Markov renewal processes
123
(limsup{ : ( ) 0}) (liminf{ : ( ) 0}) 1.
nn
n
n
PS PS
ωωω
=>= (19.9)
We then get the following theorem concerning the asymptotic behaviour of the
semi-Markov random walk (S
n
).
Proposition 19.1 If the semi-Markov random walk (S
n
) has an irreducible M.C.
and all the unconditional means
,
i
iI
are finite and, then if
is null and if for
one j,
()
1
(0)1,
j
PU
<
(19.10)
then the semi-Markov random walk is oscillating and if
is positive (resp.
negative) then the semi-Markov random walk drifts to
∞ (resp. )
∞ .
20 DISTRIBUTION OF THE SUPREMUM FOR
SEMI-MARKOV RANDOM WALKS
Let us consider a semi-Markov random walk ( )
n
S with an irreducible M.C. and
all the unconditional means ,
i
iI
finite. We are now interested in the
distribution of the following supremum:
01
sup{ , ,...}.MSS
(20.1)
For
>0, under the assumptions of Proposition 19.1, it follows from this
proposition that for all i of I and all real x:
0
()0.PM xJ i
== (20.2)
This is also true for
=0, as the positive (J,X) process (( , ), 0)
nn
Hn
> is regular
(see Pyke (1961a)) meaning that it has only a finite number of transitions on any
time interval.
Now for
<0, from relation (5.21), we get:
0
() ( ) (1 ) (),
ijij
j
xPMxJi Mx
υ
=≤==−
(20.3)
where
ij
M
⎤
=
⎦
M
is the matrix of renewal functions for the process
(( , ), 0)
nn
Hn
> .
From
Proposition 7.2 of Chapter 5 of Janssen and Manca (2006), we know that:
lim ( ) 1, .
i
x
xiI
→∞
∀∈ (20.4)
We also see that
(0) 1 , .
ii
iI
−∀∈
(20.5)
Nevertheless, to be useful, the “explicit” form (8.3) requires us to know the
kernel of the positive (J,X) process (( , ), 0)
nn
Hn
> or the functions H
ij
given by
relation (7.8). Unfortunately, this is very difficult except in very particular cases.