_,=u)~._l(u)d.+Ow(X), (11)

l u [<nmX

where 101 < ~ because, for p s if', wehave p(x-u)-p(x)for I . I - - o ( I x l ) and p(:~-.)=O<p(x)) for I"1 = o ( I x l ) 0 Ixl ~ oo uniformly withrespect to e x ~ S e.

Condition (3) means that

f ( x ) - f ( x - u) = Fxu + v,

for l u l l s , lul=o(l~l). Consider three cases of the behavior of F x

I~l=o(IFx.I) (12)

as [ x [ ~ oo.

1. Let II Fxl l -~ 0 I x I >- n'/2 In n, and u, n 1/2 ~ < ] u I< n ~/2 X, where 8 > 0 is an arbitrarily small fixed number.

For arbitrarily small fixed ~ > 0, we define the sets

as ]x I ~ ,~. Then, in (12), [ f ( x ) - f ( x - u)[ = o(n ~/2) uniformly with respect to x,

A~ = { x ~ R k" V y e A J x - y l > e } , A + = { x ~ R k" 3 y e A I x - y l < e } .

Obviously, for all arbitrarily large n ,

{~,_1 ~ nl/2A~}

whence

nl/28<]u]<nl/2 x

(13)

nl/2A-~ I Sn-I = U)1)n_l(U)du

t" < -] etTn_l "I- f(x- I~)- f(x) nl/28<M<nl/2 X

1/2--+ I e ( ~ , - I E Y/ Ns ] Sn-, =/A)bt/-l(t/)dt/" (15)

1~1/2 8,<1tt1<111/2 X

I P(f.-, nl/25<lul<nl/2x

,1~2A t ~,,-~ =") i,._l(,)e,

nl/2A~ ] Sn-1 = u)Pn_l(tt)dH = e(Tn-1 ~ nl/2A~)+Olw(X)+O2w(8-1)

= I q)B: (u) du + 01 w(X) + 02 W(~- 1 ) + 03 W(E- 1) (16) A

It is easy to understand that

1/2A+] (14)

1194

and

1/2--+

nal2g)<lu I <ni/2x

where I 0i1 < 1.

A. Yu. ZAIGRAEV

Sn-l=U)[Jn- l(U) du = P(Tn-1 ~n l /2A+)+O4w(X)+Osw(8-1)

= ~ ~8:(u)du + 04w(g ) + 05W(r -1) + 06W(E-1), (17) A

By virtue of the arbitrariness of the choice of e, ~5, and X, it lbllows from (15)-(17) that

S P(7"._, + f ( x - u ) - . r ( x ) ~ nl/2AID,,_i=u)D,,_l(u)du - j'q~8:(u)du lul<nll2 x A

uniformly with respect to x, of the theorem.

[ x l > nl /21nn, e x 6 E e ,

(18)

whence, taking (11) into account, we get the first statement

Introducing again the sets we have

2. Let IIFxH ~ 1 as Ixl . Then, in (12), I,,1=o(, '/2) uniformly with respect to x, Ixl>__ nil21nn,

and u, n t / 2 8 < ]u l< n i l 2 x , where fi > 0 is an arbitrarily small fixed number. Introducing sets A + and A~

defined by relations (13), instead of (14), we obtain the following relation for sufficiently large n:

,,2A } (19)

By reasoning analogous to (15) -(17) , instead of (18), we get

lu[<al/2 X A

uniformly with respect to x, ix I > n I/2 In 17, ex~ E E, whence, in view of (11), we obtain the second statement of

the theorem. It is easy to note that if, for a certain (k • d)-matr ix F, the limit liFx - FII ---) 0 holds, then I (F -

Fx) u I = o ( 171/2 ) and, consequently, we have F instead of F x in (19), which implies G instead of G x in (20).

3. Let I I F ~ . I I ~ as I x l ~ o o . Then, in(12), I,:l=o(n'/211511)uniformlywithrespectto x, I x l >

n 1/2 In n, and u, n 1/2 8 < [ u I< n I/2 X, where 8> 0 is an arbitrarily small fixed number. In this case, we consider

FxA instead of A. In view of (12), we have

P(~i-! + f(x-u)- f(x) ~ ni/2FxA l S,,-I =u) Pn-l(u)du ni/2~)<Jul<ni/2 X

= ~ P ( n - ' / z ~ _ , - n - ' / Z v ~ F x ( n - i / Z u + a ) [ S ,_ ,=u)~ , ,_ l (u)du . (21) nl/2~<lul<nl/2 X

A + and A~ defined by equalities (13), we note that, in this case, instead of (14) or (19),

