* This research was supported by the Office of Naval Research under
Contract N00014-67-A-0321-0002.
Reproduction in whole or in part is permitted
for any purpose of the
United States Government
ON THE TAILS OF QUEUEING-TIME DISTRIBUTIONS*
Hal ter L. Smith
Depax'tment of Statistics
University of North Carolina at ChapeZ HiU
Institute of Statistics Mimeo Series No. 830
June, 1972
ON THE TAILS OF QUEUEING-TIME DISTRIBUTIONS*
~Jal
ter L. Smith
University of North Carolina at Chapel. BiZZ
1.
INTRODUCTION
This paper grew from a desire to examine the relationship, in the familiar
GIG/I
queue, between the "tail probabilities ll of large service-times and
the "tail probabilities" of large queueing-times.
The investigation was led
to a class of functions we call of moderate growth and an important sub-class
of these we call sub-exponential. tail functions.
A significant part of this
paper is necessarily devoted to a study of these classes of functions.
The
rest of the paper then establishes the validity of some very simple, but
quite general, relationships between the probabilities of large values of
~
service-times, ladder variables (see below), and queueing-times.
We shall write
V for a typical service-time, with d.f.
for a typical inter-arrival time, with d.f.
pose that
U and
IEXI
by
X = V-U,
write
F(x)
Eu
Q(x)
>
Ev
(so that the
for the d.£. of
X and
As we shall often be considering "tail probabilities ll ,
\1.
we shall adopt the convenient notation exemplified thus:
If
U
We shall, moreover, sup-
V have finite expectations, that
queue is "stable ll ) , and we set
denote
A(x).
B(x), and
c
B (x)
= l-B(x).
be the stationary d.f. of the queueing-time, then it is well-
known (Lindley (1952»
that
oo
(1.1)
Q(x)
=
I
F(x-z) Q(dz),
x > O.
0-
However, this integral equation is frequently intractable, in spite of the
*
This research was supported by the Office of Naval Research under
Contract N00014-67-A-032l-0002.
2
existence of "theoretical" solutions.
It is gratifying, therefore, to find,
as we shall show in this paper, that for a wide and interesting class of
GIGII
queues there is a very simple asymptotic relation between
c
B (x);
QC(x)
and
for these cases
oo
1
~;
(1. 2)
c
B (u)du,
f
as
x
-+
00.
x
Before we can be more precise about the conditions under which (1.2)
holds we must discuss one or two technical matters.
function
g(x)
defined on
x
~
We shall say that a
is a function of modepate gpowth (fmg) if
0,
it is strictly positive and if, for every fixed real
functions will be discussed in some detail in §2.
and only if
g(log x)
a, g(x)
A function
~
g(x+a).
g(x)
These
is fmg if
is a function of slow growth, a class of functions long
familiar since their principle properties were laid bare in the classical
papers of Karamata (1930a), (1930b) (1933).
g(x)
We shall see that a non-decreasing
is fmg if and cnly if
(1.3)
g(x)
where
a(u)
~
f:
g(O) exp
a(u)du,
as
x
-+
00,
is a bounded non-negative function and
If, in (1.3),
a(u)
then we shall say
g(x)
a(u)
-+
0
as
can be chosen so as to be non-increasing
u
-+
00.
(a(u)
+ 0),
is a moment function OfF); the pecippocal of a MF
will be caZled a tail function (TF).
Occasionally we find it necessary to impose some restraint on the rate
of growth of a fmg.
(1.4)
A(x)
=
and we shall say
(1.5)
f:
It is convenient to write
a(u)du,
is sub-exponential (SE) if
g(x)
lim sup A(2x)
A (x)
x -+ 00
<
2
•
e"
3
Feller (1971) makes great use of functions he calls functions of reguZar
variation; we refer to his book for details.
However, we shall see below that
any function of regular variation, with non-zero index, is asymptotically
equal to a monotone such function.
It is then an easy matter to show that any
of the functions of regular variation, with strictly positive index, is asymptotically equal to a SEMF.
is a SEMF.
If
On the other hand,
Let us say
<
S
<
1,
a > 0,
exp{x/log x}
is 8ubZinear if
T(X)
is non-increasing.
0
In much of this
T(~)
the function
exp axS
is a MF, but is not SE.
is non-decreasing and
T(X)/X
Gub-exponential tail functions
p~pe~
It is shown in §2, amongst other things, that
(SETF) play an important role.
any tail integral of the form
T(x)
= foo
T(U) du
uY
x
is necessarily a SETF.
The role of the "ladder variable", introduced by Blackwell (1953), in
queueing theory is now well understood; see, e.g. Prabhu (1967).
It is nec-
essary to present a brief synopsis of this theory here to acquaint the reader
Let
with our present notation and viewpoint.
{U}, {V}
n
be independent in-
n
finite sequences of iid inter-arrival times, service-times, respectively.
Let
X
n
~
= Vn-U n ,
the event:
= 0,
So
{Sl
~
Sn
0, S2
an improper random variable
N
l
=
if no
00
P{V > U}
11"
= P{N I
that
n
-1
=0
0, ••• , Sk_l
N
l
by setting
~
for
so we shall always suppose
-+-jJ
then
11"
>
O.
Furthermore
n
= 1,2, •.• ,
0, Sk > OJ.
N1
~ happens (for any finite k).
< co},
S
n
~
= Xl +X 2+.••+Xn
=k
if
and denote by
Then we can define
~ happens and
Our problem is banal if
P{V > U} > 0;
EXn = Ev n-Eun
thus, if we write
< 0,
which implies
almost surely; from this it is easy to deduce that
and is the stationary probability a customer has to queue.
(1956) has shown that
11"
< 1
Although Spitzer
4
=
n
ex p{-
I
! pes
n=l n
n
~O]},
this formula is difficult to apply and there seem at present to be no
straightforward procedures of generaZ appZication that lead to useful bounds
n,
on
let alone precise evaluations, except when arrivals are at random
(and then
n = p,
N
l
If
we shall set
~
<
as is well-known).
is the first "ladder" var-
iable.
In his pioneering paper Lindley (1952) showed that
Q(x)
(1.6)
=
S
max
P{
n
O~n<~
~
x}
and this relation, rather than the integral equation (1.1), is the more useful in the present investigation.
then
walk
{S}
n
P(A) = l-n.
If
~
is plainly zero.
at Zeast
Let
denote the event complimentary to
A
~
AOC> happens then the maximum of the random
But i f
L~ Ak
happens the maximum of
To study. the further development of
{S }
n
beyond
n
we can,
SN
1
{S -SN }, n > Nl ,
n
1
n > N such that
l
because of the independence involved, treat the walk
a newly-started walk.
