Note on the convergence to normality of quadratic forms in independent variables

Note on the convergence to normality of quadratic forms in

independent variables

Citation for published version (APA):

Vregelaar, ten, J. M. (1987). Note on the convergence to normality of quadratic forms in independent variables. (Memorandum COSOR; Vol. 8708). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1987

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Department of Mathematics and Computing Science

COSOR memorandum 87-08

NOTE ON THE CONVERGENCE TO

NORMALITY OF QUADRATIC FORMS

IN INDEPENDENT VARIABLES

by

I.M. ten Vregelaar

Eindhoven University of Technology,

Department of Mathematics and Computing Science,

PO Box 513,

5600 MB Eindhoven, The Netherlands.

Eindhoven, April 1987 The Netherlands

NOTE ON THE CONVERGENCE TO NORMALITY OF

QUADRATIC FORMS IN INDEPENDENT VARIABLES

by

J .M. ten V regelaar

ABSTRAcr

In this paper we prove a result on the convergence to nonnality of quadratic fonns in independent zero mean variables, which is a generalization of a theorem Whittle proved in his '64 paper (Whittle '64).

2

-1. Introduction

In this paper we want to generalize a result on the convergence to normality of quadratic forms

in independent variables (cf. Whittle '64).

The problem of determining limiting distributions of quadratic forms in random variables arises when considering asymptotic properties of least squares estimators for the parameters of an ARMA-model with noisy measurements of inputs and outputs (Ten Vregelaar '87).

Let us start giving a somewhat simplified version of Theorem 1 in Whittle '64. Consider the quadratic form

where

N2

X T A x

=

L

alj Xi Xj i.;=l

XhXZ, ••• ,XN2 are independent random variables

and A is a NZxN2

-symmetric

matrix with elements ajj depending on N.

We split up A as

A

=

diag(A itA 2, ••• ,AN) + A' , where Aj are N xN blocks, i

=

1,2, ... ,N.

The splitting is supposed to be "disjunct", in the sense that N

• A 12

=

L

n

Ai

nz

+

II

A

'11

2 ,

;=1

(11'1 denoting the Euclidean matrix norm).

If IE Xi

=

0 and varXj

=

(12, i

=

1,2,' .. ,N2 (1.1a) (LIb) (1.2a) (1.2b) (1.3)

the following conditions will be sufficient for xT A X to tend to normality in distribution with increasing N:

(ia) i~ IE Xi4 >

oA,

f

(ib) ~p IE IXi 14H\ < 00 for some 0 > 0,

,

(ii) N [II Aill

]2+6

N -+CO

3

-(iii)

IA'II

N __

BA I

~

O.

Condition (iii) expresses some diagonal dominance property. For sufficiently large N the distri-bution of the quadratic form xT A x approximates the distribution of

xT _diag(A}tA

z,· .. ,AN)X ,

which can be written as a sum of independent random variables. Applying some central limit theorem, Condition (ii) guarantees, roughly spoken, the asymptotic normality of this sum. The generalization we will consider is dropping the "disjunct" assumption (1.2b).

The proof goes along the same lines as Whittle's.

2. Result

Consider the quadratic form (1.1). Again we split up

(2.1) now only assuming both A and A' (hence A h . . . ,AN) are symmetric and A h . . • ,AN are

NxN blocks.

Theorem.

The random variables Xl> ••• ,XN'2 are independent and IE Xi

=

0, var Xi

=

cr2

(i

=

1,2, ... • Nz) .

The following conditions hold: (ia)

(ib)

inf IE Xj4 >

oA ,

j

sup IE Ix; 1#3 < co for some 0 > 0,

j

the matrices A1,A

_{z•· ..}

,AN and A' in (2.1) can be chosen in such a way that

(I'}') ~ N

[H

IAi

Ai

II

]Z+<'l N-+co ~ _{0 for some 0> 0,}

and

(iii)

iAiI

/lA'1l N-+co _{-7 O.}

Then the quadratic form xT _{A x is asymptotically normally distributed, i.e.}

xT A x - IE xT _{A x} ""varxT A X

converges in distribution to the standard normal distribution,

4 -Remark..

For the special case Al

=

A2

= ... =

AN

= A,

Condition (ii) follows from Condition (iii).

This can be shown by first proving a Lemma.

Let P, Q and R be N xN -matrices with

P=Q+R. If N -+""

DR"

gPO ---7 0, then IQ

n

N::;

--, 1. IP

II

fm2f.

Since Q

=

P - R we have N

I

Q 112 =

I

P 112 +

I

R 112 - 2

:E

Pij rij . i,j=l

Applying the Cauchy-Schwartz inequality gives

I.

~

Pi} rij

I

~

I

P

III

R

I .

',J=1

The lemma follows from

N

2 2.~ Pi; rij

JQ..t

= 1 +

!.!U... _

2 ...;1 • .:..-1=.;...1 - - : - _

IPn2 _IPU2 _{liP 112}

since the modulus of the last term in the r.h.s. does not exceed : ; : .

_IJ

Therefore Condition (iii) implies

N -+"" ---7 1. so

f

[II

A

II

]2+S

=

N

[n

A

II

]2+11

=

[-J}lll

A

I

]2+S

_1_

N::;

0 /=1

HA II

UA II

IA II

N8I2 for any 0 > O.

5 -3. Proof of the theorem

The proof is given by means of three lemmas. LemmaA.

Let SN, :EN and aN be random variables with

and

then ~ is asymptotically nonnal implies SN is asymptotically nonnal.

Proof. For the special case IE aN

=

0 the proof can be found in Bernstein '26. Define with cr' N := aN - IE aN' Obviously hence N _ ~ 0 from 3.1b .

