f i9/y ^(0)d
121 consistent estimator of the spectrum, so that the joint distribution
of the ordinates is changed and these results no longer apply.
Nevertheless, it would appear that the results of this section do have some potential for application as well as being of theoretical interest.
122.
CHAPTER 6
CONCERNING TWO MARKOV CHAINS WITH STATIONARY EXPONENTIAL DISTRIBUTION
6.1 INTRODUCTION
In this chapter we show that the stationary first order
exponential autoregressive process (EAR(l)) of Gaver and Lewis (1980) is the time-reversed version of a process in Tavares (1980) which has stationary exponential distribution, and that the exponential is characterized among absolutely continuous distribution functions as the only possible common distribution for such a pair of mutually time-reversed stationary Markov chains (MC's) as is specified by (6.1.1) and (6.1.2) . We write Exp(y) for an exponential
random variable (r.v.) with mean 1/y for some 0 < y < 00 . Gaver and Lewis (1980) studied first order autoregressive M C ’s {X } defined by
n
distributed (i.i.d) non-negative r.v.'s with independent
Exp(A) if and only if £ = I C , where the {l } are i.i.d.
^ J n n n n
Bernoulli r.v.'s with P{l =0} = P and the (C ) are i.i.d.
n n
Exp(A) r .v.'s .
Tavares (1980) considered the stationary MC {Yr} with sample paths satisfying the relation
(6.1.1)
where 0 < P < 1 and the {E^} are independent identically
of X , and showed that the stationary distribution of X^ is
Y = min(Y /p,n ),
n+1 n n (6.1.2)
where 0 < P < 1 and the (r) } are non-negative i.i.d. r.v.’s with r) independent of Y^ , and showed that when the { r W are
123.
E(A(l~p)) r.v.’s then is Exp(A) .
Henceforth {X } and {Y } will he understood to refer to
n n
the sequences defined by (6.1.1) and (6.1.2) together with
12 k .
6.2 MAIN RESULTS
Result 6.2.1 If {X } and {Y } both have stationary
--- n n
exponential distributions , then they are time-reversed versions of one another.
Derivation Since the distribution of a stationary exponential
MC corresponds to a bivariate exponential distribution, which is uniquely determined by its Laplace transform, it is sufficient to show that for all 0 , cp > 0
-<j>X -0X -0Y -<j>Y
E[e ° 1 ] = E[e ° l ] . (6.2.1)
We know from equation (4.2) of Gaver and Lewis that the left hand side is
pA , (l-p)A2 A+cj)+pG ( A+(})+p0) (A+0)
The right hand side is
E[exp-( GYo+4>min(Yo/p ,Gq )) ] X/fP -0x. - A x n , e Ae dx(, r'hrX(l-p)e-X(l-p)y
dy
'o Jo — (px/p — A( 1—p ) x/p \ • 0 0 / 9which simplifies to the above expression for the left hand side,
verifying (6.2.1) . The bivariate survivor function is then
P{Y > x,Y > y)
o 1
e-A(y+(x-py)+)
which may be compared with the bivariate exponential distribution of Marshall and Olkin (1967).
125.
R e s u l t 6 . 2 . 2 I f t he s t a t i o n a r y MC's {X } a n d {Y } h a v e a
--- n n
c ommon a b s o l u t e l y c o n t i n u o u s d i s t r i b u t i o n f u nction and are m u t u a l l y
time r e v e r s e d v e r s i o n s o f one a n other, then t h e i r common
d i s t r i b u t i o n is n e c e s s a r i l y exponential. D e r i v a t i o n An e q u i v a l e n t r e p r e s e n t a t i o n of (6.1.2) is Y n p Y n + l +
(V PV
a n d since we r e q u i r e (Y -pp ) to be i n d e p e n d e n t of pY o u r n n + n+1 re s u l t follows as an i m m e d i a t e c o n s e q u e n c e of the f o l l o w i n g m o r egen e r a l r e sult, w h i c h is in the s p irit o f G a lambos (1972) (see
also §3-3 o f G a l a m b o s a n d K o t z (1978)).
t
L e m m a Let i n d e p e n d e n t n o n - n e g a t i v e r.v.'s U a n d V h ave
a b s o l u t e l y c o n t i n u o u s d.f.'s. T h e n m i n(U,V) a n d (U-V) are
i n d e p e n d e n t i f a n d o n l y if U a n d V are e x p o n e n t i a l l y distributed. P r o o f o f L e m m a D e n o t e t he d.f.'s of U a n d V b y A( *) and B (*), a n d t h e i r R a d o n - N i k o d y m d e r i v a t i v e s b y a( •) a nd b( * ) , r e s p e c t i v e l y . T h e n F( x , y ) = P { m i n ( U , V ) < x , (U-V)+ <y) x+y ( l - B ( u - y ) ) d A ( u ) + ( B ( x ) - B ( u - y ) ) dA(u) x+y
A(x) - B (u-y ) dA (u ) + B ( x ) ( A ( x + y ) _ A(x) ) .
y
By the i n d e p e n d e n c e c o n d i t i o n
F(x,y) = F(x,°°)F (°°,y)
= (l-(l-A(x)Xl - B ( x ) )) (l- B (u - y ) dA (u )) , y
126.