On estimating the parameters of an ARMA-model from noisy measurements of inputs and outputs

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On estimating the parameters of an ARMA-model from noisy

measurements of inputs and outputs

Citation for published version (APA):

Vregelaar, ten, J. M. (1986). On estimating the parameters of an ARMA-model from noisy measurements of inputs and outputs. (Memorandum COSOR; Vol. 8618). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1986

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EINDHOVEN UNIVERSITY OF TECHNOLOGY Faculty of Mathematics and Computing Science

Memorandum COSOR 86-18

On estimating the parameters of an ARMA-model from noisy measurements

of inputs and outputs by

J.M. ten Vregelaar

Eindhoven, the Netherlands November 1986

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On

estimating the parameters of an ARMA-model from

noisy measurements of inputs and outputs

I.M. ten Vregelaar

University of Technology, Eindhoven

ABSTRACT

In this paper we discuss an estimation method for the unknown parameters of an ARMA-model when its inputs and outputs are measured with noise. In case the noise is normally distributed. the consistency and asymptotic normality of the estimator (in this case being the maximum likelihood estimator) is proved under some mild conditions and a consistent expression for the asymptotic covariance matrix is given.

1. Introduction: description of the problem

In this paper we consider the following estimation problem (in system theory called identification problem). Given the ARMA-model

'1')1 =O'l'1}l-l+ .... +O'p'l')t-p+f3o~t+ .... +f3q~t-q, t

=

1,2, ... with observations YI = 'l}1

+

el XI

=

~I

+I[ .

t

=

L2, ... N (1) (2a) (2b) the problem is to estimate the parameters O'l, ... O'p ,/3o ... /3_g where the orders p and q are supposed to be known.

In genera] the true inputs and outputs

e"

'1')1 are allowed to be resp. r - and 5 -vectors (the

socalled MIMO-case, i.e. multiple input-multiple output).

The estimation is performed under the following assumptions for the measurement errors et

,It :

E

It

=

o.

t

=

1.2 •...

.N

e 1 ... e.I\' .II , ... .f N are independent,

moreover even all their (5 +r)N components are independent. VAR e

_t

= (J'21s • VAR

It

= (J'21,. •

where (]' is unknown but fixed and Is denotes the 5 X 5 - identity matrix.

When assuming zero initial conditions ('1}0

= ....

=

'I} I-p

=

0 ,

eo

= , ... =

e

1-q

=

0) we

can rewrite model (1) as

A'I')+B~=O (3)

(4)

2

A =

t

Sk ® a/.:. B =

t

Sk ® (?, J.:

1=0 1=0

are sN XsN and sN XrN matrices resp. and 7)

=

(7)[ •... •

7),'['l.

~

=

(~[,

...

,glf

are resp. sN - and rN -vectors. S is the N xN shift matrix whose (i,j) element is equal to

~i.j+l and ao:= -Is·

The measurements (2) and assumptions are reformulated as

Y=7)+e.

Ee=O

(4a)

x=g+/.

E/=O

(4b)

VAR [

i

1=

( j 2 I (s +r )N . (4c)

Let

fV

:= (0: 1, . . . ,O:p .(?,o •... ,(?,q) be the 5 X (pS +(q

+

l)r )-matrix of parameters to be estimated, we can rewrite the model and measurements once more as

D (9_{0 )}0

=

0

Z = 0

+

n , E n = O. V AR n = ( j 2 I

where

eo

is the matrix of true values of the parameters.

D := (A B) (size sN X (s +r )N),

~:=

I

~

),

z :=

I

~

I

and n :=

I

i

I

with n is the noise vector.

2. Method of estimation

(Sa) (5b)

Estimates for the parameters are obtained by the solution of the following con-strained minimization problem:

mine); II Z -0112 ,

subject to

D(9)0=0.

( 11.11 denotes the Euclidean vector norm).

(6a)

(6b)

When assuming the normal distribution for n the solutions of (6) give the maximum likelihood estimates for the parameters.

Application of the Lagrange multiplier method gives an unconstrained minimization problem for the parameter matrix 9:

mine IN(9)

where

(7) with

Qce):= DT(DDTr1D

(the factor

5~

is added for convenience). This (5 +r)N xes +r)N matrix satisfies Q

=

QT Q ' so Q is an orthogonal projection matrix with tr Q

=

rank Q = sN. Note that

(5)

3

-since A is nonsingular, DDT

=

AA T +BBT is nonsingular as well for all O.

