Lecture Notes in
Mathematics
Edited by A. Dold and 13. Eckmann
508
Eugene Seneta
Regularly Varying Functions
i!
Springer-Verlag
Berlin.Heidelberg 9New York 1976
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Author
Eugene Seneta
Department of Statistics
The Australian National University
P.O.Box 4
Canberra, A.C.T. 2600/Australia
AMS Subject Classifications (1970): 26A12, 26A48, 60E05
ISBN 3-540-07618-2
ISBN 0-387-07618-2
Springer-Verlag Berlin 9 Heidelberg 9 N e w Y o r k
Springer-Verlag New York 9 Heidelberg 9 Berlin
This work is subject to copyright. All rights are reserved, whether the whole
or part of the material is concerned, specifically those of translation,
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Under w 54 of the German Copyright Law where copies are made for other
than private use, a fee is payable to the publisher, the amount of the fee to
be determined by agreement with the publisher.
9 by Springer-Verlag Berlin 9Heidelberg 1976
Printed in Germany
Printing and binding: Beltz, Offsetdruck, Hemsbach/Bergstr.
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PREFACE
The main purpose
the basic real-variable
stated assumptions,
functions,
of these notes is to present,
in self-contained
reader wishing
to acquire
tool, irrespective
theory of regularly varying
Thus they may be used by any
a user's knowledge
of this valuable
of his field of mathematical
these aims in mind,
where possible;
manner.
With
to keep proofs simple
have been provided
the theory as well as to yield practice
analytical
specialization.
the author has endeavoured
and exercises
under precisely
to show the scope of
in the use of the material pre-
sented.
The author's
probabilistic
in the subject matter was stimulated by
own interest
applications.
theory of regularly
varying
functions
suggested by the book of Gnedenko
to be widely
recognized
among probabilists
Applications which contained elements
theory.
Unfortunately,
edition9
clear.
in probability
and Kolmogorov.
2 of Feller's An Introduction
of Volume
other hand,
the papers
with precise
difficult
of their non-existence.
de Haan's
material
of the Karamata
(and remains
assumptions
in the newer
and conditions
reader.
un-
On the
theory has been progressively
contributions
in the early 1950's,
that there is a general
impression
modest hope that these notes
in a manner somewhat
different
from
(1970a).
Apart from the presentation
discern
was
It is the author's
these gaps,
came
in 1966
to Probability Theory and Its
in which Karamata's
refined and extended since the original
will help to bridge
with the publication
for the non-expert
are so little known to prohabilists
theory was already
It subsequently
of an exposition
this presentation
highly personal,
It thus proves
role played by Karamata's
The fundamental
of the basic
an attempt by the author to provide
e.g.
It needs
w
the reader will
of less standard
and the Appendix.
to be mentioned
only to the material
theory,
a selection
also that the references
presented,
given pertain
and so cannot in any sense he regarded
as complete.
The bulk of these notes was prepared early in 1973 in the course
of an academic year spent at the Department
University.
of Statistics,
(The author takes this opportunity
G.S. Watson and D.R. McNeil
Princeton
to thank Professors
for their kind hospitality.)
The motivation
for the work was a proposed book with N.H. Bingham and J.L. Teugels,
which the present material
The author wishes
was to form the first two chapters.
to express
also his indebtedness
to
in
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IV
Professor Ranko Bojani~ in regard to materials and stimulating correspondence, and more generally, to the strong Yugoslav school of mathematicians founded by Karamata.
Finally, the author is indebted to Ms Helmi Patrikka for her
careful typing of the manuscript.
Canberra
E. SENETA, 1975.
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CONTENTS
CHAPTER 1.
1.1
FUNCTIONS OF REGULAR VARIATION
Introduction.
1.2
Fundamental Theorems.
1.3
Refinement of Definition of Regular Variation.
Characterization of Regular Variation.
1.4
The Structure of Slowly Varying Functions and
Alternative Proofs.
13
1.S
Further Properties of Regularly Varying Functions.
17
1.6
Conjugate and Complementary Regularly Varying Functions.
2S
1.7
The Definition of a Regularly Varying Function.
29
1.8
Monotone Regular Variation.
37
Bibliographic Notes and Discussion.
43
Exercises to Chapter i.
47
1.9
CHAPTER 2.
SOME SECONDARY THEORY OF REGULARLY VARYING FUNCTIONS
2.1
Necessary and Sufficient Integral Conditions for Regular
Variation.
53
2.2
Tauberian Theorems Involving Regular Variation.
59
2.3
A Class of Integrals Involving Regularly Varying
63
Functions.
2.4
A Class
of Functions
Related
to Regularly
2.5
Varying
69
Functions.
Bibliographic Notes and Discussion.
8S
Exercises to Chapter 2.
86
APPENDIX.
GENERALIZATIONS OF REGULAR VARIATION
A .1
R=O V a r y i n g
Functions.
92
A .2
S-O V a r y i n g
Functions.
97
A.3
Monotonicity;
A.4
Bibliographic Notes and Discussion.
Dominated Variation.
99
104
REFERENCES
100
SUBJECT INDEX
111
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CHAPTER 1
FUNCTIONS OF REGULAR V A R I A T I O N
i.I.
Introduction.
Regular v a r i a t i o n of a function is a one-sided,
tic p r o p e r t y of the function, which
logical
local and asympto-
arises out of trying to extend in a
and useful manner the class of functions whose asymptotic be-
haviour near a point is that of a power
such asymptotic b e h a v i o u r
factor which varies
'more slowly'
Being a local property,
point.
i.i.
R
~ > 0
(I.I)
= ~
for some
p
is defined relative
to a
is said to be regularly varying at in-
positive
A > 0 , and if for each
lim ~
than a power function.
is taken to be as follows.
