I
I.
I
I
I
I
I
I
I
REMARKS ON RENEWAL THEORY WHEN THE QUALITY
OF RENEWALS VARIES
by
Walter L. Smith
University of North Carolina
Institute of Statistics Mimeo Series No. 548
September 1967
aI
I
I
I
I
I
I e,
I
I
This research was supported by the Office
of Naval Research under Contract Number
Nonr 855(09)
DEPARTMENT OF STATISTICS
University of North Carolina
Ch~pe1
Hill, N. C.
I
II
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
•I
1.
Introduction.
In spite of its simple "raw ingredients," the theory of renewal pro-
cesses has produced many very subtle mathematical investigations and supports
a very elaborate structure of generalizations, extensions, and applications.
The "raw ingredients" to which we refer consist of an infinite sequence of
independent and identically distributed, non-negative random variables {X };
n
in the original renewal theory these random variables represent lifetimes of
successive renewals of some industrial item subject to failure.
installed at the initial instant (t
= 0)
second item is renewed in turn at time t
= Xl;
and renewed at time t
= 52 = Xl
An item is
+ X2 ; and so on.
this
Thus
the nth renewal occurs at time t = 5 = X +...+ X and the number of ren
1
n
newals to occur by time t is the maximum integer k such that 5
k
< t.
In actual applications the assumption that the {X } are identically
n
distributed may be unrealistic; there may be steady technical improvements
in the quality of the item; there may be a steady diminution in those forces
(friction, for instance) which bring about the failure of the item.
It
therefore seems desirable to explore the consequences of various assumptions
that could be made to allow of progressive improvement (or degradation) in
the quality of renewals.
A large number of different models are worthy of study in the present
context, but in this paper we shall be primarily concerned with the model we
shall call the model of sequentially geometric improvement (s.g.i.).
Let
Vex) be a fixed non-degenerate distribution function (d.f.) of a non-negative
random variable and let A > 1 be some finite constant.
The model s.g.i •
I
I.
.,
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
••
I
2
supposes that the {X } are still independent but that P {X < x} • V(x/ An) •
n
n An equivalent, but perhaps more vivid, formulation is given in the following
specification.
Model A
Let {Yn} be an infinite sequence of independent and identi-
cally distributed random variables with d.f. Vex). assumed non-deaenerate.
n
and suppose that Xn • A Y.
n
n· 1.2 ••••• for some constant A > 1.
In studying this model a certain series plays an important role.
This series is
Yl
(1.1)
Y2
Y3
Z·T+~+~+
It is important to know when this series converges almost surely; we shall
find that a necessary and sufficient condition for this is that
ClO
I
1 - Vex) dx
x
1
(1.2)
<
ClO
Notice that (1.2) is equivalent to the condition E log+y
n
<
ClO.
When (1.2)
is satisfied it follows that Z is a proper random variable with a nondefective d.f. K(x). say.
In these circumstances it is not hard to see that
a renewal process satisfying the requirements of Model A will necessarily
satisfy the reqUirements of the more general model below:
Model B Let there be a non-degenerate and non-defective d.f. K(x)
and suppose that, as n
~
ClO
•
(1.3)
at all continuity points.
We shall see that when Model B holds the d.f. K(x) must be the d.f •
of a random variable Z given by a series such as (1.1), the random variables
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
••
I
3
{Yn } of this series must have a d.f. V(x) satisfying (1.2); we shall also
see that the d.f. K(x) must be continuous (it may be singular of course).
In our work» and in stating our main result (Theorem 1) below» it is
necessary to make use of the following notation.
For all t
n(t) to be the least integer not less than 10gAt.
d(t)
=
n(t) - 10gAt.
Theorem 1
>
1 we define
We then define
We shall prove:
Let N(t) be the number of renewals to occur by time t» including
Suppose the renewal process satisfies Model B.
any that occur at time t.