In this paper we will concentrate on asymptotic properties of the so-called total least squares estimators introduced above. For this purpose we need some expres-sions for IN' and IN", resp. the column vector of first derivatives of IN w.r.t. 0 (0 can be

al

N

a

21N

seen as a vector) and the matrix of second derivatives: IN':=

(00. ),

IN":= (

ao . ao. )

I J I

(obviously Q and therefore IN are infinitely times continously differentiable). Expressions for expectation and covariance matrix will be given here as well.

By differentiating it follows

1.1\' '(0)

=

s~

ZT Q '(O)z (8a)

(the r.h.s. represents a vector, its i-th component is

s~

z T Q j (O)z :=

s~

zT

:~

z ), and

I

(8b)

( h h d . . h ( .. ) I 1 TQij (Ll) '- _1_ T

a

2

Q ) W

t e r . . s. enotes a matriX WIt 1.) e ement sN z 17 Z . - sN Z

ao

j aO

j z . e

gave an algorithm for the computation of ",·(0) and IN'CO) for given 0 (cf. Ten Vregelaar '85).

Since IN (0), IN '(0) and IN "(0) are random variables we can determine expectation and (co)variance. For arbitrary distribution of the noise vector n, satisfying (5b), we

obtain

(9a) (9b) (9c) using E

Cz

T Qz )

=

tr (Q VAR z )

+

(IE z

Y

Q (IE z ), and tr

:~

=

a~~

Q

=

0 etc.

I I

Assuming finite fourth moments for the components of n , var l,v (0), VAR IN '(0) and the variance of the elements of IN "(0) exist.

If

z

has a multinormal distribution, the relation

cov

Cz

T M I Z , Z T M 2 Z )

=

2 tr (M I V AR z M 2 V AR z )

+

4(IE z

Y

M I V AR z M 2 (IE z) holds for symmetric matrices M I' M 2' Therefore for the normal case ( n --N (0, u 2 I)) it follows

var 1,.,. (0)

=

2u4

+

4u2 &T Q (0 )&

. sN (SN)2 (lOa)

(VAR 1,,'(0))· = 1 (2u 4tr QiQj+4u 2&TQiQj&)

" I,J (sN)2 (lOb)

var(JN"CO))j,j

=

1 (2u4tr (Qij)2

+

4u2&T(Qij )2&).

(sN)2 (10c)

When dealing with asymptotic theory we distinguish the normal case and the general case. This paper deals with the normal case.

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4

-3. Asymptotical results for the maximum likelihood estimator

Under some mild conditions we will prove the consistency and asymptotic normality of the total least squares estimator (being the maximum likelihood estimator) in case the noise vector n is multinormally distributed: n - N (0. (}"2]). For notational convenience the results are derived for the single input single output (SISO)-system (5

=

r

=

1). For the time being the parametrization is supposed to be free.

It will be convenient to write

(11a)

where

H_N(8):= (S1) S21) . . . SP1) ~ S~ ... sq~) (11b)

is a N X (p +q

+

1) matrix CcL Aoki . 70 p. 241. obviously 1)

=

H N (8)9 is equivalent to D(6)S

=

0). As a consequence ST Q (6)8 = (6-6₀)T MN(6)(9-6 0) (11c) N with MN(O):= (HNUnY(DDT)-lHNCS) N

We impose the following smoothness conditions:

0) let S be a compact subset of RP+q+l containing

eo

as an interior point

(lId)

(ii) the polynomial A eX)

=

-1

+

t

a i X i has its zeros outside the unit circle for 9 E S

i = 1

(iii) the input sequence {~i

It

1 is uniformly bounded: sup; I~j 1<00

(iv) M .... ,(9) ... M(e) if N ... oo. uniformly in

9,

with M(6»OforOeS (meaning M is positive definite on S).