A function
finity if it is real-valued,
to functions where
function m u l t i p l i e d by a
regular v a r i a t i o n
The defining p r o p e r t y
Definition
function,
is that of a power
in the interval
and m e a s u r a b l e
-- < p < ".
(0
on
[A,-),
for some
is called the index of
regular variation).
A function
R(.)
is said to be regularly varying at zero if R(i/x)
is regularly varying at infinity.
at any finite point
point.
a
by shifting the origin of the function to this
It is thus apparent
that it suffices
regular v a r i a t i o n at infinity,
the words "at infinity"
tion of results
at
0
to develop
the theory of
which we shall do, frequently omitting
in the sequel.
Some exercises
in the transla-
from regular v a r i a t i o n at infinity to regular v a r i a t i o n
are given later.
Let us write
form
Regular v a r i a t i o n can now be d e f i n e d
xPL(x).
urable on
It follows that
[A,~)
(1.2)
a regularly varying function with index
L(x)
is real-valued,
p
in the
positive
and meas-
and from (i.i)
lira ~
= 1
X+~
for each
index
~ > 0
Thus
L(.)
is also a r e g u l a r l y varying function,
~ = 0 .
D e f i n i t i o n 1.2.
A function
index of regular v a r i a t i o n
L(.)
which is r e g u l a r l y varying,
~ = 0 , is called slowly varying.
with
of
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The notation
L(.)
is customarily
used for such functions
of the first letter of the French word "lentement"
the foundation
+
Karamata.
papers
Thus a function
be written
which means
of the theory having been written
R(.)
is regularly varying
because
"slowly",
in French by
if and only if it can
in the form
RCx) = xPLCx)
where
-- < p < -
and
L(.)
is slowly varying.
This
is the product
form alluded to in the opening paragraph.
Any eventually
limit as
example
x + ~
positive
is clearly
of a slowly varying
log log x
measurable
function
is
regularly varying
(others
possessing
a positive
The simplest non-trivial
log x ; any iterate
of it e.g.
is also slowly varying.
On the other hand the exponential
2 + sin x
function
slowly varying.
functions
at all; and undampened
are similarly
less obvious
not regularly
oscillatory
varying.
is involved
functions
These
are given in the exercises)
intuitive notion of what
e x , e -x , are not
such as
few examples
should provide
in the concepts
of regular
some
and slow
variation.
It should also be clear that to study regular variation,
to study the properties
1.2.
Fundamental
of slowly varying
functions
functions
Theorem
i.I.
pertaining
follow readily
1.2.
(The Representation
such that for all
x > B
Theorem).
[a,b],
If
L(.)
is a slowly
0 < a < b < ~, the rela-
h~[a,b].
Theorem).
If
L(.)
defined on
then there exists a positive number
we have
X
L(x) = exp { q ( x ) + ~
(1.3)
B
where
n
is a bounded measurable
c(t)
Notes
dt }
t
function on
§
See Bibliographic
can
of slowly
from them.
[1.2) holds uniformly with respect to
Theorem
of slowly
in that either
and most other properties
then for every fixed
A > 0 , is slowly varying,
to the properties
they are fundamental
from the other,
(The Uniform Convergence
varying function,
tion
theorems
in the theory;
be obtained readily
varying
it suffices
for most purposes.
Theorems.
There are two basic
varying
functions,
and Discussion.
[B, |
such that
[A,~).
B ~ A
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n(X)
+ c
that
([C[
e(x)
< ~),
+ 0
and
(as
E
is
a continuous
function
We shall proceed by first proving Theorem
via a sequence
Theorem
[B,~)
on
such
x + ~).
of lemmas.
The converse
i.I and then Theorem
deduction,
of Theorem
1.2
I.i from
1.2 is left to an exercise. +
For the following
lemmas
itself but a function
formed by
f(x)
We shall
f
= log L(e x)
thus assume
is real and measurable
it is rather
easier
of a kind to which
L
to work not with
can be readily
.
that we are dealing with a function
on
L(.)
trans-
[y,|
for some
f
y, and satisfying
which
the con-
dition
(1.4)
f(x + u)
f(x) +.0
The relation
Lemma 1 . 1 .
as
x + ~, for each
hoZd8 uniformly for
(1.4)
~
~ .
in a n y fixed
finite closed interval.
Proof.
We first prove
[0,I].
Suppose
the assertion
the assertion
xn § |
E > 0, {x n}
such that
each
n, satisfying
[f(x n + ~n)
(1.S)
Define
sets
Un,
Vn
with
(l.6b)
Vn={X:XE[0,2],
[f(Xm+~m+X )
tone
are clearly measurable
increasing
such that
interval
Then
3
~n r [0 ,I]
for
.
- f(Xm) [ < ~1 e
[f(Xm+~)
Vn
in the particular
by
Un={~:~e[0,2],
and
~
n , {~n }
- f(Xn) I ~ c
(l.6a)
Un
for
is not true for this interval.
sequence
of sets,
,
~g/m k n }
f(Xm+~m) I < ~1 , ~/m _> n } .
and each of
and such that
{Un},
{V n}
is a mono-
Un, V n § [0,2]
in
virtue of (1.4).
Hence
if
sufficiently
that
m(V~)
m(.)
large
= m(VN)
is used to denote the measure,
m(Un)
> 3/2, m(Vn)
> 3/2.
Let
it follows
that
V~ = V N + ~N
VN
' so
> 3/2, and note that
I
u N C [o,z] C [0,3]
v~ c [o,31
it follows
set).
+
Thus
that for any
J~
e UN
See Exercise
1.3.