Either there is no finite
o (an event with probability
(l-n» ,
{S },
n
for
ladder variables
j
~
0,
= SN
-SN • In short, the random
2 1
n ~ 0, will produce a random integer J, say, and J iid
j
Zl,Z2"",ZJ' It is easy to see that P{J=j} = (l-n)n ,
Z2
and that the maximum of the walk
(taken as zero if
Let us write
occasionally write
evidently have
as
or there is a least integer
we can define a second ladder variable
walk
is
{S}
{S}
n
J = 0).
Lr(X)
L (x)
l
for the d.f. of
more simply as
Zl+Z2+",+Zr'
L(x).
although we shall
Then, for
x > 0,
we
5
(1. 7)
Q(x)
=
I""
(l-Tr) +
(l-Tr)Tr
r
L (x).
r
r=l
This (well-known) equation is vital to our arguments in this paper.
show in §3 that if either
l-Tr
J.1Tr
LC(x)
or
I""Bc(u)du
x
J"" BC(u)du '
as
x
is a
We shall
fmg then
-+ "".
X
I "" BC(u)du
x
Then in §4, we shall deduce from (1. 7) that, i f either LC(x)
LC(x)
Conversely, i f (1. 8) holds, then both
and
must be fmg.
or
QC(x)
is
a SETF,
(1.9)
LC(x)
l:Tr QC(x),
as
x
-+ "".
We shall also show that if (1.9) holds then
LC(x)
and
QC(x)
sarily fmg (though we cannot show they are necessarily SETF).
at this point that (1.2) will hold if either
QC(x)
or
I
00
x
are necesIt is apparent
C
B (u)du
is a SETF
and that if (1.2) holds then both these functions are necessarily fmg.
From the applied viewpoint it is desirable to have usable conditions to
ensure that
I"" BC(u)du
x
condition is that
BC(x)
is a SETF.
In §2 we shall see that one sufficient
be a function of regular variation with strictly
1""0 BC (u)du
negative index ( in fact the index must be strictly negative, if
is to exist).
for
y > 1,
A more general sufficient condition is that
where
T.(x)
BC(x) "" T.(x)/x Y,
is sub-linear (as described earlier in this section).
Another sufficient condition, somewhat wider in scope, is that
BC(x)
Tail Function for which
is the func-
xa(x)
is non-decreasing (where
a(x)
tion introduced above in the discussion of "moderate growth").
Theorem 2.5, prove this condition and provide the estimate
\-le
be a
shall, in
6
A = limx~ xa(x)
in which
and
By way of an easy example:
= a/x I-a ,
a(x)
A
= co,
A/(>"-l)
is interpreted as unity if
x
Bc(x) ~ e-
suppose
A
= co.
a
for
O<a<l.
Then
and we infer
(I-a)
a
x ' - - - e -x
}.I a
In the final section (§5) of this paper, we look at the question of moments of a fairly general kind.
f
co
O S(x)Q(dx)
For instance, if a certain integral
must converge, what condition has to be met by
B(x)?
This
question has been discussed in a fairly general way by Kiefer and Wolfowitz
(1956).
Here we introduce even more general moments and also examine the
associated matter of moments of
certain conditions on
f
co
O SJ(x)B(dx)
S,
f
co
O
L(x).
The main result in §5 is that, under
S(x)Q(dx)
converges, where
SJ(x)
converges if and only if
that it be a non-decreasing fmg such that
x
~
0,
~
y
x
= f O S(u)du.
S(x+y)
~
The conditions on
S(x)S(y)
S are
for all
O.
2. FUNCTIONS OF MODERATE GROWTH
Although it is possible to infer some relevant results from the work of
Karamata (1930a), (1930b), (1933) on functions of slow growth, useful insight
will be gained by developing the basic theory of functions of moderate growth
ab initio.
Fortunately we need only concern ourselves with monotone func-
tions, and this restriction helps avoid much difficulty.
Let
g(x)
a
(2.1)
Then
e n
a
n
~
=
0,
be a non-decreasing function of moderate growth, and set
g(n)/g(n-l),
a
n
+
g(n)
0
as
=
n
n
= 1,2, ••••
+ co,
and
n
= 1,2, . . . •
7
Let us now define, for
= 1,2, ••• ,
n
=
a(u)
(n-l) ::; u < n.
a ,
n
Then
=
g(n)
Moreover, if
:;;;
(n-l) ::; x
f:-
g(O) exp
a(u)du.
n,
1
a(u)du
J:
::;
and so
f:
::;
a(u)du
a(u)du
x
fOa(u)du
::;
g(n-l)
g(n-l)
But
f:
~
g(x)
~
g(n)
g(O) e
and
~
g(n)
g(n).
~
g(n-l)
as
n
+
00.
Thus
x
(2.2)
as
g(x)
x
+
g(O) exp
with
00,
a(u)
~
Jo a(u)du
0,
and
a(u)
+
0 as
u
~
00.
This relation (2.2)
is what we principally wished to demonstrate, but it might be noticed that
(2.2) is an exact equation (rather than an asymptotic relation) whenever
x
is an integer.
Let us recall that, by the definitions in §l, a taiZ funation
T(x),
say, will have the representation
T(x)
e
-A(x) .
,
=
J:
where
A(x)
and
a(u)
holds.
~
O.
Suppose further that
Then we can find a
p,
a(u)du
T(x)
is sub-exponential, so that (1.5)
0 < p < 1,
IZx
a(u)du
If
a(u)du
such that for all large
::;
p.
x
8
For
0 s z
S ~
let us set
IjI (z)
I/J' (z)
Jxx-z
a(z)
f~ a(u)du
a (u)du
is non-increasing,
J~ a(u)du
1jI'(z) ~ 0
a(x-z)
fxx-z a(u)du
s 1z s
a(z)
Thus
J~ a(u)du
a(x-z)
=
ljI(z)
a(u)
=
a(u)du
z,
Then, for almost all
But, since
JXx-z
and hence
ljI(z)
S
1jI(~),
o
S
Z S
-%x.
This implies that
z
JX
SpJ
a(u)du
o
x-z
a(u)du,
or
A(x) - A(x-z)
Therefore, for
0 S z
S~,
e
S
pA(z).
we have
-A(x-z)
e
-A(x)
Consequently we have established the following lemma.
Lemma 2.1:
p,
K such that
If T(x)
0
is a SETF3 then there exists finite con8tants
< p < 1
and
T(x-zL
T(ic)
for all 0
S 2 S ~
and x
Note that the constant
e-A(x)
.
_<
K
sufficiently large.
K arises in the above lemma because
are only asymptotically equivalent.
T(x)
and
e"
9
Next suppose
A(x) , for the moment, to be the arbitrary d.f. of two
independent positive random variables
and, for
x
Y
l
and
Y .