Condition (ij) expresses the stability of the system: (iij) implies the output sequence (1) I

I

t~ 1 is uniformly bounded as welL Furthermore cond ition (ij) implies (the proof can be

found in the appendix):

klI~(DDT)-1~k21 for all OES. 0<k)<k₂<00 (12) which in addition with (iv) gives the nonsingularity of _{(HN (S»TH}N(8) expressing 6 is

uniquely determined from 0 given 1) = HN(8)O. We can qualify 00 as the .true parameter

vector.

We will discuss the asymptotic properties of

9.",

defined by

J,\'

(9

117 )

=

mineE e iN (e). (13)

Since (DDT )-1> 0 there exists some regular N xN matrix C (9) with (DDT )-1

=

CT _C. (We can choose C differentiable, moreover if

C

denotes

:~

. it is possible for C to satisfy

• • I

C = -CDD T (DDT I, which will be of use in the sequel). Hence (omitting the argument 0 in C and D)

(7)

5

-(14a) where

(14b)

c;,

denoting row i of C.

From CDz ... N(CD8.cr2J) it follows tl(8).t2(8).··· .t]l;(8) are independent. hence S 1(8 ).s 2(8) .... . S]\' (8) are as well. So in the normal case J N (8) and J N '(8) can be written as sums of independent random variables which is useful in obtaining asymptotical results.

Lemma 3.1

IN (8) -+ J (8) a.s., uniformly in 8. i.e. p(I N (8) -+ I (8) uniformly in 8)

=

1. where

J(8):= cr2

₊

_(8-8

0

Y

M(8)(8-80 ). (15)

Proof. From assumptions (ii). (iii) and (lOa) it follows var IN(8) -+ 0 uniformly in 8.

Chebysheff's inequality yields plim (Ix

ce)-

E IN (8))

=

0 uniformly in 8. i.e.

SUPeEe P( IJN (8)-£ IN(8) 1 ~ e) -+ O. for any e>O. Since IN(8) is a sum of

indepen-dent random variables. a.s. convergence follows (cf. Breiman '68. p. 45). Assumption (iv)

guarantees the existence of the limiting mean of I_N(9).

0

A consistency result is given by the following

Theorem 3.1

6"

is (strongly) consistent for the true parameter vector 80 , i.e.

6

N -+ 8₀a.s ..

Proof. J_{x (9) is twice continuously differentiable so}1 is continuous on

e.

Since M (8» 0 (assumption (iv)) J is uniquely minimal in 80 , The theorem now follows from the lemma

and theorem 3.1 in Linssen '80. p. 24.

0

Showing asymptotic normality needs some preparations.

Lemma 3.2

1,Y'(8) and J,.:"(9) converge almost surely. uniformly on

e

and limJ]I;"(8_{0 )}> O.

As a consequence of assumption (ii), tr

Q

2 ~ C IN and

8

T

_Q

₂₈_~_{C 2118112 where C}_I,_{C 2 are} bounded functions of 9 (the append ix c~ntains a proof).

Assumptions (ii) and (iii) imply var J...,.(8) -+ 0 uniformly in 8. As in the proof of the IV •

previous lemma, since /,,(9)

=

~ i~/i(8)

(cf. (14)), where 51.' ..

.iN

are independent. almost sure convergence follows.

The proof for IN "( 9) is similar.

Actually (cf. Linssen '80 p. 26) we obtain

(8)

6

-h.:·(90 ) ... 0 and II\'''(90)-+ 1"(90)= 2M(90 )

>

0 from assumption (iv) and (15).

The key for the asymptotic normality result is the following

Lemmo. 3.3

If the smallest eigenvalue of V AR (n T Q '(90 ) n ) tends to infinity for N -+ co then IN '(90 ) is asymptotically normal. moreover

(N EolN '(9)Ul.' '(9))T )_'hT ~ IN '(90) -+ N (0. I) (convergence in distribution).

(16)

o

By notation Eo refers to taking expectation for 9 = 9_{0 and A}-lIzT = «A lh)-If where in

general for positive semidefinite matrix A . A 'I, denotes a square root: (A

Ihl

A liz

=

A .

Proof· If· denotes

a:.

again, it is easy to verify that

I

Q=D+i>Q +Q.1.iJT(D+)T (17)

where Q.1 := I-Q is as Q an orthogonal projection matrix and D+ := DT (DDT )-1.

