AbstractThe limit of the Hellinger transform of measures related to multidimensional stationary Gaussian autoregressive processes is obtained

Fazekas, I.

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Hellinger transform of Gaussian autoregressive processes 

Computers Math. Applic. Vol. 27, No. 7, pp. 15-21, 1994 
Copyright@1994 Elsevier Science Ltd 
Printed in Great Britain. All rights reserved 
089%1221(94)E0022-C 
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Hellinger Transform of Gaussian 
Autoregressive Processes 
I. FAZEKAS 
Institute of Mathematics and Informatics, Kossuth University 
P.O. Box 12, H-4010 Debrecen, Hungary 
(Received August 1993; accepted September 1993) 
Abstract-The limit of the Hellinger transform of measures related to multidimensional stationary 
Gaussian autoregressive processes is obtained 
1. INTRODUCTION AND MAIN RESULT 
Let PI,..., PN be probability measures defined on a common measurable space (s2, F). Let P be 
a c-finite measure dominating each Pj. Then the Hellinger transform of PI,. . . , PN is defined by 
&(P,,...,P,) =/iyI ($)“‘dP, 
Q j=l 
wherecl!=(crl,..., ~~N),~~>O,...,CYN>O,C~=~=ICYJ=~. 
We remark that for N = 2 the quantity lHI,(Pl, Pz) is called the Hellinger integral of mea- 
sures PI and Pz. The Hellinger integral is widely used to study absolute continuity of measures 
(see e.g., [l-3]). 
Let, N,(b, R) denote the p-dimensional normal distribution with mean vector b and covariance 
matrix R. Let Pj denote the distribution N,(O, Rj), j = 1,. . . , N. Suppose that Rj is invertible 
forj = l,..., N and C,“=, aj RT’ is also invertible. Then the Hellinger transform of PI, . . . , PN 
is the following 
&-x(pl, * * ., fi (detRj)ai det 
j=l 
-l/2 
. (2) 
Let {&, t = 0, fl, f2,. . . } be a (multidimensional) Gaussian process with mean 0. Let Hj be 
the hypothesis that {&} has covariance matrix Cj. Let, Pj”’ denote the distribution of {co, . . . , &} 
ifHj istrue,j=l,..., N. Then 
IHI,(Pf), Pi’) = fi (det ( [Cj] i)) a’ 
j=l 
det (g&j ([Cj]E)-‘) i”‘f (3) 
I wish to thank F. Liese and Y. S. Sathe for helpful discussions. 
Thii research was supported by the Hungarian Foundation for Scientific Researches under Grant No. OTKA- 
T4047/1992 and Grant No. OTKA-1648/1991. 
Typeset by AM-l&X 
15 
16 I. FAZEKAS 
where [cj]i denotes an appropriate ‘section’ of Cj (i.e., [Cj]: = ((CJ)~,~):,(=~). We suppose 
that the inverses involved exist. 
Our aim is to study limt_,cr) (l/(t + 1)) log H, (P,‘“‘, . . , , Pi’) when the process {&} is stationary 
(under each hypothesis Hj,j = 1,. . . ,N). If N = 2 then lim~,co(l/(t+l)) logH,,I_,(P, (t), p,‘“‘> 
is Rbnyi’s information (see [4]). Suppose that the (multidimensional) stationary process {&} has 
spectral density (matrix) function fj(X), X E [-7r, ~1, under hypothesis H3 
exp(i(k - Z)A)fj(X) dX, Q k,l. (4) 
It is known that a stationary process with spectral density fj(X)fh(X) has covariance matrix 
CjCh. Moreover, if fj(X) . IS invertible for all X then a stationary process with spectral density 
(f,(X))-’ has covariance matrix (Cj)-‘. 
We need the following result due to [5]. 
LEMMA 1. Suppose that fj(X) is positive definite for X E [-.zr,r]. Then 
iiz &(logdet( [cJ]f,) = 2’,- l’=logdet(.?j(A))dA. 
7r 
(5) 
However, the above mentioned facts cannot be used directly to calculate 
&I& & log w, (P:t), . . . , qy ) 
for arbitrary stationary processes because formula (3) involves ([Cj]i)-’ and not [Cj~‘]~. 
For multidimensional Gaussian AR(l) processes we have the fAllowi;g theorem. 