2
Choose a large
~ >
set
> 2~,
Consider the events:
Y +Y > x},
l 2
Yl +Y > x},
2
E -
{~
<
Y
l
~ x-~,
E
1
-
{~
<
Yl
~ ~,
E
2
-
{~ <
~ <
E
-
{~
~,
3
Then it may be verified that
Yl '
< Y ~
2
El , E and
2
E
c
YZ},
Yl +Y
2
E
3
>
xl.
are disjoint and that
E + E + E .
l
2
3
Thus
= P(E)
I(x:~,A)
~
P(E ) + P(E ) + P(E )
l
3
Z
i.e.
(2.3)
[Ac(~)] z + 2
I (x;~,A)
At this stage suppose
x
+
00,
large
where
A(x)
AC(x)
J~
~
AC(x-z)A(dz).
is a S.E.T.F.; say
AC(x) ~ e-A(x)
is the function encountered earlier.
Then, for all
x,
(2.4)
~
e -2A(~)
2A(x)
exp - (l+p) ,
by (1.5), with
p
as before.
But
A(x) t 00,
since
as
AC(x) ~ 0,
and
0
10
2 > (l+p).
Hence (2.4) gives
....
(2.5)
0,
x ....
ex>.
Also, by Lemma 2.1, for all large
c1
J~.
A (x)
A
x,
~
Ac (x-z)A(dz)
K
J~
u
K
(l-p)
_d_A..>.(z-s),--[l-A(z)]P
[1-A(6)] (l-p).
If we combine this inequality with (2.5) and (2.3) we have the
following.
Lemma 2.2:
C
that
A (x)
is any d.f. of a non-negative roandom vaT'iabZe such
If A(x)
is a SETF then" foro any
E:
> 0
we can find
Me:)
such that
Z.un sup c 1
x ....
A (x)
ex>
As an example of the value of this lemma we prove the following useful
result which generalizes a theorem of S. C. Port ( quoted on P. 272 of Feller
(1971»
for functions of regular variation.
Theopem 2.3:
Let A (x)
and
C(x)
Let
T(x)
=
B(x)
r:
be distroibution functions and set
A (x-z)B(dz).
be a SETF and suppose that foro some aonstants
ex ~ 0"
13 ~ O~
. . ex}"
a
B (x)/T(x) .... 6
Then
cfJ(x)/T(x)
Proof:
variables.
To begin with, suppose
Choose a large
6 > 0
.... ex + B.
A and
and write
B refer to non-negative random
11
=
IX-6± JX
6
=
+
J6
x-6
0
II (x) + 1 (x) + 1 (x).
2
3
Plainly,
: ; JX
=
and, since
T(x)
~
T(x-6),
B(dz)
x-6
C
B (x-6) - BC(x)
we have
Further
and so
To deal with
Il(x),
let us set
=
G(x)
so that
If
E >
0
GC(x)/T(x)
(a+8)
>
+
0,
we can pick
~(a+8).
GC(x)
6
~{A(x)
+ B(x)},
Then
is a SET F
and Lemma 2.2 shows that, given any
so that
lim sup Il(x)/T(x) ::;;
x+
CXl
It then follows easily that
x
C
f o A (x-z)B(dz)
T(x)
But
+
a,
E.
12
CC(x) ~ (a+8)T(x)
and hence
the case
= O.
a+8
Now, given any small
distribution functions
and
Al (x)
as was to be proved.
Al(x), BI(x)
= £T(x) = BI(x)
C c (x)
I
then
C c (x)
I
~
~
Cc (x)
2€T(x),
for all
x
as
-+
£ > 0 we can evidently construct
such that
for all large
=
CI (x)
00.
x
f:
However we must deal with
x.
AI(x)
~
for all
A(x)
x,
If
Al(x-z)BI(dz)
and, by the first part of the proof,
Thus
c
= o(T(x»
C (x)
and the theorem is
proved.
To extend the result to not-necessarily-non-negative random variables,
we write
O+
=
f
AC(x-z)B(dz) +
proof will show
and, since
c
oo
C
A (x-z)B(dz) +
0+
-00
It is trivial that
JX
Jl(x)/aT(x) -+ B(O+)
as
J (x)/aT(x) -+ 1 - B(O+).
2
A (-6)
x -+
00
Jx
c
A (x-z)B(dz)
and the first part of the
Further; for
6
>
0
large,
can be made arbitrarily near unity, this implies
Thus the theorem is established.
At this point it should be clear how one can also prove the following:
CorolZary 2.3.1:
If A" B" C" T are as in Theorem 2.3.1, but one has
only the weaker hypothesis
!f(x)
=
O(T(x) )}"
=
O(T(x))
as
13
then one may concZude
For the remainder of this section we shall be concerned with displaying
functions which belong to the various classes we have introduced.
In parti-
cu1ar, we shall be concerned with finding whether properties of a tail function
T(x)
I
are preserved by
00
x
when this tail integral exists, and
T(u)du,
we shall also look at the question of estimating this tail integral.
Let T(x)
Theorem 2.4:
be a non-negative non-increasing function in
00
L (0 3 m) and set TI(x) = 1 T(u)du. Then:
1
x
(i) TI(x) is a frrrg if T(x) is a fmg.
(ii)
Proof:
hence
is a tai Z function if T(x)
TI(x)
Tr(x)
(i) If
T(x)
is a fmg, then
00
~
Ix T(u+l)du
=
Tr(x+l).
is a tai Z function.
T(x)
T(x+l)
~
as
This is enough to show
f.m.g.
(ii) If
where
T(x)
a(u) + 0
is a tail function, then
as
u
+
00.
J:
T(x) ,... exp
-f
Hence, if
e
~
a(e+u)du
(u)du
Xo
J:
0, v
~
~
0,
a(e+l+u)du,
or
v
r+
6
~
a(u)du
f6+I+V
a(u)du.
6+1
Therefore
or
J:
f:
e
e
-IeEl+va(u)du
s
dv
-/~a(u)du
If we use the notation
A(x)
dy
s
J:
J:+ 1
= IXo a(u)du,
e
e
+l +v a ( u )du
_/ ee+1
-/~+la(u)du
then we have:
dv,
dye
x
+
00
TI(x)
and
is a
14
I; e-A(y) dy
(2.6)
e
e-A(y) dy
J0+1
oo
-A (0)
e
-A(6+1)
Let us set
-A(y)
00
16+1 e
= co -A(y)
l6 e
<p (0)
~(O) +
Then
1 as
6
= roo
)e
H6)
By (2.6) we see that
is non-increasing.
dy
and
+ co,
pI (0)
dy
<pI (e)
~
e
-A (e)
e
-A(O+l)
co
e- A(y) dy
e-A(y) dy
l 6+1
<P(O), <p(1),
so the sequence
0,
~(2),
rf we refer to the discussion at the start of this
section, leading to (2.2), we see that our results imply that
where
A is some constant and
~
6(u)
O.