Therefore E IN '(90 )

=

0 and var ~ IN '(fJo)

=

N E 0 IN 'UN ')T

Since

z

=

S+n. COY (n T Q '(90 ) n ,ST Q '(90 ) n)

=

0 and

:f.

8

=

HN (S ).i (column of

I

HN (8)), we obtain

NVARJN'(9 0)=

~VAR(nTQ'(90)n)+4(J'2MN(90)'

(18)

Assumption (iv) gives VAR ~J,,/(9o)

>

0, for N sufficiently large. Consider now for fixed (p +q

+

l)-vector A

}...r~I

.• /(eo)= 1 (n TQ'(9_o)n)+ IN-XT _(STQ'(9

0)n). (19)

Since the terms of the r.h .s. of ( 19) are uncorrela ted and

IN-XT (8 TQ'(9₀)n )-N(O,4u2AT M_N(9₀)X) asymptotic normality of the first term

suffices to obtain asymptotic normality for)..T ~J, ... ·'(Oo).

In the appendix it is shown that the assumption in the lemma gives

1 T ( T ' ( 9 ) ) (XIVAR (nTQ'(Oo)n)X)

~X n Q 0 n AN 0, N '

X T

en

T Q '( 9_{0 )}n )

Le. ( T ( T (ll) ) )'h ... N (0, 1) in distribution.

X V AR n Q' I ] 0 n A .

Applying the multivariate central limit theorem in Varadarajan '58 yields the lemma.

0

Now we are able to prove

Theorem 3.2

Provided the assumption of lemma 3.3 holds. we have that (N _{E oI N}

·U

_s ·)T)_lhT EoIN" ~(eN-OO) ... NCO,I). or equivalently

(9)

- 7

Proof. As a consequence of tl1eorem 3.1 and lemma 3.2 the limiting distributions of

-m

J/,.;'(90 ) and

m

J"(00)(ON-9o) are equal (Linssen '80. p. 27 lemma 3.3). Lemma 3.3 implies

(N EOJN'UN'Y )_'hT J"(9_{0 )}

mea

_{N - ( 0 ) ...} NCO,I). (20)

Since Q ii :=

06;P.~0.

= pri

+

(pii l . where

) I

pii = QiDiTCDDTrlDiQJ..-D+(DiD+Di+Dj D+Di)Qi -D+DiQiDjTCD+l (21)

. h Dr d . aD

Wlt enotmg

aB. .

I

it follows from (9c) that

Furthermore _{EOJN" U"(OO»-l}-+ I and

(NEoJ",,'UN'lr'/zT EOJ_{N "}

U"(9

_0»-1

(NEoh·'U,\.yYhT ... I . (22) The theorem now follows from Linssen '80. p. 26 lemma 3.2 with XN the l.h.s. of (22)

and YN the l.h.s. of (20).

0

Another expression for the asymptotic covariance matrix of

m

(0

N - (_{0 )} can be obtained as follows. We conclude from (18) that

NEoh"UN'Y = 40'4.6..1\' (9_o)+4O'2M,I\'

C9

o) •

where the (i.j) element of .6..\.(0_{0 )}is

14 cov(nTQiCBo),t.nTQieBo)n)= 1NtrQiQi

40' N 2

=

1...

_{tr DiT(DDT)-lDiQi} f 9 II

N or

=

17 0' i .j

=

1.2 ... p +q

+

1. (23)

As a result

m

(6

N -eo) AN (0.0'2 M,\' (Oo)-l{J +O'2_.6.

N

(9

0) MA,

CO

o)-1

D.

(24)

Finally, a consistent estimator for the asymptotic covariance matrix of m(ON-90 ) is given in this section.

According to (20)

m

(6 , ....

-90 ) AN (0. U "(90 »-1 N E oJI\' 'UN')1 U "(90»-1) holds. As a consequence of 1".;"(0) ... J"(9) a.s., uniformly in

e.

'[" is continuous in eo and 9 J\' -+ 90 a.s. we obtain

(10)

8

-A consistent expression for N Eoh' 'UN

'Y

can be found by considering again (cL (14))

1',:

J,\"(O)

=

~

1:5,'(0) ,with 51'(0), ... ,5j\"(9) independent.

j = 1

Writing 5j(0)=zTQi(0)Z • with Qi:=DTCTdCi,D it follows

E 5j '(00 )

=

0

.