LIMIT THEOREMS FOR CONDITIONAL DISTRIBUTIONS WITH REGARD FOR LARGE DEVIATIONS

for all sufficiently large n. Using (22), by reasoning analogous to (15)-(17), we derive from (21)

S P(]F'n-i+f(x-u)-f(x) ~ nl/2Fx A [ ~Sn-l=u) Pn-l(U) du nl128<lul<nl12 x

- I p(.-,/2~o_, ~ ~ ( _1/2 + A) i ~._, : , , )~, ,_,( . )d, , nl12 8<lul<nl12 X

= P(n- i /2Tn- i ~ Fx( n-l/2 S'n-I + a))+Olw(X)+O2w(5-1) �9

where

and

Finally,

We represent

e(n-l'2~. 1~ F~(.-'/2L_I+AD : I, +12+13,

I 1 =

12 =

P(n-l/2Tn_ 1 ~ Fx(n-1/2Sn_ 1 + A), n -1/2 Sn-1 ~ (-A)-c),

z - 1/2 Z, ~ P(n-1/2 ~,_, ~ Fxtn J,,_i + a), n-i/2 Sn_l ~ (-A)+ ),

13 = e( l / - l /2T#z_ ' E Fx(F/-1,2 an_ 1 -'1- A), n-'/2Sn_ 1 ~ (-A)-E, n-1/2Sn_ 1 ~ (-A)+).

We study separately the asymptotic of every term in (24). In view of (1), we get

I 1 = p(n-l/27"n_l~Fx(n-l/2~,_1+a) ln-1/2S,,_l~(-m)-~)p(n-1/2~Sn_l~(-m)-~)

- P (n -1 /2L- l~ ( -a ) - e ) - fq)B(U)du+Ow(a-1), 101<l A

12 =

<_

P(n-1/2Tn_ 1 ~ Fx(n-1/2Sn_ 1 + A) l n-l/2~71z_l f~(-A)+ ) P(n-l/2~Tn_l ~(-A) +)

-1/2- A) In-1/2S,,_I~(-A) +) = 0(1). P(n-1/2Tn_ 1 ~ Fx(rl Sn_l +

13 = P ( n - l / 2 ~ z _ , E F x ( n - l / 2 L _ l + A) I n - l / 2 L _ 1 ~ ( - A ) ; , n-1/2Sn_ ' ~ ( - A ) + )

X P(n-I/2~S,,_I ~(-A)~, n-l/2L_ , ~ ( -A) +)

< p(n-l/2Sn_l~(-A)-~, n-1/2Sn_l~(-A)+~)= 0w(e- l ) , 101<1.

Relations (25)-(27) hold uniformly with respect to x, Ix I__> n 1/2 In n, ex~ ES.

1195

(23)

(24)

(25)

(26)

(27)

1196 A. Yu. ZAIGRAEV

Combining (11) (with regard for FxA instead of A) and (23)-(27), we obtain the third statement of the

theorem by virtue of the arbitrariness of the choice of X, e, and & Theorem 2 is proved.

Consider examples of the application of Theorem 2. Assume everywhere that the distribution of a random

variable ~ with values in R d is absolutely continuous with density p �9 P, ~'> d + 2. Suppose that E~= 0 and B

is a covariance matrix of ~. The given statements hold uniformly with respect to x , Ix [ > n 1/2 In n, and, in the

multidimensional case, also with respect to e x �9 E E for fixed e > 0.

Conditional Distribution o f Sample Mixed Moments under the Condition for the Sample Mean. Let

I I I I

= Z i, -- n el l 4 < - .

i=1 i=1

Consider first the case d = 2. We take f ( z l , z2) = (Z 2, Z 2, ZIZ2) T.

The limit distribution of sample mixed moments /~ll, /)22, a n d b12 under the condition n~ = x, where x

lies in the domain of large deviations, somewhat differs depending on the location of the components x 1 and x 2.

For example, we present two assertions. Let q = (/~11 - b l l , b22 - b22, /~12 - b 12) T. If both components of the

vector x lie in the domain of large deviations, then, for A c R 2, we have

p(nTl_(x2 2 T n~=x) = fq)B(U)du(l+ (l)), X 2 , x l x 2 ) �9 n l /2FxA [ o A

where

2x I 0 ]

F x = 0 2x 2 .

x2 xl )

If Ixll _ n j / 2 Inn, i.e., the first component of the vector x lies in the domain of large deviations and Ix21 - o(171/2), then

P ( n ~ l - ( x 2 , 0 , O) T �9 n'/2FxA [ n - ~ : x ) = ~q)B(U)du(l +o( l ) ) A

with the same matrix F x.

