Page 253 - Contributed Paper Session (CPS) - Volume 1

CPS1304 Manik A.
and {Mt} an i.i.d. geometric sequence with
[ = ] = (1 − ) , j ≥ 0.

Then, the marginal distribution of { } is given by

[ = ] = (1 − ) , ≥ 0, 0 < < 1.

The probability distribution of is

(1 − + ), = 0
( = ) = { (1 − )(1 − ) ≥ 1.

The marginal mean, variance and pgf of { } are

( ) = /(1 − ),

2
( ) = /(1 − )

and
1
Φ () = (1 − ) (1 − )⁄ , || < .

The conditional mean and the variance are

( | ) = + (1 − )

−
−
1 −
and
1
( | − ) = (1 − ) − + (1 − )(1 + ) (1 − ) 2 .

The conditional pmf (analogous to conditional pmf in GINAR(1)) is given by

− − −+1
(1 − ) ∑ ( ) (1 − )

( = | − = ) = =0 (2)
+ ( ) +1 (1 − ) − , ≤ ,

{(1 − ) − (1 − ){ + (1 − )} , > .

The conditional pgf is given by,

1 − + (1 − )

| () = (1 − + ) ( ) , = [ ]. (3)
−
1 −
+ −

From (3) it can be shown that,
1−
| () → () = as → ∞.
+ − 1−

If { } is a process defined as in (1), then the autocorrelation function of the

process { } is given by

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