This is enough to show that
Tr(x)
is, indeed, a tail function.
We shall occasionally need to know if
Tr(x)
will be sub-exponential;
the most we have achieved on this matter is the following.
Theorem 2.5:
decreasing; let
Let
T(x)
uCJ.(u) t A ~
be a tail function suah that
co
as
A
u t
. . , D·
where
A/(A-1)
co.
is interpreted as unity if
as
A=
x+
co.
is non-
A > 1,
Then, if
T(x)
CJ.(x) ,
uCJ.(u)
co
Purthe~ore,
(if
A > 1)
TI(x) is sub-exponential.
(Note that it will be shown later that
A ~ 1,
necessarily, if
Tr(x)
exists. )
Proof:
Since
T(x)
is a tail function,
T(x) ...., e-A(x),
e"
15
where
and
a(u) t O.
A(x)
+ ~
a > 0).
ultimately, for some
as
+ ~
x
~(y),
is non-decreasing from
(for we can show
A(x)
a(u)
>
is strictly increasing,
say, such that
=
A(11 (y»
A(x)
ua(u)
Thus, since
we can introduce an inverse function
Since, also,
a(u)du
We are here to assume that
which it follows easily that
a/u,
J:
=
A(x)
y.
is differentiable, it follows that
l1(Y)
is likewise,
and
11 ' (y)
=
0(x)
=
1
Let us set
e
-A(x)
and
0 (x)
r
so that
~
Tr(x)
as
0 (X)
r
= f~
0(y)dy,
x
+ ~.
x
From its definition, one can see that
unbounded as
y
+~.
l1(Y)
is strictly increasing and
Thus a change of variable gives
= f~
e-A(u) du
x
=
-Y
m
fA(x)
e
dy
a(l1(y»'
At this point, let us set
x(x)
Thus
=
e
-A(x)
a(x)
1 f~
= ----
a(x)
A(x)
e -y dy
•
16
(2.7)
0 (x)
I
x(x) - 1
~
=
fA(x)
=
J0 e
e-y+A(x) {
a(x)
- 1} dy
a(~(y»
~ -y {a(x)
a(~(y+A(x») - 1
}
dye
The expression in braces can be written
a(~(A(x»)
a(~(y+A(x»)
and, because
=
_ 1
is increasing and
~(y)
sion is non-negative.
Thus
We wish to show that
y
y
0 (x)
I
~
(x)
0
+
y (x),
y
a(x)
is non-increasing, this expres-
x(x) for all
as
dominated by some suitable function of
x
y,
say,
+ ~
x.
and that this convergence is
thereby allowing us to take limits
under the integral sign in (2.7).
Now, by definition of
we have
~(y),
f
=
y
~(y)
a(u)du.
0
Thus
(2.8)
1
=
f
~(Y+1)
a(u)du.
~(y)
However, we have that
(2.9)
1
~
~(y)
a(11(y»
ua(u)
r(Y+1)
is non-decreasing.
Thus (2.8) yields
~
~(y)
=
~ (y)
a (~ (y» log 11 (y±1) •
~
Let us first suppose that
that
~(Y+1) ~ ~(y),
wa(w)
(y)
+ ~
as
w
0
as
+~.
Then (2.9) and the fact
prove that
log ~ ~(y±1)
(y)
+
,
i.e.
~(y+1)
,...
~(y).
y +
~,
17
~(y)a(~(y»
But
a(~(y»
Since
so that
is non-increasing, it follows that
and hence that
S(u),
~ ~(y+l)a(~(y+l»,
a(~(y»
say, such that
is a function of mild growth.
S(u)
~
0,
S(u)
a (~(y»
e
-/~S(u)du
e: > 0,
Therefore, given an arbitrary
a(~ (y»
a (lJ (y+z»
for all sufficiently large
0 as
+
as
Y
+
0
as
for all large
and
+~,
y +
00.
z
for all
~
ly+zS(u)du
e
y
(l+e:)
~
(l+e:) e e:z ,
~
0,
y.
From this inequality and the fact that
y (x)
u
Consequently there is
a(lJ(y»
is a fmg, we have that
as desired; we also have that
~ (l+e:)ee: y
o ~ yy (x)
x +
00,
y.
Thus we can appeal to dominated convergence to deduce from
(2.7) that
8 (x)
r
~
X(x),
as
x
This proves the first part of the theorem if
ua(u)
Suppose next that
for all large
u.
Thus
+
A
If
< ~.
A(x) - Allog x
+
+ ~.
A=
A
<
-00,
00.
then
1
and so
0_
~
A log x,
~
A log
as
A
x + 00,
so
A(lJ(Y»
~(y),
as
y
AI
T(x)
11 T(u)du,
With this established, let us suppose
A(x)
x
00
This contradicts the assumed convergence of
necessarily.
ua(u)
+ 00,
>
1.
< AI <
+
1
00,
as
so
A ~ 1,
We have
18
i.e.
log ll(Y) ....,
f'
or
and hence, for
z
~
0,
(2.10)
Y -+
00.
In particular, (2.10) implies that
before, for any small
e-y/Aa(ll(Y»
is a frog and so, as
0,
£ >
a(ll (y»
a(ll (y+z»
for all large
y.
Since we can choose
small enough to ensure
£
£
+ llA
< 1,
this inequality and (2.10) allows us to deduce from
0 (x)
I
x(x)
-
foo e- Y
-
0
a(A(x»
dy
a(ll(y+A(x»)
that
0 (X)
r
x(x)
-+
f:
A
e-y(l - l/A) dy
= A-I'
by dominated convergence.
as
x
-+
00,
This completes the second part of the theorem.
To complete the theorem we need the following:
Lemma 2.6.
If
ua(u) -+
00"
then
"tog
a(x) = o(A(x))",
as
x -+
00.
e"
19
Take
Proof of Lemma:
A(x)
T very large, then
J:
,
a(u)du + Ta(T) log(xtT).
Hence
lim inf
But
Ta(T)
~ Ta(T).
A(x)
log x
x-+ oo
can be made arbitrarily large.
Thus
The lemma follows from the fact that, since
l/x
for all large
Now, when
x,
implying that
xa(x)
=
-
= Thus, i f
T(x)
is non-decreasing,
A(x) - log a(x) + 0(1)
A(x) {I
o(l)},
is SE so. that
x-+
2
<
oo
,
follows that
log T (2x)
r
lim sup
()
x -+
log Tr x
2,
<
00
also.