;
N
sufficiently
such that
large,
u - UN ~ VN
"
UN nV~
~ ~
(the empty
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For this
(l.7a)
If(xN+~ )
(1.7b)
If(xN+~N+ ~ -~N)
f(xN)
1
I
<
~E
I
f(xN+~N) [ < ~ e
by
(l.6a);
by
(l.6b);
or
equivalently
i
f(xN+~N) I < ~ c
If(xN+~)
Putting
(l. Ta) and
inequality,
(l.?b]
together
I f(x N + u N)
a contradiction
to
For the case
by
f(x)
U
~
=
[a,b]
Lemma
[X,X'],
- f(x]
v =
~=~ . e [0,I]
~X(X
~ ~)
By L e m m a
taking
- f(y)
(~-a)/(b-a),
i.i,
JX
for
x
- f(x)
so
f
b > a, define
f(-)
+ f(x-a)
that
- f(x)
y § -
~
x § -
;
i8 bounded on every interval
[X + k - I, X + k]
is b o u n d e d
for any
y
in the
interval
[X,X+I]
,
+ 1
and c a r r y i n g
i IfCx+z)1
for p o s i t i v e
f
V~[O,l]
this
argument
further
we
[X + i, X + 2]
+ 1 i IfCX) l + 2
integer
If(x) I i If(x)I
Corollary.
that
I < 1 , x ix,
If(x) l
inequality;
on
Ifcx)l
have
such
x = X, X + ~ = y
by an e l e m e n t a r y
on
[a,b],
.
If(y) l ~
We thus
= f(y+~)
sueh that
IfCx+~]
obtain
interval
X' > X .
Proof.
Thus,
< c ,
Then
(x-a)/(b-a),
1.2.
- f(x N]I
of an a r b i t r a r y
= f((b-a)x)
y
side of the t r i a n g l e
(I.S).
f(x+~)
where
as the d o m i n a n t
we o b t a i n
that
+ k
and so on
is integrable ~ver
and m e a s u r a b l e
k
thereon).
[X,X + k]
[X,X']
D
for any
X'
> X,
(since
it
www.pdfgrip.com
Lemma
1.3.
if
X
is
as
in
1.2,
Lemma
then
for
x > X
,
X
f(x)
= c(x)
r
+ f
X
where
and
c
and
c(x)
~
are
measurable
~ c(]c I < =),
Proof.
For
x ~ X
~(x)
write,
and
§ 0
using
x+l
f(x)
= f
x
~
Lemma
on
[X,X'],
any
(f(x)-f(t))dt
if we i n t r o d u c e
> X,
1.2,
X+I
+ f (f(t+l)
X
new n o t a t i o n
X'
|
x
x
Then
bounded
as
f(t))dt
+ f
f(t)dt
X
by p u t t i n g
respectively
X
= ~(x)
+ f
~(t)dt
+ c
X
it f o l l o w s
that
r
= f(t+l)
6(x)
= f
- f(t) § 0
as
t § ~
from
(1.4)
,
and
x+l
1
(f(x)
f(t))dt
= f
X
-~
in v i r t u e
of L e m m a
1.4.
For
0
as
i.I.
c(x)
Lemma
(f(x)
f(x+~))d~
O
:
all
X
Hence
a(x)
x
+
c
> X~
-~
o~
the p r o o f
.
is c o m p l e t e
if we put
m
, for
some
X~ > X
,
n
X
(1.8)
f(x)
= c*(x)
+ f
r
X~
where
and,
ca
and
moreover,
~*
have
~
is
the
properties
Let
f*(x)
= f
Take
f(x)
~
in
Lemma
X
~(t)dt
= f
X
(1.9)
and
c
continuous.
X
Proof.
of
(f(t+l)
f(t))dt
, so that
X
- f*(x)
= c(x),
+ C
as
X
+
oD
> 0 ; then
X+~
f~(x § ~)
- f*(x)
= /
(f(t+l)
f(t))dt
X
= ;
(f(y+x+l)
O
Now for
y
in
[0,u]
f(y§
.
1.3~
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f(y+x+l)
and,
by L e m m a
i.I,
- f(y+x)
+ 0
uniformly
f*(x+u)
This
is true
argument
for
true
Clearly,
some
X*
all
> X
f*(x)
any
for
= f(x+y+l)
~ > 0
v < 0
for
; trivially
of Lemmas
X
such
y
- (f(x+y)
; hence
as
- f(x))
x +
+ 0
; hence
replacing
f(x)
true
1.1-1.3
so for
for
~ = 0
every
are now
; and by a s i m i l a r
~.
applicable
to
f*
, with
,
X
f*(x)
= ~*(x)
+ f
~*(t)dt
+ c*
X*
where
we
can
take
e*(t)
which
= f*(t+l)
is c o n t i n u o u s ,
since
-
f*(t)
f*(t)
is.
Hence
from
(1.9)
X
f(x)
= c(x)
+ f*(x)
= c(x)
+ 8*(x)
r
+ f
+ c*
X*
which
gives
Remark.
the
result
By r e p e a t i n g
of times,
we
far along,
All
"undesirable"
into
c*(x),
bounded
on
Theorems
vely
by
for
x > 0
Theorem
the
we
has
about
finite
I.i
the
which
1.2 now
/
exp
still
with
(1.8)
stage
already
is
where
that
from Lemmas
so that
e*(t),
x +
I.i
and
f(x)
order.
accumulated
it is m e a s u r a b l e
as
in the
number
suffi-
specified
increasingly
limit
mentioned:
x)}
an a p p r o p r i a t e
of any
say only
a finite
follow
{f(log
lemma
derivative
at any
we may
transformation
can
representation
behaviour
and
of this
a continuous
intervals,
, L(x)~
9
the p r o c e d u r e
can o b t a i n
ciently
the
required.
1.4
and
respecti-
= log L(e x)
i.e.
Representation
take
= c*(log
n(x)
x)
,
= c*(log
E(x}
x)
since
log x
x e,
c*(t)dt
X*
where
B = exp
Corollary
tion
(1.3)
X*
f
B
(log y)
Y
dy
.
to T h e o r e m
where
=
n
Any function
1.2.
and
c
defined and having representa-
have the properties
stated is slowly
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varying.