Then (a) for any integer value of j (positive, zero, or negative),
(1.4)
P{N(t) - n(t)
j} - K (A-j-d(t»
>
-+
as t
0,
-+
00.
(b) for any integer value of v > 1,
ffi!.tl.
E ~(t)
(1.5)
-
,1 v
~
-+
as
0,
(c) if there is a d.f. W(x) < PiX
n
if, for some integer v
(1. 6)
~
t -+
00.
< AnX} for all
n
1»
oo
J1
(log x)v [1 _ W(x)] dx < 00
x
it follows that
(1. 7)
where
I[N(t) - n(t)]v - R (d(t»
V
-+
0,
as t
-+
00»
and all x, and
I
I.
.
,
I
I
I
I
I
I
I_
4
Concerning Model A we shall prove the following •
Theorem 2
If the renewal process satisfies the requirements of Model A and
if (1.2) holds. then parts (a) and (b) of Theorem 1 again hold.
If the d.f.
Vex) is such that for integer v > 1
I~ (logx x)v
(1.9)
[1 - V(x)]dx <
~
1
then part (c) of Theorem 1 also applies.
A feature of these results, at first surprising, is that, under
suitable conditions, the distribution and moments of N(t) - 10gAt do not
approach a strict limit as t
-+ ~
periodic function of 10gAt.
An important special case of our results is
that, as t
-+
but approach more and more closely a
~,
(1.10)
I
I
where
I
In more familiar problems in renewal theory such periodic behavior of the
I
I
I
I
••
I
(1.11)
renewal function tN(t) is a consequence of a lattice-structure of the variabIes {X}.
n
In the present problem no such lattice-structure need be in-
volved.
Of course, before the present results can be applied, the computation of the remainder functions R (d) is necessary.
v
Whilst this does not
seem to be a difficult numerical matter, a theoretical study of these functions is desirable; time and space compel us to postpone this study, however •
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
I·
I
5
The methods of this paper will deal with more general set-ups.
For
instance one might imagine a more general version of Model A in which
n • 1,2,·",
for some constant y.
Only an increase in algebraic complexity is involved;
no changes in the essential argument are needed.
Some results along the present lines were presented by the author in
an invited paper, presented at the meeting of the Society for Industrial and
Applied Mathematics in November 1965 at Seattle, Washington, entitled "On
renewal theory with improving or degrading quality of replacements."
The
results then available were much less complete than the ones now given.
2.
Consequences of Model A.
It will be most convenient to prove Theorem 2 first, and then in the
next section we shall discuss modifications of our argument which are necessary for the establishment of Theorem 1.
In this section we shall therefore
suppose Model A applies, and we begin by discussing the relationship between
the distribution functions K(x) and V(x).
Stieltjes transform of a d.f. V(x) thus:
Lemma 2.1
The d.f. K(x)
= P{Z
~
As usual we denote the Laplace -
V*(s).
x}, where Z is defined by (1.1), is non-
defective if and only if (1.2) holds.
Proof
By reference to the zero-one law, we see that convergence of (1.1) is
00
almost sure if and only if the infinite product TIl V*(A
does not diverge to zero).
This occurs if and only if
00
L
1
{I - V*(A
-n
)} <
00,
-n
) converges (i.e.
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
I·
I
6
which is equivalent to
JWo W
(x) {I
0
- V(x)}dx < ~
where
But it can be shown (by comparison with suitable integrals) that for some
finite constants A2
> Al > 0,
and for all x
> 0,
Thus the lemma is proved.
Lemma 2.2
If Vex) and K(x) are the distribution functions already defined
then for v = 0,1,2,"',
J
~l (log x) v . {I _ K(x)}dx < ~
x
if and only if
f
OOl
Proof
(log
- - xx)- v+l {I - V(x)}dx
< ~
Let {Z } be an infinite sequence of independent and identically disn
tributed variables with d.f. K(x).