This establishes that
Finally, when
as
Ilog a(x)1 Slog x.
lim sup A(2x)
A (x)
it
xa(x)
= o(A(x»,
is unbounded, the first part of the theorem shows
log Tr(x)
by the lemma.
log x
ua (u)
-+
T(x)
T(2x)
is SEe
Tr(X)
A <
....
00,
we can see that, as
2x
a(u)du
x
e
f
-+
e
A log 2
i.e.
T(2x)
-A
T(x).
'"
2
=
foo T(u)du
Thus
2x
=
2
Joo T(2u)du
x
x
-+
00,
x
-+
a(x)
00.
~
20
Therefore
log T (2x) ..., log TI(x), and TI(x) is again proved to be SEe
I
,(x) is a non-negative non-decreasing function such that ,(x)/x
When
is non-increasing, we call
,(x)
a sub-linear function.
Such functions have
proved useful in a previous investigation of the author (Smith (1969».
It is
an easy exercise to show that a sub-linear function is also sub-additive:
,(x+y)
xl, (x)
Also:
,(x) + ,(y),
S
is sub-linear if
,(x)
x > 0,
y
>
O.
is sub-linear.
A useful class of functions, in discussing tail probabilities, are of the
,(x)/xY,
form
y > 1.
for
functions of the form
class.
l/,(x)x Y;
such functions are included in the first
In connection with that class we have the following.
Theorem 2.7.
linear and
Y > O.
co
y-1
Let T(x) ..... ,(x)/x\
Then
T (x)
then TI(x)
Ix T(u)du"
'1 (x)/x
There is nothing to be gained by considering
"
Proof:
where
as
x
-+
co"
where
,(x)
is sub-
is a fmg.
is a sub-exponential, tail function of the fom
'1 (x)
is another sub-linear function.
If we set
x (x)
then, since
T(X)
and
X/T(X)
x
are non-increasing,
y-l
(x+l) y-l
>
-
X(x+l)
X(x)
(x+l) Y
From this we see that
X(x+l)
X (x)
and consequently
Tr(x)
T(x)
is also a fmg.
is a fmg.
-+ 1
,
as
x
-+
co,
By Theorem 2.4 (i), it follows that
e"
21
Let us set, when
y > 1,
I
oo
=
•
x
(u) duo
y
u
Then
Xi(x)
Xr(x)
Since
.(u)/u
is non-increasing we shall establish that
xi(x)/xr(x)
is
non-decreasing if we can show that
xy - l
foo
.(u)u- y du
=
p(x),
say,
x
is non-decreasing.
But
p' (x)
=
(y-l)x
~
(y-l)x
y-2 Joo
-1
.(u)u Y du - x
.(x)
x
y-2
foo
• (x)
u- y du - x -1 • (x)
x
~
~ x-I .(x) - x-I .(x).
Thus
p(x)
is, indeed, non-decreasing and so
d
-dx log Xr(x)
is non-decreasing.
Consequently, for any integer
n
i.e.
(2.11)
~
xr(n)
xr(n-l) •
rf we hark back to our preliminary results on moderate growth, we see that we
can set
xr(n)
XI (n-l)
b
=
e
n
,
say,
22
where
b
n
+
0;
conclude that
b +
n l
then (2.11) shows that
Xr(x),
and hence
Let us define (still for
~
Tr(x)
x
y-l
=
Jx
oel
Then the argument just above shows that
dx
'(1 (x)
x
b ,
n
Xr(x),
and we are allowed to
is a tail function.
y > 1)
'(1 (x)
d
S
'(
(u) du.
y
u
'(lex)
is non-decreasing.
=
and the right-hand side of this equation is non-positive
y > 2;
then, by the monotonicity of
~
['(1 (X»)
dx
x
(y-2)x
'(l(x)/x
'(1 (x)
y S 2.
Suppose
'(u)/u,
y-4
'(x)
Joo du
-2
y-1 - x
'(x)
x- 2 '(x) - x- 2 '(x).
is non-increasing.
We have shown that, when
=
where
if
x u
$
Thus
Also,
is sub-linear.
y
>
1,
'(lex)
y-1
x
A particular consequence is that
1
2
y-l
and hence
~
1
y-1'
2
From this it is a routine deduction that
log Tr(x)
and, hence, that
Tr(x)
is SE.•
Of course, in matters of the present sort, the "functions of regular
variation" introduced by Karamata (1930a), (l930b), and exploited, in particular, by Feller (1971), are of great importance.
Before we close this
23
section we shall show how they fit in with the present scheme.
-0.
X
Sex),
where
0.
>
0
and
Sex)
If
T(x)
is a function of slow growth, we say T(x)
is a function of regular variation (frv) of index
Since~
-0..
as is now
well-known from the work of Karamata, it is true, that
I
Sex)
.
o.x
S (u)
OC>
0.
as
(1+0.) du,
x u
x
-+
00,
we may aZways oonveniently suppose suoh a frv to be striotly deoreasing
(a similar argument shows that when the index is strictly positive a frv
can be supposed strictly increasing).
The example
= 23 +
Sex)
sin/(log x)
shows, however, that a function of slow growth (which this
Sex)
is) need not
be asymptotically equal to a monotone function; thus our convenient supposition
does not extend to the case of zero index.
e
Suppose now that, for
0. > 0,
Sex) ,
0.
x
T(x)
x
-+
00.
Let us set
oo
X(x)
so that
X(x) '" T(x),
as
x
-+
00.
=
0.
I
~
x u
1+0. du,
Then
X' (x)
=
o.S(x)
X
1+0.
and so
Consequently,
"-
that
X(x)
log X(x)
is ultimately non-increasing and the conclusion
is a tail function follows as for the case involving sub-linear
functions, above.
Indeed, the proof that
X(x)
is sub-exponential can be
24
constructed similarly, if we use that
Theorem 2.8.
If T(x)
negative index, then TI(x)
S(2x)
~
S(x).
Thus we have
is a function of regular variation with strictly
is a sub-exponential tail function.
3. RELATIONSHIP BETWEEN BC(x) AND LC(x)
Let us write
service-time and
F(x)
for the d.f. of
X
= V-U,
where
V is a typical
U a typical (independent) inter-arrival interval.
Then we
have
Theorem 3.1.
Conversely, if
Proof:
Since
If either
(3.1)
holds then both
c
B (x+6)
6
(x)
and r (x)
6
are [mg.
is any large positive constant, (3.1) gives
is given to be of moderate growth~
If BC
choosing
C
B
is a fmg~ then
is non-increasing, this immediately yields
On the other hand, if
since
or r(x)
We base our argument on the equation
BC(x)
a)
BC(x)
~
c
B (x).
But
large enough.