The proof is simple and is left to the reader;
one consequence worth noting,
large
x
(1.3) states
that for sufficiently
we may write a slowly varying function in the form
L(x)
where
in that
there is however
M(x)
= M(X)Lo(X )
is positive,
along and approaches
measurable,
a positive
a particularly well-behaved
b o u n d e d in intervals
limit
M
as
far enough
x + ~ ; while
slowly varying function,
L (x)
is
o
so that as
x +
L(x) ~ M L o(x)
where
~dt}
X
L ~ (x)
where
have
e(t)
the
is
continuous
representation
(1.10)
~(t)
where
the
prime
There
which
is
is
an e l e m e n t a r y
positive,
xg'(x)/g(x)
and integrate
but
t § ~
In fact
we
)
important
and has
converse
continuous
to
this:
derivative
any
for
function
x > B ,
~ 0
for
To see this put the left hand side
g , finally using the corollary above.
(i.ii)
will be regularly varying,
Refinement
as
B, and satisfies
the right hand side of
1.3.
zero
itself:
a derivative.
x § | , is slowly varying.
= e(x)
g
e(t)
0
indicates
defined,
(1.11)
and approaches
for
= x L'(x)/Lo(X
for some positive
as
e x p { fB
=
is, more generally,
of index
If
p , -~ < p < ~ , then
p
of D e f i n i t i o n of Resular Variation.
Characterization
of Regular Variation.
The defining relations
(i.i) and
(1.2) of regular and slow v a r i a t i o n
can be much w e a k e n e d without changing the theory.
sometimes
of use in applications;
ring them is to demonstrate
restrictive
Such refinements
however the chief purpose
that the r e l a t i o n
are
in conside-
(i.I) is not nearly as
as it at first appears.
A p r e l i m i n a r y result,
of w h i c h we shall have need in the sequel
is the following.
Lemma 1.5.
Suppose
a function
R , defined,
measurable
and positive,
www.pdfgrip.com
[A,~)
on
, for
(1.12)
A > 0
some
, satisfies
lira R R @ = r
X-~m
for
each
X
is f i n i t e
> 0
in a c l o s e d
and positive
, and for
Proof.
Let
some
finite
u > 0 .
it follows,
for
any
and
The
left-hand
the
Repeating
limit
Lemma
1.6.
that
(I.12)
measure,
Proof.
f(x)
exp
x+|
RR@
r
(1.13)
for
9 a S~
.
, since
that
>
be
then,
for
[a/b,b/a],
k-i
since
times,
a/b
y < xla < bla
and
say,
< I, b / a
by this
for a set
limit
for
such
y,
,
de-
it
follows
that
~(X)
any p o s i t i v e
value
interval.
of Lemma
S
> 1
1.5 are p r e s e r v e d
of positive
is f i n i t e
except
X , of positive
and positive.
Then
the
1.5 p e r s i s t s .
easier
x
then
have
, a set
o
0
covered
, all
f(x
in
the c o n d i t i o n s
the
We
defined,
y
>
,
and
It is s l i g h t l y
~ S
each
> o
merely
of Lemma
for
r
eventually
holds
,
argument,
+
Suppose
r162
is well
~(y)
= log R(e x)
{~}
X/y ~ b
for
R(u165
R(Xx/y)
= lim
by
on w h i c h
conclusion
X > 0
X~[a,b]
a ~
; i.e.
the o r i g i n a l
will
fixed
satisfying
limit
for y c [ ( a / b ) k , ( b l a ) k ] ,
~
each
R(Xx/y)
R(~'J
y
R(u
of
for
~(x)
, where
holds
is
y ~ xlb ~ a/b
noting
taking
(1.12)
Then
R(Xx/~)
=
and
> 0
t(~(Xx/~))
lim R ( ~ x )
x+~ R ( x J
exists,
, 0 < a < b < ~
interval.
~(X)
Then,
R(xx) =
R-K-CKY-
[a,b]
interval
on this
+ ~)
to w o r k
, ~(z)
that
f(x)
of p o s i t i v e
= log
as
with
the
~(e T)
transformed
for
T
such
x +
+ ~(~)
measure.
Then
if
v E S*
forms
that
where
www.pdfgrip.com
f(X+T+~)-f(x)
[] f ( X + T + ~ ) - f ( x + ~ ) + f ( x + T ) - f ( x )
+ r
as
x + -
; and
so
(1.13)
D = {~; ~ = ~ + ~,
according
T , ~ g S *}
interval
I .
for
all
mises
of L e m m a
The
shows
must
1.5,
r
Theorem
Lemma
1.3.
1.6,
Proof:
r
and
so its
have
varying
has
positive
defined
, where
measure.
in this
(if n e c e s s a r y )
the
conclusion,
way
Now
contains
of this
the
section
form
10
to L e m m a
lim ~
1.6 we
=
r (x)
following,
so the
defined
R
since
p
have
~ > 0
for e a c h
it
considered
sense.