£ log+Zn
< ~
Then we see from Lemma 2.1 that
if and only if E Z A-n converges almost surely; this is equivan
lent to the almost sure convergence of
...
I
I.
I
I
I
I
I
I
I
Ie
I
I
I
I
I
I
I
.I
7
Pursuing the same argument as in the proof of Lemma 2.1 leads us to the
condition
~ ~l(x){l - V(x)}dx
o
<
00
where
However, there will be constants B2
B1 (1 + log+x)
(1
Thus
t
log+Z
n
<
00
+
x)
>
Bl
>
0 such that, for all x
>
0,
B2 (1 + log+x)
< ~l(x) <
(1+ x)
if and only if
00
[
1
(log x){l _ V(x)}dx <
x
or equivalently, if and only if S(log+Y )2 <
n
00
'
00.
This argument can be continued sequentially to prove the lemma.
example, we have that t(log+Z )2
n
converges almost surely.
of
<
00
For
if and only if
This is equivalent to the almost sure convergence
I
I.
I
I
I
I
I
I
I
Ie
8
where there are n(n+l)/2
y's in the numerator of the nth term.
to requiring
J:
~2(x){1 - V(x)}dx < ~
where
One can show that, for some constants C2
+
Cl(l + log x)
(1
+
2
x)
'
••
I
>
Cl
0, and for all x
>
+
C2 (1 + log x)
< ~2(x) <
(1 + x)
>
0,
2
Thus the lemma is proved.
Theorem 3
Proof
If Vex) is not degenerate then K(x) must be continuous.
Let a p
a 2 ,··· be distinct points such that P{Y = a } > 0 and let
i
n
be the set of all these a-points.
I
I
I
I
I
I
I
This leads
It is not hard to see that if
p{y £~} < 1 then K(x) must be continuous.
n
Thus we may suppose that
p{y £,} = 1 and hence thatJ) has at least two members.
n
Suppose there is a 8 such that p{Z = 8} = b
>
O.
Then
Y
n
--=
Thus we can find a finite m
m
L
i=l
p{Z
>
2 such that
Y
ai .
n
8 - -}p{y
= ail = c
--=
An
n
An
In what follows we shall let a
>
o.
1 < i < m, and a = a l •
m
i = a i +l ,
min
p{y = ail
n
= ail
n
iii = l<i<m p{y
,
Let
~
I
I.
I
I
I
I
I
I
I
I_
9
then
w>
0 and
y
n
--=
>
Thus
we.
(2.1)
Let there be k distinct values taken by the numbers (a
m.
Then k
~
which is zero.
I
I
I
I
I
I.
I
I
i
- a ) for i
i
= 1,2,
2, and we write Y , Y2 , ••• , Yk for these values, none of
1
Then (2.1) implies (if we make due allowance for possibly
repeated terms) that
(2.2)
V
Suppose we choose the integer v so that A exceeds every number IYi/Y
also that A-V is less than every such number.
I
we.
j
I and
Then, if rand s are integers,
the equation
implies that Iyi/yjl
= A(r-s)v,
and this requires that r
Let"')'r be the set of points
Yl
Y2
A
A
Q+...
rv' Q+...
rv'
Then (2.2) shows
=s
and that i
= j.
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I.
I
I
10
and we have that tl' j(2' ... are disjoint sets.
conclusion P{Ze:L l'r1 =
This leads to the impossible
Thus the theorem is proved.
co.
We are now ready to prove Theorem 2.
In the discussion that follows
we shall, as already stated, write net) for the least integer not exceeding
log t and define d(t)
net) - log t, so that 0
=
~
d(t) < 1.
We shall be
first concerned with proving some lemmas about the following functions of n
and d; we define
W (d)
n
(n-l) (
)
n-d
j V-I ~{Xl + X + •••+ X
> A
1.
1
2
n-j -
T (d) = j-_L
n
For large integer C
>
0 we also need the following multilations of the above
functions.