A(6)
we have from (3.4) that
can be made arbitrarily near unity, by
This remark, with (3.3) proves (3.1).
25
r
If
b)
is given to be of moderate growth 3 we have, from (3.4),
that
FC(x-Li)
for all large
x.
BC(x) A(Li)
~
C
C
F (x-Li) '" F (x)
Since
i t should be clear that a proof
such as in (a) again applies, to yield (3.1).
a)
r (x)
If
A(Li) < 1.
'"
If (x)
as x
3
-+
Let
00.
Li
be any constant such that
Then (3.2) gives
~
FC(x)
BC(x)A(Li) + BC(x+Li)Ac(Li)
and so
1
~
A(Li) + AC(6) lim inf FC(x+Li)
x -+ 00
FC(x)
This shows that
C
But
F (x)
is non-increasing.
Hence
C
C
F (x+6) '" F (x).
arbitrarily large the theorem is proved.
6
0
such that
o<
6
<
Li '
O
A(6 ) = lour choice of
0
However, if
k
If
6
can be chosen
If, however, there is a least finite
6
is restricted to the range
is any fixed integer it is obviously true that
FC(x+kLi) '" FC(x).
Thus the proof is clear.
a
Co!'O ZZary ;3. 1. 1. If either fOB (u)du
00
both are finite.
If either
f; Ba(u)du or
or
a
is finite then
f; r(u)du is a fmg 3 then
they both are3 and they are asymptotiaally equal.
asymptotiaaZZy equal then they are fmg.
00
f 0 F (u)du
ConverselY3 if they are
26
Proof:
'The Corollary follows from (3.2) which gives
The arguments applied to (3.2) in the proof of the theorem can now be
applied to (3.5).
c
~
10 B (u)du is the mean service-time.
Note that
Before we proceed further, let us explain a notational convention we shall
adopt.
ever
If
I
H(o),
say, is a measure function, we shall write
II H(dx).
is an interval, to denote
shall find it convenient to write
finite.
H(x)
In terms of the random walk
I
in place of
{S}
I
= (-~,x],
H(I),
when-
however, we
when it is
described in §1, we shall now in-
n
troduce the usual renewal measure; if
If
H(I),
is any finite interval of the real
axis,
=
H(I)
L
P{S
n=l
This series is convergent because the walk
I}.
€
n
{S }
n
is transient (Chung & Fuchs
(1951».
Let us define another measure, involving the improper random variable
N,
1
the suffix of the first ladder variable:
G(I)
=
L
n=1
Then
L(x)
G(I)
S
H(I)
(3.6)
and so is also finite for any bounded interval.
Notice that
L(x),
= P{SN
1
~
xlN
1
~L(I)
P{Sn€I & N1 >n}.
< ~}.
the d.f. of the first ladder variable, is given by
Thus if
I
c
=
=
L
n=l
P{Sn€I & N1 >(n-1)}
(O,~)
is a finite interval,
27
co
L
= F(I) +
n=l
= F(I)
1:co
+
P{Sn+1 E1 & N1 >n}
F(I-z) G(dz).
Also, for any bounded interval
I,
co
(3.7)
H(I)
co
l
=
n=l
PiS €I
n
= G(I) +
N >n} +
1
&
co
n
l
l
n=l r=l
l
PiS EI & N sn}
1
n
n=l
P{SnE! & N1=r}
co
= G(I) +
= G(I) +
L
t=O
P{SN + EI & N1 <co}
1 t
~L(I)
+
~
If we replace the interval
f:
H(I-z) L(dz).
I
by
I-t
in (3.7),
where
t
is large and
positive, we obtain
~
(3.8)
H(I-t)
= G(I-t) +
~L(I-t)
+
~
I:
H(I-t-z) L(dz).
It is possible for the random variables
almost surely multiples of some fixed
with no loss of generality, that
newal theory that if
I
= (0,1)
w > 0;
w = 1.
then
{x }
n
to be lattice, that is:
in this case we may suppose,
Thus it follows from standard re-
H(I-t)
is a bounded function of
and, in both the lattice and non-lattice case,
H(I-t)
where
II
= lEXn I.
-+
1
II '
H(I-t-z) L(dz)
and thence to deduce from (3.8) that
G(I-t)
t -+
co,
It is then routine to show that
f:
(3.9)
as
-
as
t -+
co.
1
-+
;'
as
t -+
co ,..
t
28
For any
(3.10)
for all
£ > 0
(1-'lT)
G(I-t)
t
~(£)
we can therefore find an integer
-
].l
£
With the present choice of
~ ~(£).
(3.6) that, for any integer
L
I
we can then obtain from
v > 0,
JO_oo n~="ly F(I+n-z) G(dz)
co
'IT
such that
~
L(I+n)
n=v
which is the same thing as
(3.11)
'lTL c (v)
J:oo FC(v-z) G(dz),
~
rL~
co
~
J-r
-(r+l)
{(1~1T)
~
c
F (v-z) G(dz)
_
£} Joo
FC(u) duo
v+~
f
Suppose that
co
x
c
F (u)du
is a fmg.
Then (3.11) yields, since
£
is arbitrary, and
C
L (x)
(1-'lT)
(3.12)
'IT].l
Conversely, if we suppose that
LC(x),
rather than
fmg, ue can stUl deduce (3.12) from (3.11) by using that
c (,,) as v -+ 00.
L
is monotone,
f
00
x
c
F (u)du
is a
LC(v-~-l) ,..,
y
If, in (3.6), we let
I
= (x,oo)
we obtain
Hence, if we use the fact that the renewal measure
pose that, when
I
= [0,1),
G(I-t)
for all
t
~ ~(£),
we have
(1-'lT) +
].l
£
H dominates
G and sup-
29
(3.14)
c
lTL (x)
Let us, temporarily, write
c = fO
H(dz),
-6
necessarily finite.
(3.15)
Then (3.14) gives
c
1TL (x)
At this point, we need:
Lemma 3.2.
If
f; r (u)du
00
/
Proof:
If
x
r(x)
c
is a fmg~ then
+
0,
x
as
+
00,
F (u) du
T is a large positive constant,
foo FC(u) du
2:
TF c (x+T),
x
FC(x)
since
is non-increasing.
Thus, since
1
S;
But
T'
T can be arbitrarily large, so the lemma is proved.
This lemma enables us to deduce from (3.15) that if
f
00
x FC (u)du
is
fmg.
(3.16)
lim sup
x + 00
LC(x)
/00 FC(u) du
S;
(l-lT )
lTll
x
However, we also wish to show that (3.16) follows from the alternative
premiss that
Lc (,....)
w
",'s
a JI
~mn.
~
V
I n thi s case, we have, f rom.
