Some generalizations

6 . 1 . Inversion in terms o f $ (t) and its derived processes

In chapter I I I we have considered the inversion of a smoothing relation in the case of stationary processes of continaous parameter. In theorem 3. 2 we obtained the necessary and sufficient conditions for each of the processes to be a smoothing of the other. This was seen to be applicable.only to instances of processes with restricted types of spectra. To enlarge the class of relevant 481

K. iNAGABHUSHAI~AM, The primary process of a smoothing relation

processes, inversion was attempted not as a single smoothing of the given ~(t)- process, but as the limit in the mean of a sequence of processes each of which was a smoothing of the resulting process (see theorem 3.3). Here we shall consider yet another way of obtaining a solution of the primary process as a combined smoothing of the resulting process and its derived processes.

Let /(u) be bounded and the associated F (2) exist and be bounded, and let the smoothing relation

o v

f x ( t - - u ) / ( u ) d u = ~(t)

be consistent, and consequently let the given ~-process have derived processes d~(t) (as explained in chapter I) up to order p. These will be denoted b y ~ - , d 2 ~ (t) dP_~(t)

dt 2 , ". , dt v ]. The present purpose is to seek inversion of the smoothing relation to obtain X(t) in the form

P d r ~ (t) X ( t ) = ~(t--v)q~(v)dv + ~br

r=o d t r

- - o r

where ~(v) is bounded and

~(2) = ; e - i ' ~ ( v ) d v

- - o o

exists and is bounded. Each of the derived processes is given b y

o o

d ~ ( t ) f eita(i2)rdZ~(2)

dV 9

so that, if X (t) has the above representation, then

p

X ( t ) = eita {~b(A) + ~ b r ( i 2 ) ~} dZ~(2).

B y just adopting the same line of argument a8 employed in proving theorem 3.2, we can prove the following

T h e o r o m 6. t . I / ~(t) is a given coxtinuous parameter stationary process which is continuous in the mean and i] the smoothing relation

o O

f x ( t - - u ) ] ( u ) d u = ~ ( t )

- - 0 o

is consistent and ne~cessitates the existence o/ the derived processes o/ ~(t) up to the order p, then /or obtaining the inversion as

ARKIV F()R MATEMATIK. B d l nr 32

~ dr~(t) X(t) = f ~ ( t - - v ) cf(v)dv + br

_~ r=o dt r

it is sufficient that i) L2(X) = L2(~) and

ii) f i l l - - F ( 2 ) { q ~ ( ~ ) + ~ b , ( i 2 ) r } 2 d a ~ ( 2 ) = O ,

- - a r r = O

where F (,~) has the usual meaning.

The following example is of particular interest in that it has been already considered in connection with theorem 3.3.

Example:

Let

l ( u ) = ~-I~l and let the smoothing be consistent. Then

2 F ( , ~ ) - 1 + ) 3

the set Q of real zeros of which is empty. Then the third condition of con- sistency gives that

; 1 [2da~(X)= ; (1 +),2)2da~(2)

_ ~ I F ( ~ ) - ~ 4

is finite. From this it follows that the ~-process has derived processes up to the second order. Let us put

(v) = 0 so that r (~) = 0, and choose

b l = 0 , and b o = - - b 2 - 1 . Then

2

1 - - F (~) {~b (~). + ~ b, (i 2)~} = 0.

r = 0

Further since Q is empty,

mx (Q) = 0, i.e., L2 (X) = L, ($).

Hence by theorem 6.1

d ~ ~ (t) X (t) = l ~ (t) __ 89 dt 2 Therefore the sequence of processes

{ X r ( t ) / - ~ f ~(t--v)q~r(v)dv}

constructed in 3. 3 converges in norm to the right side of (B).

(B)

483

K. NAGABHUSHANAM, The p r i m a r y process o f a smoothing relation

I t is clear t h a t we can generalize the smoothing relation itself into the form d ~ X ( t )

f +

- - c , O

and seek inversion in a similar form, viz.,

~ d r ~ ( t ) X ( t ) = ~ ( t - - v ) q ) ( v ) d v ' 'u

_~r ~=0 d t r

Then the conditions will read as i) L2 (X) = La (~) and

ii) f ] l - - { F ( 2 ) + ~ a , ( i 2 ) ' } {r + ~ b ~ ( i 2 ) ~ } ] 2 d a ~ ( 2 ) = O.