~P , for some
> 0
the pre-
Under the conditions
Theorem).
the form
~(V)+r
I
is the
, and
as
transformations,
hold.
in the p r e v i o u s l y
assumes
According
u E D
Inverting
(Characterization
~(x)
D
is d e f i n e d
theorem
must
be r e g u l a r l y
S*
for
r
§
, ~ , T ~ S*
fundamental
that
where
t h e o r e m +, a
f(x)
~ c I , where
u = v + T
defined
Hence
f(x+~)
where
is w e l l
to a w e l l - k n o w n
a closed
+ ~(~)
of
satisfying
o
X-~
Then
for any
y > 0
,
R(~x)
and
so,
letting
for each
positive
x, y > 0 .
real
numbers,
limit
of m e a s u r a b l e
these
conditions
last
we
of Lusin's
This
for
the only
proposition,
proof
R-r(D--
= r
shall
Theorem
is the H a m e l
a function
solutions
instructive
since
are
to give
it is done
give will
functional
r > 0
also
in the p r e s e n t
of the
form
a simple
as an
setting,
being
It is known*
infrequently
serve
equation
, which,
is measurable.
functions,
It is, h o w e v e r
The
-
x § r (x)r (~)
(1.14)
R(x~x)
_
" R--T~-
k p,
direct
(1920, T h 6 o r ~ m e VII)
and R o s e n t h a l (1948,
that
proof
in e l e m e n t a r y
which
with
is the o r i g i n a l
pp. 1 1 6 - 1 1 8 ) .
a
of this
text
books.
of the use
Egorov's
memoir.
under
-" < 0 < |
illustration
+
, Steinhaus
e.g. H a h n
on the
a pointwise
www.pdfgrip.com
i0
T h e o r e m + and S t e i n h a u s ' s
theoretic
tools
(already used),
for the p r e s e n t
theory
t i c e d by several
authors.
u s e d in n u m e r o u s
other probabilistic
given
in p r o b a b i l i s t i c
Theorem
fying
(1.14) for
for all
mary
X > 0
functional
(i.15)
r
The
i8 necessarily of form
x,y
, where
equation
of m e a s u r a b i l i t y
do so initially.
is p a r t i c u l a r l y
it solves
we shall
(1.15)
x k = x, k = l , . . . , n
(1.16)
r
r
(1.17)
for p o s i t i v e
easy
to solve
if the a s s u m p t i o n
and we shall
show that the m e a s u r a b i l i t y
i m p l ies
it c o n t i n u i t y .
+ r (x n)
, we o b t a i n
, we have
(1.16)
= me(y)
integers
r
y=l
It is custo-
= r
nr
Putting
transforms
= he(X)
x = (m/n)y
for p o s i t i v e
and m e a s u r a b l e .
+ Xn) = @(Xl) + . . . . .
and if we put
whence from
, (1.14)
implies
@(Xl+ . . . . .
If we put
x)
is r e p l a c e d by one of c o n t i n u i t y ,
Subsequently,
(1.15)
= log r
is finite v a l u e d
~
and the fact that
First,
~(x)
X p , -~ < p < ~
form.
(1.15)
on
m,n
.
Thus
= ~((m/n)y)
= (m/n)~(y)
, we o b t a i n
r
= re(l)
rational
r .
Putting
x = y = 0
+
See B i b l i o g r a p h i c
are
are n e v e r
= r
~
this
but p r o o f s
equation
+ r
to w o r k w i t h
of the p r o p o s i t i o n
connections,
measure-
as has b e e n no-
finite, measurable, positive, and satis-
r
With the t r a n s f o r m a t i o n
to C a u c h y ' s
versions
to be the n a t u r a l
sight,
texts).
A function
1.4.
Proof.
(Further,
appear
at first
Notes
and D i s c u s s i o n .
in
(i.15) y i e l d s
of
www.pdfgrip.com
,) , s o
x)
at
(I.17)
any
holds
point,
it
~,(x)
.oh
is
evidently
:ore ( 1 . . t 5 ) ,
Further,
it.
any
is
also
readily
that
t
for
that
~
m(I-F)
<
I
such
checked
the
tt
positive
a number
(1.17j
x = O;
and
that
is
~(x)
of
~,
clearly
then
is
~(>
number
t:
c
is
exists
-2
Fn
is
nn > 0
, such
satisfying
r
is
It
negative
0).
x
compact,
exists
that
is
follows
a closed
and
the
closed
restricted
uniformly
Lusin
set
of
= x~
but
to
is
merely
any
closed
now i m p t i e s
F ~
measure
~p
r
of
t
such
I-F
subsets
,
{Fn;'
to
Fn
continuous.
of
~s con-
Hence
there
that
and
clearly
(1,.t5),
of
(1.15);
case
relative
The theorem
there
solves
satisfies
cor~tinuous
, such
x
in which
measurable
!~!
n-1
<
< !~n
Let
0 < ~n < ~n = ~!~n(nr~ ' n - ' ~ ' ) "
+ ~n ~ Fn
< 2n -2
c Fn
form;
a sequence
> # -- n
since
~+6
for
= const,
this
I@(~a+8) @(~)1
providing
also
.
where
to
m(Fn)
and
from
is
Thu.s t h e r e
r
that
a_
problem
length
restricted
tinuous,
is
to
, whose
any
for
solution
measurable,
interval,
follows
= -~.(--x)
co.r, t i n u o u s
~'e now p a s s
assumed
= 0
y = -.x
r
hence
r
= x,(~)
true
putting
for
measurable,
The
and
of
6n
set
be
of
measure
a fixed
~ g Fn
at
re.tuber
such
most
n-2
that
+ '~n '
that
-1
I '~"( ~ + 6 n )
when
Now
~
is
let
number
contained
G =
of
the
U
in
{7
*(~)l
a set
E i , the
j=! i:j
~Si s.
EnC
set
Putting
< n
Fn
of
Ki
~
such
that
belonging
I - Ei
, H =
m(En)
to
all
I - G
> ~ - 3n -2
but
, it
a finite
follows
e~
that
H =
I - G =
~
~
Ki,
the
set
of
points
in
I
belonging
to
in-
j=l i=j
finitely
many
of
the
re(H)
Ki
i m
Thus
(0)
Ki
i=j
for
each
Hence
for
j = l , 2 , . .... ; so
any
seauence
'
that
~n}
~
<
3i -2
~Z
- i=j
finally,
, where
re(H)
6n
= 0
, so
satisfies
that
0 < ~n
re(G)
= Z
<
'
~n
:
www.pdfgrip.com
12
that
follows
(1.18)
lim ~(m+6n)
~((~)
=
n.+~
for a l m o s t
Now,
every
let
I
fixed numbers
x = ~o
- m
and if
~
from
w
in
I
be the i n t e r v a l
satisfying
and let
[ml,~2]
Then
ml < m < mo < ~2
~
and
~ be
o
from (1.15), t a k i n g
' Y = m + ~n
is take n o u t s i d e
a subset
of m e a s u r e
= ~(eo-e)
+ lira r
= q,(%-~)
+ r
I , it follows
zero of
(1.18)
lim r
)
i.e.