(3 12) t'aa t
30
Lc(x)
for all sufficiently large
x.
LC(x)
and so, since
{(l-U)
. UJ.l
- e:} fco
Thus, if
T > 0
~
~
FC(u)du
x
{(l-Tr)
- e:}
1rll
is fixed,
C
T F (x+T),
LC(x+T) ,... LC(x)
=
FC(x+T)
lim sup ~~....::.!x -+ co LC(x+T)
=
lim sup
x -+ co
~ l
T
FC(x+T)
-'--~..;:....<..
LC(x)
{(1-1r) _ e:}-l.
1rll
Thus
FC(x)
0,
-+
as
LC(x)
x
-+
co,
and we can infer from (3.14) that
lim inf --=l=-x -+ co LC(x)
If we reason along what should now be familiar lines we see that (3.15)
necessarily follows.
Thus, when we couple (3.12) and (3.16) and use Corollary
3.1.1 we have part of the following.
Theorem 3. 3.
or
co
C
f x B (u)du
is of moderate growth as
x
-+
co"
then
as x
-+ co.
Conversely" if (3.16) holds then both sides of (3.17) are func-
tions of moderate growth.
In particular (3.17) holds if BC(x)
is a fmg.
To complete the proof of this theorem, we must show that (3.17) implies
that both sides of this asymptotic equation are fmg.
Now the derivation of
31
(3.15) has not depended on assumptions about growth-rate.
From it we have,
(3.18)
This inequality, with (3.17) and Corollary 3.1.1, yields
s
1
Since
is arbitrary, we can thus infer that
€
0,
-+
x -+
00.
This allows us then to deduce from (3.17) and (3.18) that
c
1
6 > 0,
Thus, for any
(1-~) + €~ lim inf L (x+~-l)
s
(1-~)
by choosing
x
-+
6,
~»
LC(x)
lim sup
x -+ 00 LC (x+6)
But
C
LC (x)
LC(x)
00
S
1.
is non-increasing, so we are forced to the conclusion that
C
L (x) ,.., L (x+6),
1. e. that
c
L (x)
is a fmg.
The theorem is thus proved.·
4. RELATIONSHIP BETWEEN LC{x) AND QC{x)
A(e)
If
[0,(0)
is a measure function on
we shall denote its Lap1ace-
Stie1tjes transform thus:
oo
A*(s)
and take
(4.1)
s
Q*(s)
=
J0-
to be real and positive.
=
1 - ~
1 - ~L*(s) ,
e
-sx A(dx),
From (1.7) we then have
32
in a familiar way.
(4.2)
J:
e-
sx
But
QC(x)dx
=
1 - Q*(s)
s
and a similar formula holds for
LC(x).
From (4.1) we obtain
and this, in view of (4.2), gives the useful relation:
Alternatively, (4.3) can be obtained from (1.3) by direct calculation with
no reference to transforms.
An immediate and obvious consequence of (4.3) is the following.
Lemma 4.1.
Fop aZZ x
~
0,
and aonsequentZy,
One can also obtain from (4.1) the equation
which yields the second useful relation:
(This can also be obtained, but tediously, by direct calculation.)
Let us note, in passing, the rather obvious fact that
(4.5)
QC(x)
S
(l-n),
x ~
o.
33
Let us now set
=
X(x)
QC(x)
LC(x)
and
1\
X(x)
sup
=
X(x).
O~y~x
Since we define distribution functions to be continuous to the right, it
follows that
that
1\
X(x)
X(x)
is also continuous to the right.
is continuous to the right.
Then we can find a sequence
+~.
Suppose
{x. }
J
It is then easy to show
X(x)
such that
is unbounded as
x.
+ ~,
J
From (4.4) and (4.5) we then have the inequality
c
1l'L (x) + 1l'(1-1l') {Lc (x-~) - Lc (x)} + 1l'
(4.6)
+ 1l'Q
from which, on setting
x
c
(x-~),
= xj
and dividing by
J
x-~
~
c
L (x ),
j
c
Q (x-z) L(dz)
we have
c
(4.7)
X(x )
j
~
1l' 2 + 1l'(1-1l') {L ·c(Xj-~)}
L (x.)
J
+ 1l'X(x.)
J
Xj
Jb.
c
+ 1l'X(xj
) {
Let us now assume L(](x)
L
{L
-~
c
(x. -z)}
J
L c (x )
j
L(dz)
~X(~)} .
L (x.)
J
to be a SETF.
We can then appeal to Lemma 2.2
to deal with the convolution on the right of (4.7) and obtain (since
c
L (x-~»,
(1-£1l'-1l')
which, since
£
lim
j + ~
X(x.)
J
is arbitrary, shows
lim
j +
X(x )
00
j
1l' 2 + 1l'(1-1l')
LC(x) ~
34
This contradicts the hypothesis that
finite upper bound for
If we let
a
If we let
x
j
is unbounded.
K be a
If we let
we can deduce from (4.6) that
X(x)
x(x)
(4.8)
x(x )
X(x)
4
a,
= lim
x
~
sup
X(x)
co
then
X(x)
~
6+€
for all sufficiently large
in (4.8) run through an unbounded sequence
{x.}
such that
J
we can infer (with another appeal to Lemma 2.2) that
The arbitrariness of
2+
a
~
€
then allows the deduction
~
~(l-~)
+
~K€
+
~(a+€).
c
lim sup Q (x)
X 4 00 LC(x)
This result, coupled with Lemma 4.1 shows that, when
LC(x)
is a
SETF,
SUppose ne:t:t that we start instead with the premiss that Qa (x)
From Lemma 4.1, given any small
SETF.
0 > 0 we can find
6(0)
that
(1;~j
for all
x
~
6.
(1+0)
Then, from (4.3), we can infer that
(QC(x-A) - QC(x)} + (1+5)
+
[l~~j
C
L (x-6) Q(6).
J:-
A
QC(x-z) Q(dz)
is a
such
x.
35
If we now divide by QC (x),
and
Lemma 2.2, we find
where
£(~) ~
as
0
Thus
~ ~~.
L c(x)
lim inf ~=x + 00 QC(x)
(since
that
Q(~) +
~ +
1 as
m).
~
1-1T
-1T
This result, coupled with Lemma 4.1, proves
(l-1T) QC(x) ~ 1TL c (x).
Lastly, suppose we start with the premiss that
From (4.4) we see that, for any fixed
e > 0,
Hence, from (4.7),
1
We may suppose
~
C
(l-1T) + 1TL(e) + 1TLc (e) lim sup Q (x-e)
x + 00
QC(x)
LC(e) F 0,
and deduce
C
lim sup Q (x-e)
x + 00 QC(x)
But
QC (x)
is non-increasing.
enough to establish that
like1iwise.