THEOREM 1. Let {yt} be an m-dimensional stationary Gaussian AR(l) process with zero mean 
yt_+l = Ayt +c~+I, t = 0, +I, 12,. . . , (6) 
where {Ed} is a sequence of independent standard Gaussian random vectors. Let Hj be the 
hypothesis that A = Aj, j = 1,. . . , N. Suppose that the eigenvalues of Aj are inside the unit 
circle for all j. Let 
f,(X) = ([I - Aj exp(iX)lTII - Aj exp(ix)])-l (7) 
be thespectral density of the process and let Pj” denote the distribution of (~0,. . . , yt} if Hj is 
true, j = 1,. . . , IV. Then 
log det f.j (X) dX + logdetccYj(f,(A))-‘dX 1 3 (8) j=l 
where (1~ = (~1,. . . , QN), ~1 > 0,. . . , @N > 0, c,“=, aj = 1. 
Here, and in what follows, T denotes the transposed of a matrix while 1 denotes the unit 
matrix (of appropriate size). If N = 2, Theorem 1 is known for any multidimensional stationary 
Gaussian process (see [S]). 
The proof of Theorem 1 is given in Section 3. In Section 2, a stronger version of Theorem 1 is 
proved for scalar prosesses (Theorem 2). 
Hellinger Transform 17 
2. SCALAR AR(l) PROCESSES 
Let {&, t = 0, fl, f2,. . . } be a real stationary Gaussian AR(l) process. Suppose that under 
hypothesis Hj 
b+i = aj 6 + Et+i, t = 0, fl, f2,. . . ) (9) 
where {Q} is a sequence of independent standard Gaussian random variables, laj( < 1. Sup- 
pose that the mean of the process is 0. The covariance matrix and its inverse are of the form 
(see [7, p. 1311) 
cj = -L 
1 - a; 
1 
*. *. f. 
. . . 
. . 
. 1 aj a; 
. . 
. aj 1 aj '.. 
a; Cij 1 “. 
*. 1. *. . . .I 
. . . . . . 
. . . l+aj2 -aj 0 
. . -aj 1 + aj2 -aj ‘*. 
0 -aj l+a,2 ‘.. 
. . . . f. 
* 1 
The covariance matrix of (6,. . . , &} and the inverse covariance matrix have the form 
1 0 . . . -aj 0 
-aj 1 +a3 '.. *.. i 
0 Cj I> i -l= LX’ 0 . . '_ '. 0 I . 
*. . . +a; 1 -aj 
0 . . . 0 -aj 1 
It is easy to see that det ( [Cj]i)-1 = 1 - a;, but according to (3)) we need 
d=det (gaj ([Cj]E)-‘). 
BY (12) 
d = det [Clf2 - 2a2 det [C]im3 - 4a4 det [Cl:-“, 
where C = C,“=i ajCjrl and a = Cy!r ojaj. 
The spectral density of the stationary process {&} defined by (9) is 
fj(X) = (1 - 2aj COSX +a:)-l. 
The stationary process with spectral density 
gj(x) = --!- = 
.fjCx) 
1-2ajCOSX+f_$ 
has covariance matrix Cj:‘. Furthermore, the stationary process with spectral density 
g(A)=~CXjgj(A)=~cuj 
j=l j=l .fjCx) 
. (10) 
(12) 
(13) 
(14) 
(15) 
(16) 
has covariance matrix C = C,“=, ajCjT1 . 
18 I. FAZEKAS 
We need the following lemma (see [8, p. 761). 
LEMMA 2. Let f(A) b e a spectral density function on [-n, SK] and let C be the covariance 
matrix corresponding to f. Suppose that f’ exists and satisfies the Lipschitz condition 
If’(h) - f’(h)1 < Kp+l - x21”, (17) 
whereK>O, O<p<l. Then 
JLs det( [Cl:) 
I 
{ exp (& [y log f(x) dA> }‘+’ = Lf (18) 
exists and 1 < Lf < 00. 
As spectral densities defined by (14) and (15) satisfy Lipschitz condition (17), we can apply 
Lemma 2. Using (18) and (13), we obtain 
where kg is a finite constant. For the first, term on the right hand side of (3), we have 
Thus, we have proved the following theorem. 
THEOREM 2. Let {&} b e a zero mean stationary Gaussian AR(l) process, &+I = a& + Et+l, 
t = 0, fl, f2,. . . , where {et} is a sequence of independent standard Gaussian random vari- 
ables. Let Hj be the hypothesis that a = aj, where laj 1 < 1, j = 1,. . . , N. Let fj(X) = 
(1 - 2aj cos X + aj) ’ -’ be the spectral density of the process and let Pjt) denote the distribution 
of{<o,... ,&} ifHj is true,j = l,..., IV. Then 
lim 
IHI, P,(“),...,P$) 
( 
t-+m {exp ((-1/4n) s_‘,” log (fly=, .fj*‘(X) c3N,1 Qjf;'(x)) dJf)}t+l = 
L, (21) 
whereLisafiniteconstantanda=(al,..., ON), CYI>O ,..., (YN>O, CjN_l(Yj=l. 