- - o o s = O r = 0

6 . 2 . Vector processes

Suppose

(Xr(t))

to be a set of k stationary processes such that L2 (Xr)• L2 (X,), r ~ s. Let k 2 smoothings (Lt~) be given, and let

L ~ r X r ( t ) = ~ ( t ) ,

the repeated index r standing for summation from 1 to k. Such a summation convention will be used in what follows. We now suppose the vector process (~l(t)) is made up of k stationary processes which are stationarily correlated with each other. The (~r(t))-process and the k 2 smoothing operators (Lrs) are assumed to be known. The,l the problem of inversion will consist in obtaining the primary vector process (Xr(t)). I t is not proposed to go into any details of this and related problems here, except to point out that t h e y are more or less analogous to those treated in this thesis. Corresponding to each smoothing operator Lrs we have as before a function F(~t) which we shall denote by F~,(2). The set of real zeros of F~s(2) is denoted by Q~,, while the set of real ), for which

det IF,, (~) ] = 0

is denoted by Q. Also with the smoothing operation Lrs we associate the constant [~a(~,)l, the sum of the weights of the smoothing, in the discrete para- j z meter case

Cr, = and

f/(,.)(u)du

oO ⁱⁿ the continuous parameter case, ],,(u) being the weight - ~ function used in the smoothing L r s . "~

ARKIV F61~ MATEMATIK. B d 1 nr 32 The determinant ]Or,[ is denoted b y A. We shall denote the k mean values of the given resulting vector process b y (M,r). Then the first condition of consistency of the smoothing vector relationship is seen to be that

if

(M#r)

is not the null vector, A =~ 0. (i) The method of 1.9 combined with the assumption of the orthogonality of the Hilbert spaces of the components of the primary vector process leads us to the relation

dZ~

(2) = Fs z ( 2 ) d Z x

Z

(2) (A) from which we have as the second condition of consistency that for each fixed s m~s (Q~i' Q~2 . . . Q~) = O, s = 1, 2 . . . k. (ii) (Note that s (being fixed in any equation) is not a summation symbol).

When ~ e W - - Q , we can solve the linear equations (A) and obtain

dZx~ (2) = O~, (2) dZ~ s

(2).

The third condition of consistency will be that W-Q

is composed of mutually orthogonal processes of finite norm.

When these conditions are satisfied, we obtain the primary process belonging to the Hilbert space of the given resulting vector process uniquely (in the sense of norm) as

Xr(O = f eU'~Grs(2)dZ~s().).

W - Q

6. 3 . C e r t a i n p r o c e s s e s o f b o u n d e d n o r m

Lastly let us consider the case where the resulting process ~(t)is assumed to be of bounded norm but not stationary, is of continuous parameter, and is adjusted to have its mean value function

M~(t)= 0

for all t. Further let this process h a v e KARHUNEN'S representation

oO

= f a)dZ(2).

~ o 0

We take the smoothing relation to be as before

oO

f X ( t - - u ) / ( u ) d u = ~(t).

- - OCJ

485

K . NAG_&BHUSHANAM,

The primary process of a smoothing relation

In our a t t e m p t s to recover the primary process we have in our previous work obtained b y use of Fourier transforms a linear relationship between the random spectral functions of the two processes from the given smoothing rela- tionship. As this is no longer possible, we shall specify the problem of inversion as follows :

to find

X(t)

of bounded norm and having the representation

O0

X(t) = f ~(t, 2)dZ(~).

- - o o

In this specification we have

Lz(X)=

L2(~ ). Also the mean value of

X(t)may

be taken to be zero for all t. We m u s t then have

] ~ (t, ~) ]2 d a (~) bounded. (i)

- - o 0

Also from KARItUNEI~'S theorem of 1.7, it is necessary t h a t

Then

so t h a t

~ ] f ~(t--u,~)/(u)du 2 d(~(2)

is bounded.

- - o 0 - - o o

(ii)

f =

- - o o - - o 0 - - c ~

= ~(t)

o o

= f fl(t, ]t)dZ(,~)

~ o o

f ] fl(t,,~)-- T ~ 2 d(~(~l)=0

(iii)

- - o 0 ~ o 0

for all t.