(i.19)
lim ~ ( ~ o + ~ n ) = ~(~o)
n-~
for e v e r y
mo
satisfying
sequence which
I
and must be c h o s e n
any null
sequence
in a c c o r d a n c e
lim sup r
n§
It is p o s s i b l e
to s e l e c t
n.
< ~i
not q u i t e
of p o s i t i v e
(1.20)
0 < @
ml < ~o < ~2
is a p p a r e n t l y
with
numbers,
Here
{6n }
is a p o s i t i v e
since
it d e p e n d s
0 < 6 n < ~n
Let
on
{@n }
be
and s u p p o s e
) > lim inf r
n§
a subsequence
) .
{en}
' and
{e n}
from
such that
i
1
lim r
i§
1
) = lim sup r
n§
=
b y (1.19).
A similar
lira inf
so we have
Since
"
arbitrary,
{0 n}
argument
(%)
gives
the result
that
~(~o+Sn ) = ~(~o ) ,
a contradiction
is a r b i t r a r y ,
right-continuous
r
) ,
to
(1.20);
except
at any p o i n t
mo
and c o n c l u d e
e a c h limit
that
@n > 0 , it follows
' as
I
can be a r b i t r a r i l y
is
r
~(mo).
is
chosen.
www.pdfgrip.com
13
To obtain
left continuity,
(x)
and h e n c e we o b t a i n
tinuous
~o = -xo
1.4.
' which
x~
entails
first,
alternative
rather
w
strong
these
; then from
(1.15)
that
~
xo
Functions
that
~
right
is arbitrary.
conat
9
and A l t e r n a t i v e
the definition
properties.
is
is left continuous
Proofs.
of a slowly varying
The p u r p o s e
and s e c o n d l y
of this
section
to consider
some
proofs.
have the
to
One p o s s i b i l i t y
for
e(t)
as r e q u i r e d
However,
clear
it
is
we may s t i l l
s(t)
the
function
as c o n t i n u o u s ,
with
that
since
more d e e p l y ,
We b e g i n b y c o n s i d e r i n g
desirable
implies
the proof,
seen in
to explore
= ~(-x)
m a n n e r as b e f o r e
o f S.lowl?' Va r y i n ~
We h a v e a l r e a d y
is
~(x)
: ~(x+y)
This
completes
The S t r u c t u r e
function
+ ~(y)
in similar
at any point
put
, gotten
(1.3).
expressed
that
from Lemma 1 . 4 ,
i f we d o n ' t
get
a (simpler)
measurable
insist
p r o o f o f Lemma 1 . 3 s l i g h t l y ,
by w r i t i n g
-i < x+~
= m
f
(f(x)
x
(1.3)
in fact
for
is
allows
sometimes
L(-)
itself.
us to t a k e
it
Theorem's statement.
on t h e
representation
and b o u n d ed ;
It
in terms of
by t h e R e p r e s e n t a t i o n
still
f(x)
representation
~(t)
continuity
by u s i n g
let
of
s(t),
Lemma 1 . 3 ,
us g e n e r a l i z e
x ~ X , for
any
the
~ > 0
x
f(t))dt
+ f
{f(~+t)
- f(t)}dt
X
X+O)
+
It follows
as before
fX
f(t)dt
that we can take
~(t)
;
{f(e+t)
c(x)
=
~(log
- f(t)}/e
and so
using the
fact
that
f(t)
x)
= ~
= l o g L(e t )
; so t h a t
Xo (= e ~) > t
, we h a v e t h e r e p r e s e n t a t i o n
boundedness,
but not necessarily
(1.21)
which
E(x) _ log 1 ~o l o g
is a simpler
expression
1 log
continuity,
(1.3),
of
f o r any f i x e d
number
with measurability
e(x)
g i v e n by
L(•~
-L'TiT-
than that entailed
by using Lemma 1.4,
and
www.pdfgrip.com
14
and w i l l
be made
use
rariness
of
that
It is not
~o'
difficult
representation
will
become
form
continuous
define
of r e p r e s e n t a t i o n
to see
that
even w i t h
already
is e s s e n t i a l l y
from
the
derivative
fl(t)
at the
c(x)
satisfies
c(x)
E
but
[A,~)
end of
w
if a s l o w l y
(i.ii).
is as follows.
, where
continuous
non-unique;
of the
is far
arbit-
f r o m unique.
required,
in any
case
the
this
sequel.
discussed,
for c o n t i n u o u s
for continuous
on a c c o u n t
kind
(1.3)
t > y : log A
It is clear,
this
apparent
We have
simple
of later.
that we
varying
A general
Take
f(t)
is the d o m a i n
of
can
simple
L
a
with
construction
= log L(e t)
L
get
function
for
, as b e f o r e ;
and
by
t-n
(1.22)
fl(t)
= f(n)
+ 6(f(n+l)
f(n))
f
u(l-u)du
,
o
for
> ~
n ! t ! n+l
.