(} .

since tr Qi = ae-tr Qi

=

1

=

0 (for all 6) and

1

.

,

a

8T Q (90)8

=

oT {2DT CT•i Cj.D

+

DT a9. (CT., Ci»D}o

=

0 1

where· denotes

a:

for arbitrary j E {1,2 ... p +q

+

I}.

1

Therefore

N

N EOJN'UN')T

=

~EOj~15j'(0)(5i'(6))T.

Furthermore. again from (14) we obtain 5i '(0_{0 )}

=

2 ti (6₀)ti '(60 ) and E ti (00 )

=

0 implying

cov (ti '(6_{0 ).}ti (6₀

»)

= E 5i '(0_{0 )}

=

0

and because of normality ti (00 ) and t; '(00 ) are independent. i

=

1.2 ... N . (Actually li(6) and t;'(O) are independent for a116 .)

Consequently

2 N

N EOJN'UNY =

4~ EOi~/J'(0){t,'(6»T.

From the strong law of large numbers

~

itl

It; '(0,\' )(ti '(O,\,))T - E'oti '(6)(t, '(9»T

I-

0 a.s ..

Then

N E:

0

I

_N

'U ,\'

'Y

is consisten tl y estimated by

A 1 N A A

40-2 N

i~/;

'(6 N)(l j '(6 N)i

where

(25) Obviously frorp J",(6)-J(6) a.s., uniformly in

9.

J is continuous in

9

0 , and ON-OO a.s. we obtain IN(O,,,) - J(60 )

=

0-2 a.s ..

N A A

The (k.1) element of

Lt

_i'(6_N)(t/(6_N»)T is

(11)

9

-where ~:= QJ..z (the vector of residuals). Therefore we have proved

Theorem 3.3

A consistent expression for the asymptotic covariance matrix of

m

(9

N - (_{0 )}is given by

4;2

ci

N ,,)-1

MN

(8

)(j,,,',)-l where

;2

is given by (25), A A IN" := IA' "(6 N ) A A 1 A T -1 A MN(B):= N(HN(B)) (DDT) HN (8)

le=e

N CcL (lld)). Remark.

For the very special case p

=

q

=

0 (no dynamics). IN '(6_{0 )} is a scalar and the assumption in lemma 3.2 is satisfied:

a

40" 4 var nT !lfl Q({3o)n

=

- - - - N -+ 00 if N -+ 00. V/J (t

+

(3 From (24) it follows

m(a

._fl ) AN (0 O" 2 N (1+fl

2+

O"2N /JA /JO •

tTt

/JO

t

(cL Linssen '80. p. 51 (4.23)). 4. Discussion

The asymptotical results of the previous section follow from writing IN (6) and

1.'11,' '(0) as sums of independent variables. which allows for applying some law of large numbers and some central limit theorem. In general (dropping the normality assumption for the noise) 1.101 (6) and the components of

It.;

'ce)

can be written as sums of random variables. say

N

LIL,

Y,2 wherecov (Yi.Yj )= O"2B,,j'

, 1

It is our aim to extend the asymptotical results to the general case mentioned above. Furthermore we want to generalize the results to the MIMO-model. not assuming zero ini-tial conditions. and allowing for relations between the parameters to be estimated.

(12)

10

-Appendix

Following Aoki '70 we give some bounds on DDT for the SISO-model (cf, (12)).

As a consequence of the stability assumption (ii) (the polynomial A (A.)

=

-1

+

t

Of i A.;

i=1

has its zeros outside the unit circle) we recall from Aoki '70, p. 245

The stability assumption yields the lower bound. Since DDT

=

AAT +BBT it easily follows

PII ~ DDT ~ P3I

and

(Al)

The orthogonal projection matrix Q being Q = DT (DDT

Q...L

=

I-Q

ID we can write for

Q...L = ET(EET)-IE

where E:= ( -BT _A_-T ₁₎_(size_{N x2N).}

Since EET

=

1

+

B T (AA r )-1 B we obtain

k 31 ~ (EET _)-1_~_{1. O<k}_3~_1.