Consider now the general case. Let f j ( z ) = 2 z j f i j ( z ) = z i z j , z � 9 d , j = l , 2 . . . . . d , i--- 1,2 . . . . . . j - l , and ^ . ,

q = ( b l l - b 11 . . . . . bdd - b dd, /~12 -- b 12 . . . . . blcl - t3 ld, b23 - b 23 . . . . . b(d_l) d - b(,~_l) d) ~. Assume that all

components of the vector x lie in the domain of large deviations. For A c R ~, we have

. . . . .. x ) = j" , , (u) du (1 + o / l ) ) , ,Xd , X l X 2 , ' " , X l X d , X2x3, " , Xd - I d E n l l 2 f x A I A

where

LIMIT THEOREMS FOR CONDITIONAL DISTRIBUTIONS WITH REGARD FOR LARGE DEVIATIONS 1197

c 2x 1

0

X2

F x = x d

0

0

0

0 ... 0 0

0 ... 0 2x d

x 1 0 ... 0

0 ... 0 x 1

x 3 x 2 ... 0

X d 0 . . . X 2

0 . . . Xd_ 1 X d

In the following two examples, we consider the case d = 1. Denote

17

= = n -1 k bk E~ k, qk Z ~ i -bk' i=I

k > l .

Condi t ional Distribution o f Sample Momen t s under the Condit ion f o r the Sample Mean. Assume that E~ 2m

< oo for certain m ~ N. Let f j ( z ) = z j, z e R , j = 2 . . . . . m. The vector q = (q2 . . . . . q m ) T, under the condition

n~ = x, has the degenerate distribution located along the straight line

Z 1 X m/~ Zm - 1 X

2x 2",fn mx m-I m-,/n

and

2.,#n - - < Y l n~ = x) = dP(b21/2y),

where ap(.) is the distribution function of the standard normal law.

Thus, the averaging with respect to the sample momen t of the lowest order results in the degenerate distribution

for the vector of sample moments o f higher orders.

Condi t ional Distribution o f the First 2m Sample M o m e n t s under the Condi t ion f o r the ( 2 m + 1 ) t h Samp le

Momen t . Assume tha t E~ 4'n+2 < oo for certain m ~ N. Let f j ( z ) = Z "j/(zm+l) , Z ~ R , j = 1,. . . , 2m, and q =

)T C R 2m, ( ~ 1 . . . . . 112m . F o r A w e h a v e

P ( n q - ( ( x + b2m+l)l/(2m+l), . . . , ( x+~'~'2,,,+l,~2m/(2m+l)'~T, ~ nl/2 A nq2m+ l = x ) = f q)D(U)du(l + o ( l ) ) , A

2m where D = I bi+j + bib j i j=l" In contrast to the previous example , the averaging with respect to the sample momen t

of the highest order results in the nondegenerate distribution for the vector of sample moments of lower orders.

1198 A. Yu . ZAIGRAEV

4. Limit Conditional Distributions for Sums of Projections of a Vector onto Given Subspaces

L e t {ui, 1)j}il"=l ~=l be a linearly independent system of vectors in the space R d, k + s <-- d. Denote (S~, u) =

( (S . , u 1 ) . . . . . (S~, uk)) T and (S n, v) = ( (S n, v 1) . . . . . (S . , v,.)) ~.

u k We introduce the following notation: g ( y ) = h ( ey ) [y I -~, y ~ R a, U = [(u i, j)[i,j=~, V = ](v j, u ~ls k t/ j = l i = l i s the

orthogonal projector onto the subspace {x ~ Rd: (x , ui) = 0, i = 1, 2 . . . . . k }. Assume also that the vectors s v s {P.vj}j= 1 are linearly independent and denote Q = I(Puv i, j)li,j=l"

If x e R ~ and y ~ R s, then denote by (x , y) the (k + s) -dimensional vector whose first k components are

the components of x and whose last s components are the components o f y.

Theorem 3. / f p ~ P, Y > d + 2, then

P ( ( S ~ , v ) - V U - ' t ~ I x I Q ' / 2 A I ( S , , , u ) = t ) = Sq~x(y)dy( l +o(1)) A

for A c R s uniformly with respect to t, It I_> n In n, where x = U - l / 2 t , q,'x (y) = Ke~ q((ex' y) ) ' y ~ R s'

q ( z ) = Iz[ d-k-s-v S g (T(e=,w) )dw, z ~ R k+s, Rd-k-,s

and T is the orthogonal (d • d)-matrix determined by the vectors {u i, ~j}~'=l ~=1 whose explicit f o rm is given in

the proofi

Remark 1. The form of the conditional density is substantially simplified if k + s = d. In this case, q( z ) = g ( T z ) . II; in addition, u i = e i and ~)j = ek+ j, i = 1 . . . . . k, j = 1 . . . . . d - k, where e i is the i th unit vector in R d

then q( z ) = g(z) . In particular, we have q ( z ) = p ( z ) for a density p of the form (2).