(1.23)
for
, and
all
n ~ no
, where
is the s m a l l e s t
no
integer
Since
f{(t)
n _< t _< n+l
f~(t)
:
6(f(n+l)
-
, it f o l l o w s
is c o n t i n u o u s ,
f(n))(t-n)(1-{t-n})
that
for
all
9 f~(n)
n _> n o
:
0
,
that
and
If~(t)l ~ ( 3 / 2 ) l f ( n + l )
f(n) l
for
n
< t
m
-
< n+l
-
Also
]fl(t)-f(t)l
for
n < t < n+l
!
Now,
as
n
§
~
If(n)-f(t)I
, where
~ ~ ~(t,n)
, f(n+l)
~
n § ~
for
n
< t
< n+l
by
(1.24)
in
, f(n)
the
-
-
16(f(n+l)-f(n))(t-n)~(l-~)l
is c o n t a i n e d
(3/2)If(n+1)
§ 0
; and
f(t)
§ 0
uniform
as
convergence
-
w.r.
Thus
f(t)
§
as
0
t
, fi(t)
+ |
§ 0
.
in
[0,t-n],
fCn) l
t + |
f(t)
Lemma 1 . 1 .
fl(t)
l +
f(n)
f(t+~)
established
f(t)
If(n )
+
, we have
to
v
in
that
[0,1]
of
www.pdfgrip.com
15
t ~ no ,
Now, for
t
= fl(no)
fl(t)
fl (u) du
+ f
n
so t h a t i f
we p u t
log
for
t
o
~ no
Ll(et ) = fl(t)
LI(X)
= exp
, we g e t
{fl(lOg
x)}
, x L exp n o
const 9 exp { flog
n
x f ~ (u) du }
o
x f{Clog t)
exp { f
t
K
const,
, = K say
dt }
Put
c(x)
=
so that
that
x > B
C(x)
C(x)
is
~ 1
say.
a measurable
as
x + |
function,
, from
defined
(1.24);
so
for
C(x)
is
x L K
bounded
and
such
for
Hence
X
L(x) = exp {n(x) + f
e(t~
dt}
B
where
(r
n(x)
~ 0
and
by
(1.2s)
e(t)
(I.24),
e(t)
are
as
as
required
t § -),
= f~(log
t)
by t h e
Representation
Theorem
and
,
t i
B
say
This reasoning has a number of important consequences; for first
it follows that numerous
fl(t) can be constructed in similar manner,
merely by replacing the integral
X
6 f
u(1-u)du
0
in (i.22) by the indefinite integral of some other suitable probability
density on [0,i], (suitable in that it will render f~(t) continuous).
Secondly we have the following :
Lemma 1.7.
If L(t) ie a slowly varying function which is eventually
non-decreasing (non-increasing), then the continuous
E(t) in its representation for sufficiently large values may be taken as satisfying
(t) L 0 (~ 0)
www.pdfgrip.com
16
Proof:
If
is
; and so
f(t)
whence,
L
from
fi(t)
will
at the Representation
step of Lemma
L
itself follows
directly
fact
this
boundedness
we have given, was apparently
Representation
Indeed,
0
,
on
finite
the
This last,
property
if established
intervals
function
not realized
9
.
condltlons
a consequence
is
form that
in the general
for some time in the histori§
on
a substantial
L(.)
obstacle,
to obtain the
prior to the above argument.
sufficiently
far) directly
from
1.2
(boundedness
(1.4) without
on
the
(which can then be deduced as a consequence@).
For
let
S n = {~ > O; - ~U --< f(x+u)
From
Theorem
one can arrive quite easily at Lemma
intervals
agency of Lemma i.i
>
far enough along.
and this presented
auxiliary
Theorem,
(i.e.,
the intermediate
from the uniform convergence
of a slowly varying
various
enables us to
section.
cal evolution of the theory,
necessitating
9
that a slowly varying func-
states
or from the Representation
that
of the definition
invoking
then so
fi(t) L 0 (! 0),
c(t) ~ 0 (~ 0)
directly
1.2, which effectively
(as we have shown)
The
(decreasing),
satisfy
Theorem using only Lemma I.i
without
Theorem)
in the manner of the present
a
satisfy
i8 bounded on any finite interval
of course,
finite
increasing
(1.23) will
it is clear that the above construction
Uniform Convergence
tion
given by
(1.25) , e(t)
Finally,
arrive
is eventually weakly
(1.4),
there is an
it follows
no
that
such that
U S = (0,~),
n=l n
Sn
f(x) _< a~
and since
has positive
Lebesgue
, V X --9 n}
L
.
is measurable,
measure.
Now it
o
is easy to check for any fixed
~i + u2 e S n .
Thus #
Sn
n
, that if
contains
Ul,U 2 g S n , then
a half-line,
(T(a),-).
Thus for
, which
is tantamount
O
all
~e(T(a),~),
f(no)
- a~ ~ f(no+~ ) ~ f(no)
+ a~
to the required.
We have already mentioned
tation
(1.3) where
of times.
e
Indeed more
in w
that we may obtain the represen-
is in fact differentiable
is true
any specified number
in this vein as we now state
:
Such as continuity of L(.), which of course implies boundedness on
closed intervals.
See Bibliographic Notes and D i s c u s s i o n to this
chapter.
@ L4tac (1970a,b).
# Steinhaus (1920) TheOr~me VII.
* Adamovi6 (1966).
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17
For a given slowly varying function
L(x)
on
[A,=)
there exists
another, infinitely differentiable, slowly varying function
L l(x)
with
the following properties :
1~
L l(x) ~ L(x)
, as
2~
Ll(n)
, all integer
-- L(n)
x § | ;
n
sufficiently
3~
If
L
is ultimately
monotone,
4~
If
L
is ultimately
convex
then so is
then so is
large
;
L1 ;
L1
II
Propositions
tion carried
1.6).
of this kind can be obtained
out in this section
The infinite
proposition
above
probability
density
(of which
differentiability
follow
readily
on
by the kind of construc-
a further
and parts
by choosing
1~
example
and
in (1.22),
3~
is Exercise
of the
in place of the
[0,i]
x
f
u(1-u)du
,
o
the density
x
Jr exp - {u(l-u)}-idu /
o
is here replaced
(Proposition
2~
1.5.