Hence
QC(x)
1.
$
C
Q (x-e)
~
C
Q (x),
as
is a fmg and so, by (4.7),
x +
m.
LC(x)
This is
must be
We have thus proved:
Theorem 4.2.
If either LiJ(x)
or QiJ(x)
is a sub-exponential tail func-
tion then
(1-1T) QiJ (x)
,.. 1TLiJ (x)~
X
+
00.
ConverseZy~ if this asymptotiiJ equation hoZds~ then both
are of moderate growth.
QiJ(x)
and LiJ(x)
36
5. ON MOMENTS AND THEIR EXISTENCE
In this section, we consider the relationship between the distribution
functions
B(x), F(x), L(x) and
Q(x)
with reference to the moments that may
exist; we consider moments of a quite general sort.
Our main result will be
in terms of the moment functions (M.F.) introduced in §l.
However, some of
our subsidiary results can be expressed in terms of a more general class of
functions:
we shall say
is a scaZe function if it is non-negative,
S(x)
non-decreasing, and satisfies the ordeq condition
large positive x.
scale function then so is
Theopem 5.1.
If S(x)
~
is a scale function).
SJ(x)
and onZy if f~ S(x)F(dxJ
Let
= O(S(x-I»
for
It is easy to see that a moment function is a scale func-
tion (but not vice vepsa: eX
Proof:
S(x)
<
Also, if
S(x)
=
fX S(u)du.
def o
is a saaZe function then
f~ S(x)B(dx) <
K> 0
J:
00
if
00.
be a median of the inter-arrival distribution.
there must be a finite
is a
such that
S(x)F(dx)
=
Then
S(x) ~ K{I+S(x-~)}. Now
ES«V-U)+)
~ W;S«V-U»,
and so
J:
On the other hand,
S(x)B(dx)
S(V)
~
J:
=
ES(V)
~
K{l + 2
S(V-U),
S(x)B(dx)
J:
S(x)F(dx)}.
so
~
J:
S(x)F(dx).
The theorem follows from these last two inequalities.
Theopem 5.2.
(5.1)
J:
S(x)L(dx)
If S(x)
< co
is a scaZe function,
e-
37
if and onZy if
J:
(5.2)
SJ(x)P(dz) <
SJ(x) =
where
Proof:
(5.3)
a>~
~ S(u)du.
From (3.13) we have
~Lc(x)
= FC(x) +
n
FC(x-z)G(dz)
f
Jn=O -(n+l)
-n
J-(n+l)
lXI
~
L FC(x+n)
FC(x) +
n=O
r
G(dz)
00
~
FC(x) + C
FC(x+n)
n=O
~
where
FC(x) +
c Joo
C is the bound introduced just prior to (3.15).
OO
Jo
Since
FC(u)du,
x-I
Sex) = O(S(x-l»
Sex) Joo FC(u)dudx <
x-I
Thus (5.1) holds if
lXI.
this holds if
and an integration by parts shows the latter is implied by (5.2).
Next we observe, again from (3.13), that
~LC(x) ~
Y
FC(x+n+l)
n=O
J-
n
G(dz)
-(n+l)
L
C
P (x+n+l),
n=6
for some suitably large
--
o<
C<
(l-~)/~
6 > 0,
where
C can be any real satisfying
and we have appealed to (3.9).
~LC(x) ~
Thus
C Joo
pC(u)du.
6+x+l
38
If (5.1) holds we can thus infer that
A further appeal to the defining property of
S(x)
and an integration by
parts then shows (5.2) must hold, and the theorem is complete.
To complete our discussion of moments we must discuss the lImoment re1a. tionshipll between L(x)
and
This is not so s:lJnple a matter as those
Q(x).
we have studied earlier in this section.
Theorem 5.3.
Let Sex)
be a saale funation of moderate growth satis-
fying the aondition
for aU
xl
2:
0"
x
2
2:
O.
Then
I; S(x)L(dx)
and
I; S(x)Q(dx)
converge or
diverge together.
Note:
S(x)
automatically satisfies these conditions if it is a MF.
Let us first suppose
Proof:
ES(Z)
where
= J~
S(x) L(dx)
o.
Z is a typical ladder variable.
<
~,
Then, in view of (1.7), our first
task is to prove the convergence of
~
(5.5)
L
n
r
ES(Zl+"'+Zr)'
r=l
where, as before,
{Zj}
are iid ladder variables.
We observe that
ES (r+z 1+ •• '+Zr)
=
where
ES(1+Zl)R 2R3 ···Rr ,
39
S(j+Zl+" ,+Zj)
=
SO-l+Zl+. ,+Zj_l)
Thus
where
At this point, we need the following
Lerruna 5.4.
If
is as in Theorem 5.2,
sex)
ES(Z-HJ;)
Since
Proof of Lermna:
x
-+ co,
for every fixed
y
S(x)
~
O.
as
1,
-+
Sex)
x -+
is of moderate growth
Thus dominated convergence proves the lemma.
-+
necessarily, since the
We can now observe that
co, uniformly with respect to
Z's
are ladder variables).
Zl+",+Zr_l
Thus, given
arbitrarily small
sr
Thus, for
r
-+
S(y).
S(x)
r
S(y+x)/S(x)
But, by (5.4)
S(y+x)
tends to unity, as
co •
~
(1
+
£)
large,
(1+£)ES(1+Z1)R 2 ···R r _ l
=
(1+£)ES(1+Z1)R2···Rr_2Sr_1
(1+£) 2ES(1+Z )R ••• R _ ,
I 2
r 2
(> 0"
£ > 0
1,
as
40
and so on.
(5.6)
It should now be clear that
=
lim sup {ES(Zl+••• +Zr)}l/r
1.
r~=
(Note that (5.4) ensures
ES(x)
~
1
for all
~
x
0.)
~
the convergence of (5.5), whatever the value of
Plainly (5.6) implies
(0 <
~
< 1).
The completion of the theorem is trivial, for (1.7) shows that
(l-~)~L
c (x).
Clearly this implies that any moment finite for
Q is necessarily finite for
L.
We conclude with two notes:
(i) The functions satisfying Theorem 5.3 are
discussed in Smith (1969) where it is shown, in particular, that if
x Y ,(x)
for
Y~ 0
and
,(x)
sub-linear, then
equivalent to such a scale function.
so in
SJ(x);
(ii) If
N(x)
=
O
S(x)Q(dx) <
S(x) is a scale function then
this observation is relevant when combining Theorems 5.1,
=
if and only if
f
=
=
is asymptotically
5.2 and 5.3 to draw the conclusion (under suitable conditions) that
f
N(x)
O SJ(x) B(dx) <
=.
41
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(1930b)
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