Using (Al) we are able to prove (d. lemma 3.2) the following

lemma A.1

. 2 . (}

Q ~ C 121\! • O<C <00. where Q

=

ao

Q for any i.

t

Proof. From (17)

Q =

p+pT holds where P

=

D+ DQ..L.

For arbitrary 2N - vector z we have z T

Q

2 Z

=

II Qz 112.

From (Al) and

DT D

~ Iuv (in case of free parametrization) it follows

IIPzl12 ~ k21DQ Zll2 ~ k₂11Q...L z 1l2 _~_k211z112.

Furthermore we conclude from (Al). DDT ~ IN and (DDT_)-2_~_klIN_that

IIpTz1l2~ IIDT(D+lzI12~ II(D+)TzIl2~ klllDzl12~ C₁11z112 2

where C (= k /( 1

+

t

I Of; 1+

t

I (3 j I) .

t = 1 j 0

(A2)

(A3)

Since II Qz II ~ II pz II +II pT z II. II Qz 112 ~ C II Z 112 holds. where C := (..[k';+

.jE-;i.

yielding the lemma.

(13)

-11

Finally. it remains to show Ccf.lemma (3.3)) that lNAT(nTQ'(Oo)n) is asymptotically normal if n --N (0. (1'2]) for any (fixed) vector A .

For this purpose we need

lemma A. 2

Let n --N (0. (1'2]) and J-L 1 • . . . • J-L2X are the eigenvalues of a symmetric matrix RN then

n T RN n is asymptotically normal if and only if 2)

----=::-:-,,---.:;~- -+ 0 for N -+ 00 •

Proof· We can write RN as

RN = PNT ANPN

where P_Nis orthonormal and AN = diag (J-L l' . . • • J-L 2N ).

2N 2N

Then n T RNn

=

L

J-Li (P.,."n)i 2 =:

L

J-Li X; .

i::::: 1 j = 1

Since PNn --N (0. (1'2]), X 1, . . .

.x

211' are independent and identically distributed

(actu-X·

ally I

X

2 ).

(1'

Now the lemma follows from the Lindeberg-Feller central limit theorem (cL Serfting '80.

p.29).

0

p+..g..+l .

Let us define

R

N:=

L

A;Q' (00 ), and suppose its eigenvalues are J-Ll' . . . ,J-L21','

i= 1

2_ IIR:r;wIl2

Then maXi J-Li - max,.. ... 0 IIwll₂ .

P +..g..+l

Since II Rj\' w II ~

L

I A i I II Q i w II ~ K II w II. where K does not depend on N (from (At). i::::: 1

cf. proof of lemma A.n. we obtain maXi J-Li 2 ~ K2.

which tends to infinity, when the smallest eigenvalue of VAR n T Q '(6_{0 )}n does. As a consequence the eigenvalue condition in lemma A.2 is satisfied for R.\· , yielding the asymp-totic normality of

IN

AT (n T Q '(0_{0 )}n ).

Remark.

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12

-tr (Q,)2= 2-tr DiT(DDT)-IDiQ-"- ~ 2kJk3I1DiETII2.

For free parametrizations II Di ET 112 ~ k 4N holds (again k 4 denoting some constant not

depending on N ). so the elements on the diagonal of VAR nTQ'(90)n tend to infinity,

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13

-References

Aoki '70

Aoki M. and Yue

p.e.,

On certain convergence questions in system identification, SIAM J. Control, vol. 8, no. 2, 239-256, 1970.

Breiman '68 Breiman L., Probability,

Addison-Wesley, Reading, Massachusetts, 1968.

Linssen '80 Linssen H.N.,

Functional relationships and minimum sum estimation, Doctoral thesis,

Techn. Univ. Eindhoven, 1980.

Serfiini:'80 Serfiing R.J.,

Approximation Theorems of Mathematical Statistics, John Wiley and Sons, New York, 1980.

Ten Vrei:elaar '85 (in Dutch) Ten Vregelaar J.M.,

Algoritme voor het schatten van de parameters in Arma-modellen met meetfouten op de in- en uitvoer,

Memorandum-COSOR 85-10, Techn. Univ. Eindhoven, 1985.

Varadarajan '58 Varadarajan V.S.,

A useful convergence theorem, Sankhya, 20, 221-222, 1958.