Therefore S'N is asymptotically nonnal and thus SN is, since

Remarks.

(3.1 a)

(3.lb)

o

1. It is easy to verify that for SN

=

:EN + aN we have varaN N _ . ~ 0 If and only If . var:EN

varaN N _ _{~ 0, y applymg the auchy-Schwartz mequ} b . C . ali _ty fi _{or ran om vana es.} d . bI

varSN

2. Unlike Whittle's proof, IE aN = 0 does not necessarily hold now.

We associate Lemma A and the theorem by defining

LemmaB. Introduce If 1. d >0, ~ 6 ~

2.

~p

_,

IE Xi4 < co, IIA '. N _ 3. ~ 0

gAl

then N-iOO ~

o.

~. From Relation (18) in Whittle '64 (p. 106) we obtain

varSN ~ min(2,d)oAIA 112 •

and Theorem 2 in Whittle '60 (p. 302) implies

varO'N =s; K s~p

_,

IE x,4gA 'y2 for some constant K . Consequently, K~p IE

x/

N , IlA'H2 .::; 0

oA

min

(2,

d)

IIA

112 provided Conditions 1-3. Remark..

In case the vector x has a multinonnal distribution, (x - N(O, 0'2l)

holds.

varO'N 20AilA

'f

UA'112 varSN = 20'411 A 112 =

IIA

112

(3.3)

[]

Let us consider now the quadratic fonn ~. From (2.1) and (3.2) it is obvious that EN can be written as

N

EN =

L

Yj • Y 1. Y 2 •••. ,Y N independent. (3.4)

,=1

The Liapounov central limit theorem applies to obtain asymptotic nonnality for EN (cf. Serfting

7 -LemmaC.

If for some 8 > 0, the conditions

1. £ IYj- £ Yi 12+8 < 00 for i

=

1,2.··· ,N,

2.

L£

N [ IY:.-£ Y:.I ]2+8 _{I I} N ... _{~ 0} i-I

-

~

L

N _varY i .=1 'i:w - £ 'i:w

are satisfied, then converges in distribution to the standard normal distribution. "'var'i:

w

Now we are able to prove the theorem in Section 2. Since 'i:w

=

SN - ('IN,

varLN

=

varSN + var('JN - 2COV(SN,('JN) =

[ var('JN COV(SN.('JN) ] =varSN 1+ -2-...;..~..:;.;.. varSN varSN holds, where (from Cauchy-Schwartz). Then [ I y. -

IE

y. I

]2+3

N [ • • =

L

£ ...Jvar LN ;=1 _ r.::=::-;:;-"varSN Hence N _IYj-IEYjl 2-H1

LIE

...JvarLN .=1

=

N _{[ IY, - IE Y, I var('JN}

LIE

"'var SN 1+ j=l varSN -2 1 COY (SN, ('IN) varSN

From Lemma B, the Conditions (i) and (iii) of the theorem are sufficient to obtain

so the Lh.s. of (3.6) tends to 1 for N ~ 00 (using (3.5».

(3.5)

(3.6)

(3.7)

8

-Y i=Xin.jXj. --T

A.--with

X=

Xi

columnvector of N components .

Applying Theorem 2 in Whittle '60. p. 302, again (s

=

2+a). yields

s; C ~p IE IXi 14+26IAd2+S < 00 for all i

•

(C is some constant). so Condition 1 in Lemma C is satisfied.

Combining this with (3.3) gives for the nominator of the l.h.s. in (3.6):

N 2+S

]

2+3

~.IE I Yj - IE Yj 1

=

s;

(var SN

lHl.l2

C

s~p

.IE Ix,I4+26 N

[IA-II

]2+3

s;

(min~2,d»1+3/204+26 ~ I~

I

Provided Conditions (i). (U) of the theorem, the nominator of the l.h.s. of (3.6) tends to 0 for

Hence the denominator of this term, being the product of the nominator and the l.h.s. of (3.6) itself, converges to 0 for N -+ 00, The conditions in Lemma C are satisfied, we have proved so far the asymptotic normality of LN'

The theorem is proved now by applying Lemma A, since (3.tb) is a consequence of (3.7), according to Remark 1 at Lemma A.

4.

Discussion

The theorem will remain valid, if we replace (2.2) by

IE Xi

=

0, varXj

=

crl

where

0< as; crls; b < 00 • i = 1,2, ... ,N2 •

Furthennore, it is possible to split up the set of integers more generally as

, r

{l,2, ... ,N2}

=

U g, , .. 1

where gj are distinct sets consisting of a varying number of integers (cf. Whittle '64).

Condition (ia) only excludes some pathological distributions like

JP(Xj = 1)

=

t

JP(Xj

=

-1)

=

t

which satisfies IE

xl

= IE Xi4 = 1.

References

Bernstein '26 Bernstein, S.N., Sur l'extension du theoreme limite du calcul des

probabilires aux sommes de quantires dependantes, Math. Ann., 97, pp. 1-59, 1926.

Serfiing '80 Serfting R.J., Approximation Theorems of Mathematical Statistics, John Wiley and Sons, New York, 1980.

Ten Vregelaar '87 Ten Vregelaar, I.M .• Asymptotic properties of some estimation method for an ARMA-model with noisy measurements of inputs and outputs, Memorandum-COSOR, Eindhoven University of Technology, in prepara-tion.

Whittle '60 Whittle, P., Bounds for the moments of linear and quadratic fonns in independent variables, Theory Prob. Applications, 5, pp. 302-305, 1960. Whittle '64 Whittle, P., On the convergence to nonnality of quadratic fonns in