C
Wn (d)
C
T (d)
n
Lemma 2.3
co
= j~C
j
(n-l)
(v-I)·
P{X 1 + X2 + ••• + Xn+ j
(v-I)
= j~C
j
<
P{X + X2 + •••+ X
> An-dl.
1
n-j -
On the sole assumption that V(Ot) < 1 we can, for any e: > 0,
choose large enough C so that
for all large n, 0 < d < 1.
Proof
If L1(n- j ) Xi ~ An-d then Xi ~ An-d for every i = 1,2,···, n+j.
Y ~ A(n-d-i) for every i
i
(2.3)
= 1,2,···,
n+j.
Therefore
(n+j)
p{X + X + ••• + X
< An-dl <
II
V(An- d- i ).
1
2
n+j
i=1
Thus
I
I.
I
I
I
I
I
I
I
I_
11
Now there must be constants
Furthermore
,(n-d-i'
~ ~
A
~
>
0, n
>
0, such that V(x) < n < 1 if x <
if and only if i
>
n - d -
10gA~'
~.
Thus, for large j,
and so
< e:
Lemma 2.4
, for large C.
Under the assumption that V(O+) < 1 and that (1.2) holds, as
n -+ "",
~--
W (d)
-+
n
W (d) _
""
r
j(V-l) K(A-(j+d»
j-l
uniformly with respect to d, 0 < d< 1, and the limit W",,(d) is necessarily
finite.
I
I
I
I
I
I
I.
I
I
Proof
That Wn(d)
-+
W",,(d) as n
-+ "",
follows from the fact that
which is a consequence of Lemma 2.3 and the fact that Model A implies
Model B when (1.2) holds.
of d
>
O.
Lemma 2.5
Hence the uniformity follows.
If Model B holds (and, in particular, K(x) is non-defective),
.then for any e:
(2.4)
The functions Wn (d) are non-increasing functions
>
0 we can choose C so large that
lim
sup
n-+oo
T~ (c3)
nV
< e:
'I
I.
I
I
I
I
I
I
I
I_
12
Proof
n-d
> A
I.
I
I
}.
Thus
< e:
for all 0 < d < 1, if C is large enough.
Lemma 2.6
If Model A holds, with l(log+y )V+l
n
<
~, then for any e:
>
0
we can choose C so large that
(2.5)
< e:
for all 0
Proof
I
I
I
I
I
I
Evidently,
~
d
~
1, and all large n.
We have that , if Sn = X1 + •••+ Xn'
P{Sn_j ~ An- d }
= P{AYI
Y
+ •••+ A(n-j) Y n j
= p{_1 +
A
< 1 _
Y
2
~
+ ••• +
A
>
An~d}
Y
(n-j) > Aj-1-d}
A(n-j)
K(A(j-l-d».
Thus
The fact that the right-hand member can be made small by choice of C follows
from the convergence of
f~1
(l°xg x) (V-I, )
[1 - K(x»)dx
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I.
I
I
13
which is assured by our assumptions and Lemma 2.2.
Lemma 2.7
(i) Under the assumption that Model B holds, as n
(ii)
If Model A holds with t(log+ y )(V+1)
n
<
~ ~,
~, then as n ~ ~,
E
T (d) ~ T_(d) j(V-1) [1 _ K(A(j-d»]
n
j=1
uniformly with respect to d, 0 $ d < 1, and the limit
T~(d)
is necessarily
finite.
Proof
Part (i) follows easily from Lemma 2.5.
Part (ii) follows from
Lemma 2.6 and the fact that Model A implies Model B in the same way as
Lemma 2.4 was proved.
functions of d.
Note that the functions T (d) are non-decreasing
n
Let us now turn to the proof of Theorem 2.
[j
v
We have that
--v
- j-l ]P{Sn(t)+j ~ t}
n(t)-l
- j~1
= S1 n (d)
[j v - j-l'1P{Sn(t)_j > t}
- S2n (d), say.