Proo f In what follows, without any specification, assume that ]x [ > n 1/2 In n and, as before, n ~ oo.

Denote P = P((Sn , v) - VU-l t ~ U-1/z t Q1/2A I (S,, u) = t ) . Since (S,,, u) belongs to the domain of large

deviations, we may say the same about the sum $1,. Therefore, Theorem 1 implies that

Let

' k The vectors {ui}i= 1

holds, whence

P = P ( ( ~ , v ) - V U - l t ~ U-1/2t QJ/2A I ( ~ , u ) = t ) ( l + ~

k P Z u- l~2 u i = ( ) i juj , i = 1,2 . . . . . k.

i=1

are orthonormal and, consequently, the representation

k

i=1

(28)

LIMIT THEOREMS FOR CONDITIONAL DISTRIBUTIONS WITH REGARD FOR LARGE DEVIATIONS

(~ , v ) - v u - l ( ~ , u ) = ( ~ , p . ~ ) .

1199

Substituting (29) in (28), we get

(29)

P = P((~, Puv> ~ U-1/2t Q I/2A I (~, u) = t)(1 + o(1)). (30)

s

Vi ( Q - I / 2 ) u = ii J ' i = 1,2 . . . . . s , j=l

{Ui, " k U - 1 / 2 t " 1)j}i= 1 s j=l and the vector x =

Introducing the orthonormal vectors

lbr the orthonormal collection of vectors

deviations, we derive from (30) that

lying in the domain of large

P = P ( ( ~ , v ' ) e l x l A I (~ ,u ' )=x ) ( l+o (1 ) ) .

Let f21 be a joint density of ({, v') and (~, u') and fl be a density of ({, u'). Obviously,

IxlS f4,(Ixly, x)dy. f l ( X ) Z

P(<g,v')elxJA I ( g , u ' ) = x ) =

We find the distribution of (~, u'). We have

I (h(ez) + w(Iz I))Z(I z I)I zl-~ dz, t

( Z , U l ) < X 1 . . . . . (Z, bt~.)<X k

P((~, u') -< x) =

(31)

(32)

x l xk

f "" f lyOld-k-2dyl'''dyk I (h(eT'(eyo,Z))+WOYoI(I+Iz _ ~ _ ~ R d - k

Differentiating with respect to x 1 . . . . . x k and using the properties of slowly varying functions, we get

P((~, . 3 -< ~)

The change of variables y~+ 1 = ]Yo Iq . . . . . Yd = lYo IZd-k (where Y0 = (Yl . . . . . Yk)) results in the expression

X 1 x k

_e~ - c o R d - k

(h(er3,) + w(lY D)/(lyD lYl-V dyk+l . . . dYd.

where l( t) slowly varies as t --+ ,,~. �9 p

Assume that T' is an orthogonal matrix of the form T' = ( u I ... u k D r ) , where D is a ( ( d - k) x d)-matrix p i

such that Du i = O, i = 1 . . . . . k, and DD r is the identity matrix. Alter the change of variables z = Ty, we ob- tain

1200 A. Yu. ZAIGRAEV

f l ( X ) - [x[d-k-Yl(IxD f (h(eT,(ex,z)))((l+lz[2)-Y/2)dz(l+o(1)). Rd-k

(33)

Analogously,

+ Z 2 -g12d f21(t, x) = (Ixl2+ltl2)(cl-k-s-7)/21((lxl2+ltl2)l/2) [. h(eT(e(x,t),z))( 1 [ 1) z(l + o ( 1 ) ) ,

Rd-k-s

p �9 p �9 p f

where T = ( u t . . . u k v 1 . . . v s F r ) and F i s a ( ( d - k - s ) x d ) - m a t r i x s u c h t h a t Fu i = 0 , i = 1 . . . . . k , Fv j = 0 ,

j = 1 . . . . . s, and F F T is the identity matrix. Consequently,

f2~(I xly, x) = I x f -k-s-V(l + I v12) (d-k-~-v)/2 l(Ixl) [. h(ev(e(,,x,y,,z))(l + l z l 2 ) - ~ / 2 d z ( l + o ( 1 ) ) .

Rd-k-s

Substituting (33) and (34) in (32), and further in (31), we obtain the statement of the theorem. Theorem 3 is proved.

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