When (iii) furnishes a solution of a (t, ~t) which satisfies (i) and (ii), it follows t h a t

o O

X(t) = f o~(t,~)dZ(X)

~ o 0

exists as an element of L2 (Z) = L2 (~), and constitutes a solution of the p r i m a r y process.

Suppose further t h a t a function ~ (v) of the real variable v can be found such t h a t

f ?

- - r - - o o

for all t, and

is bounded.

ARKIV FOR MATEMATIK. B d 1 nr 32

Then it is seen t h a t

X(t) = ~ ~(t--v) cf(v)dv.

- - 0 0

A d d i t i o n a l n o t e

The object of appending this note is to make explicit what is meant by taking the expectation of the process obtained b y integrating a stationary pro- cess. Let us s t a r t by considering the case where the s t a t i o n a r y process X (t) has the mean value zero~ and the resulting process ~ (t)is given by the integral smoothing

~(t) = X(t--u)/(u)du = 1.i.m. f X ( t - - u ) / ( u ) d u .

_ _ r a - - - - ~ - - ~ a

b - ~ - ~

Then

c ~

~(t)= f eitaF(2)dZx(2),

where the random spectral function

Zx(s)

is such t h a t

E (Zx(s))=

0. I n this case

E(~(t)) = O.

We now turn our attention to the case in which the mean value of the process

X(t)

is a constant

Mx ~ O.

In this case we define the integral

a s

f x ( t - - u ) ] ( u ) d u

f ( X ( t - - u ) - - M x ) / ( u ) d u + f M x . / ( u ) d u ,

where the first of the integrals is the integral of a stationary process of mean value zero in the sense already explained, and the second one is an ordinary infinite integral. Thus in this case we have

~(t) = ~ eitaF(2)dZx(,t) + Mx . f l(u)du.

- - ~ - - c o

Taking the expectation of both sides we have

Mr = E (#(t)) = M x . f / (u) d u.

- - 0 0

487

K. NAGABHUSHANAM, The primary process of a smoothing relation

R E F E R E N C E S . C h a m p e r n o w n e , D. G . [1]: S a m p l i n g t h e o r y a p p l i e d to a u t o r e g r e s s i v e s e q u e n c e s , J . R o y . S t a t . Soc., Series B, Vol. 10, No. 2, 1948. - - C r a m ~ r , H . [1]: O n h a r m o n i c a n a l y s i s in c e r t a i n f u n c t i o n a l spaces, A r k i v f. M a t . A s t r . F y s . , 1942. - - - - , [2]:

P r o b l e m s in p r o b a b i l i t y t h e o r y , A n n a l s M a t h . S t a t . Vol. 18, 1947. - - D o o b , J . L . [I] : T i m e series a n d h a r m o n i c a n a l y s i s , P r o c . B e r k e l e y s y m . o n M a t h . Star. a n d P r o b . , [2]:

T h e e l e m e n t a r y G a u s s i a n processes, A n n a l s M a t h . S t a t . , Vol. 15, 1944. - - - - , [3] : S t o c h a s t i c processes d e p e n d i n g o n a c o n t i n u o u s p a r a m e t e r , T r a n s . A m e r . M a t h . Soc., 1937. - - F r i s c h , R . [1] : O n t h e i n v e r s i o n of a m o v i n g a v e r a g e , Sk. A k t . T i d s k r i f t , 1938. - - G t r s h i e k , M . A . a n d H a a v e l r n o , T . [1] : S t a t i s t i c a l a n a l y s i s of t h e d e m a n d for f o o d ; e x a m p l e s of s i m u l t a n e o u s e s t i m a t i o n of s t r u c t u r a l e q u a t i o n s , E c o n o m e t r i c a , Vol. 15, No. 2, A p r i l 1947. - - G r e n a n d e r , U . [1] : S t o c h a s t i c processes a n d s t a t i s t i c a l inference, A r k i v f. M a t . , 1950. - - H a r m e r , O . [1] : D e t e r m i n i s t i c a n d n o n d e t e r m i n i s t i c s t a t i o n a r y r a n d o m processes, A r k i v f. M a t . , 1 9 5 0 . - H o p f , E . [1] : E r g o d e n t h e o r i e , E r g e b n i s s e d e r M a t h . , Vol. 5, No. 2, 1937. - - H u r w i c z , L. [i]:

S t o c h a s t i c m o d e l s of e c o n o m i c f l u c t u a t i o n s , E c o n o m e t r i c a , Vol. 12, 1944. - - K a r h u n e n , K. [1]:

l~ber lineare M e t h o d e n in d e r W a h r s c h e i n l i c h k e i t s r e c h n u n g , A n n . Ac. Sci. F e n n i c a e , A I 37, H e l s i n k i , 1947. - - - - - - , [2] : ~ b e r die S t r u k t u r station/~rer zuf~lliger F u n k t i o n e n , A r k i v f. Mat., 1950. , [3]: L i n e a r e T r a n s f o r m a t i o n e n station/~rer s t o e h a s t i s c h e r Prozesse, D e n 10.

S k a n d i n a v i s k c M a t e m a t i k e r K o n g r e s i K ~ b e n h a v n , 1946. - - K h i n t c h i n e , A . [1]: K o r r e l a - t i o n s t h e o r i e d e r station/~ren s t o e h a s t i s c h c n Prozesse, M a t h . A n n . , 1934. - - K o l m o g o r o f f , A . [1] : G r u n d b e g r i f f e d e r W a h r s c h e i n l i c h k e i t s r e c l m u n g , E r g e b n i s s e d e r M a t h . , Vol. 2, No. 3, 1933.

- - K o o p m a n s , T J . [1]: Serial c o r r e l a t i o n a n d q u a d r a t i c f o r m s i n n o r m a l variables, A n n a l s M a t h . Stat., Vol. 13, 1942. - - M a n n , H . B . a n d W a l d , A . [1] : O n t h e s t a t i s t i c a l t r e a t m e n t of linear s t o c h a s t i c difference e q d a t i o n s , E c o n o m e t r l c a 11, 1943. - - M o r a n , P . A . P . [1]:

T h e oscillatory b e h a v i o u r of m o v i n g a v e r a g e s , Proc. C a m . Phil. Soc., Vol. 46, p a r t 2, A p r i l 1950. - - M o y a l , J . E , [1]: S t o c h a s t i c P r o c e s s e s a n d S t a t i s t i c a l P h y s i c s , S y m p o s i u m o n S t o c h a s t i c P r o c e s s e s , J . R . S t a t . Soc., Series B, Vol. 11, No. 2, 1949. - - O r e u t t , G . H . a n d C o e h r a n e , D . [1]: A s a m p l i n g s t u d y of t h e m e r i t s of a u t o r e g r e s s i v e a n d r e d u c e d f o r m t r a n s f o r m a t i o n s i n r e g r e s s i o n a n a l y s i s , J . A m . Star. A s s n . , vol. 44, 1949. - - P a l e y , R . a n d

W i e n e r , N . [1] : F o u r i e r t r a n s f o r m s i n t h e c o m p l e x d o m a i n , N e w Y o r k , 1934. - - P r e s t , A . R . [1]: N a t i o n a l i n c o m e of t h e U n i t e d K i n g d o m 1870--1946, T h e E c o n o m i c J o u r n a l , M a r c h 1948. - - S l u t s k y , E. [1]: T h e s u m m a t i o n of r a n d o m c a u s e s a s a source of cyclic processes, E c o n o m e t r i c a 5 {1937). - - W i e n e r , N . [1]: T h e e x t r a p o l a t i o n , i n t e r p o l a t i o n , a n d s m o o t h i n g of s t a t i o n a r y t i m e series w i t h e n g i n e e r i n g a p p l i c a t i o n s (Wiley). - - W o l d , H . [1]: A s t u d y in t h e a n a l y s i s of s t a t i o n a r y t i m e series, Thesis, S t o c k h o l m , 1 9 3 8 . - - - - , [2]: O n t h e i n v e r s i o n of m o v i n g a v e r a g e , Sk. A k t . T i d s k r i f t , 1938.

Tryckt den 13 april 1951

Uppsal~ 1951. Almqvist & Wiksells Boktryckeri AB

Dans le document The primary process of a smoothing relation By K. NAGABHUSHANAM (Page 61-68)