Let us first suppose only that Model B holds.
Lemma 2.4 and Lemma 2.7 (i) that
v
I[N(t) - n(t)J ~ 0
[n(t)]V
.
Then it follows easily from
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I.
I
I
14
or, in other words,
lli1tl.
; tEet)
-:J~V
This is the first part of the theorem.
+
v+l
e(log Y)
<
n
<lO
-+
o.
Let us now suppose that
for some integer v > 1.
Then
it
follows from Lemma 2.4
and Lemma 2.7(ii) that
e'[N(t) - n(t)]v
-+
j!l [jv - j_Iv]KO.-(j+d»
E
j=l
as t
-+
00
[J'v - j-l v][l _
through a sequence which keeps d(t)
=d
fixed.
K(A j - d )]
However, since
the convergence is uniform with respect to d, we can infer that
t[N(t) - net)]
as t
-+
00
v
- Rv(d(t»
-+
0
continuously, where
This completes the proof.
3.
Consequences of Model B.
In this section we shall discuss the modifications necessary to be
made in the arguments of the previous section in order to establish Theorem 1
under the hypothesis that Model B holds (with, be it noted, K(x) nondefective).
Let us write Gn (x) for the d.f. of X.
n
Then, from (1.3),
I
I.
I
I
I
I
I
I
I
15
Thus
and so
G*[_s,_]
n An+ l
Since V*(s)
-+
I.
I
I
K*(s)
= V*(!i)
K*(f)
A
, say.
1 as s+O+, it follows that V*(s) is the Laplace-Stieltjes
transform of a non-defective d.f. V(x), and
n l
G (A + x)
n
(3.1)
-+
V(AX) ,
as n
-+
00,
at continuity points.
Further,
I_
I
I
I
I
I
I
-+
~ V*[~j]
j=l
1\
=
Therefore
K* (s),
n
since K*(S/A )
-+
1 as n ~
00.
From the proof of Lemma 2.1 it then follows
that V(x) must satisfy (1.2) if K(x) is to be non-defective.
Suppose we now introduce the "domination" assumption
W(x) < G (An x)
n
(3.2)
where W(x) is such that (1.6) holds for some integer v > 1.
tions imply that V(x) > W(x) for all x and so that
(3.3)
J (logx x)
OOl
v [1 _ V(x)]dx <
00.
These assump-
I
I.
I
I
I
I
I
I
I
I_
16
From Lemma 2.2 we deduce that
converges, and from Theorem 3 that K(x) is a continuous function of x.
The proof of Theorem I now can be delineated.
In order to save
space we shall only list the changes necessary in the arguments of the last
section.
In the proof of Lemma 2.3 we replace (2.3) by
n+j
An - d }
<
1o__TIl Go(A n - d ).
1
In view of (3.2) we have, for an arbitrary
£
>
pix +...+ X
<
n+j -
1
for all sufficiently large n; we use
Lemma 2.3.
~
n 1
0, Gn(A + ~) ~ (n +
£)
and n here as in the proof of
The rest of that proof goes through as before with n replaced
by (n + E).
The proofs of Lemmas 2.4, 2.5, and 2.7 need no change, nor does the
I
I
I
I
I
I
I.
I
I
final argument at the close of Section 2 in which the theorem is established.
All that remains is to comment upon Lemma 2.6.
Let
{Yn } be
an infinite sequence of independent and identically
distributed random variables with distribution function W(x) and let K(x)
be the distribution function of
Then Lemma 2.6 applies to a Model A based upon the {y}.
n
n
But the domination
assumption (3.2) implies G (x) > W(X/A ) for all x, and hence that
n
-
n-j
A
I
I.
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I.
I
I
17
This observation should be sufficient to show that
(n-1)
L
j=C
P{X +...+ X
1
n-j
>
n-d
A
}
can be made arbitrarily small for all large n by choice of C.