Relation (21) can be rewritten into the following form 
!ir & logH, (P;?. . . , P$) = 2 
+a 
J ( 
log fi f,“j(X) &&‘(x) ClX. (22) _ 
= j=l j-1 
3. MULTIDIMENSIONAL AR(l) PROCESSES 
Let {yt, t = 0, fl, 912,. . . } be an m-dimensional stationary Gaussian AR(l) process. Suppose 
that under hypothesis Hj 
yt+l = Ajyt + Et+l, t = 0, fl, ~t2,. . . , (23) 
Hellinger Transform 19 
where {Q} is a sequence of independent standard Gaussian random vectors. We assume that 
the eigenvalues of Aj are inside the unit circle. Suppose that the mean of the process is 0. The 
covariance matrix and its inverse are the following block matrices 
-. . . . . 
. . . I+ATAj -A; 0 
. . - Aj I + ATAj -A; ‘.. 
0 - Aj I+AjTAj *.. 
. . *. . . 
where Sj = ~,“=,AYAJ”. 
The covariance matrix of {ys, . . . , yt} and the inverse covariance matrix have the form 
L9j 
SjA,T . . . S.AT@-‘) S.A-!-t 
AjSj Sj ‘.’ 3 3 S3A+4 Sj+ 
i i ‘.. i ! 7 
A3-1Sj A~-2Sj ... Sj Sj Al 
AfSj 3 3 ‘.. AtrlS. AjSj sj 
I+AjTAj *.. ‘.. : 
0 I> cj; -l= 
(24) 
(25) 
(26) 
1. (27) 
PROOF OF THEOREM 1. First we shall prove that eigenvalues of ( [Cj]:) -’ are bounded from 
zero. Let X denote an eigenvalue of [Cj]:. By Gersgorin’s theorem (see [9, Chapter XIV, Theorem 
61) 
(28) 
As the spectral radius of Aj is less than 1 the series c& A; is absolute convergent. Therefore, 
2llSjll C,c IIA:ll 5 K, where K is a finite constant not depending on t. As (Sj - XI) -I is 
symmetric its spectral norm is equal to its spectral radius. Therefore, (28) implies that there 
exists a constant K,, (not depending on t) such that 1x1 5 KO for an arbitrary eigenvalue of 
[Cj] i. So eigenvalues of ([Cj] i) -’ are bounded from zero. As ( [Cj] i) -’ are positive definite 
there exists an L > 0 such that 1x1 > L for each X being an eigenvalue of C,“=, oj ([Cj]i)-l 
(see [lo, Theorem 3.6.31). 
20 I. FAZEKAS 
Let us consider the following partition 
(29) 
where W is of size m x m while U is of size tm x tm. Adding C,“=, ojA:Aj to the bottom 
right hand block of C,“,l aj ([~j]i-‘) -’ we obtain U. Hence, 1x1 > L for each eigenvalue of U, 
whence 1x1 < l/L < cc for each eigenvalue of U-l, where the upper bound does not depend on t. 
Taking into account the special structure of matrices in (29) we can see that the eigenvalues of 
W - VTUvlV have an absolute bound. Therefore, 
)i% i logdet(W - VTU-‘V) = 0. 
Now, consider the following partition 
(30) 
(31) 
where Uo is of size m x m while Wo is of size (t - l)m x (t - 1)m. We know that 1x1 > L for each 
eigenvalue of U. As U is symmetric, we can apply [ll, Theorem 9.C.11. We obtain that 1x1 > L 
for each eigenvalue of Wo. Therefore, 
Ji% i logdet(Uo - V,W;‘VoT) = 0. (32) --i 
Now 
det (gaj (ic$,)-‘) = det U det(W - VTUdlV) 
By (30) and (32), 
= det W, det(Uo - VoW{‘Vz) det(W - VTUelV). 
(34) 
if the limit on the right hand side exists. But 
w, = &j ([c,-‘];-‘) ) (35) 
j=l 
that is W,J contains sections of the inverse covariances C.,:‘. Equations (3), (5), (34), and (35) 
imply Theorem 1. 
1. 
2. 
3. 
4. 
5. 
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Hellinger Transform 21 
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Hellinger transform of Gaussian autoregressive processes

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