Prpperties
Further
These
most
and the discussion
Theorem
of useful
easily
Varying
of regularly
Convergence
imply a number
speaking,
exp-
{u(l-u)}-idu
o
: L(e n) = Ll(en))
of Regularly
The basic properties
in the Uniform
by
1
Functions.
varying
functions
are embodied
and the Representation
secondary
deduced with
properties,
Theorem.
which
are,
generally
the aid of the Representation
of some of these
is the purpose
Theorem,
of the present
section.
Before
already
(and hence
tervals
many
proceeding,
been deduced
a regularly
sufficiently
applications,
integrals
involving
we recall
in w
varying
far along
what
regularly
important
defer
it to the next chapter.
and extensive
In the sequel
the symbols
important
that a slowly
function)
the real
is of interest
ently
functions.
that one most
namely
varying
to merit
is bounded
line.
Also,
we mention
L, LI, L 2 , denote
separately,
in-
that in
behaviour
this topic
discussion
has
function
on all finite
is the asymptotic
functions;
property
varying
of
is sufficiand we
slowly varying
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18
1~ .
Vo~
y > 0
any
, xYL(x)
+ -
, x-YL(x)
a8
Proof.
We give a p r o o f
the r e p r e s e n t a t i o n
of
for
L(x)
xYL(x)
X
§ 0
~
~
; the other
, we h a v e
that as
case
is similar.
Using
x §
X
XYL(x)
~ const
exp
{y log x ~ ~
E(t) it } "
B
X
const,
where
X
is c h o s e n
as is p o s s i b l e
exp
{y l o g
sufficiently
~(t) § 0
since
x (_~
y log x + fX
large
as
(t)
t
x + _~
at
}
t > X ,
so that for
t § |
I~(t) l
9 ~/z
Now
dt > y log x
x 1
f X [ dt
(y/2)
= (y/2) log x + const.
-~
This
completes
the p r o o f
~
X
as
->
of the p r o p o s i t i o n .
~
R
II
log
2~ 9
Proof.
L(x)/log
Using
the
x + 0
as
representation
log L(x)
= n(x)
x § |
for
+ f
L(x)
x s (t)
B
Now,
let
[r
~ > 0
< ~ .
be
Thus
an arbitrarily
for
x
, for
dt
sufficiently
large
.
t
small
number;
then
for
t !
X m X(~),
x 9 X
X
I~
~
dt I <_ ~ log
x
+
const.
X
so
that
since
for
n(x)
x
> X ~ X(6)
is b o u n d e d
the p r o p o s i t i o n
30.
L~(x)
for large
follows,
,
for
any
x .
Since
6
is a r b i t r a r i l y
small,
g
a
satisfying
-~
<
a
<
~
,
LI(XJL2(x)
,
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19
LI(X) + L2(x) are 8lowly varying.
is slowly varying.
Proof.
There
about
the
positivity
are
sum
and
and
only
the
two
L2(x)
If
non-trivial
composition
measurability
propositions
of two
hold
slowly
x
as
+ ~
§ |
LI(L2(x))
to be p r o v e d
varying
here,
functions:
the
trivially.
LI(XX ) + L2(xX )
L 1 (X) + L2 (x)
Ll(/X) t
Ll(X)
= 7
= (1+~1 (x, X))
where
for
t
L2(~x) f
E l ( x ) +L 2 (x]
fixed
X > 0
= 1 +
I
LI(X)
L1 (x)+L2(x)
, r
§ 0
z
Lz(x)
+~
t
L1 ( x ] +L 2 (x)
1
as
I
(1+r
+
L1 (x]+L2(x)
x § |
, i = 1,2
}
,
] Li(x) }
i=l r
~Li(xl+L2(x)
Z
§ 1
as
x § |
, for
each
~ > 0
since
0 <
For
fixed
Li(x)
< 1
L1 (x) +L2 (x) --
~ > 0
Lz(x~) ]
LI(Lz(X~))
=
L
1
[L2(x)L2-L-j-~j /
LI(L2(x))
Ll(L2(x))
= Ll(L2(x)(1 + r
and
since
Theorem
L2(x)
applied
§ =
to
as
L1
x + =
(since
, it
follows
1 + r
from
§
i)
the Uniform
that
the
Convergence
above
is
L1 (L2 (x))/L1 (L2(x))
-- 1
for each fixed
~ > 0 .
H
4~ 9
U(X) ~ L(x) , ~(x) ~ L(X)
as
x
§
|
where
for
any
fixed
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20
y
9
, L
0
and
are specified for
xY~(x) =
B
x ~
by
{tYL(t)}
sup
b < t < x
- -
xYL ( x )
:
u
inf
{tYL(t) }
X < t < ~
(where B is taken sufficiently
representation (1.3).)
Thus,
as a c o n s e q u e n c e ,
equal
to a non-decreasing
ally
index,
for w h i c h
xY~(x)
, being
~(x)
and
Proof.
~(x)
For
the
above
monotone
are
xYL(x)
, with
regularly
formulae
and
L(x)
large e.g. for
finite
are
fixed
to be given by
y 9 0
varying
constructions.
valued,
are
, is a s y m p t o t i c -
function
with
the same
(xY~(x)
clearly
and
measurable,
whence
also.)
x > B
1 < L(x)/L(x)
Sup
:
{tYL(t)}/xYL(x)
.
B < t < x
Suppose
there
exists
a sequence
of p o s i t i v e
numbers
{x r}
, xr
that
such
(1.27)
1 < lim
E(Xr)/L(Xr)
.
r+~
Then
there
exists
(1.28)
for
the
a positive
~ > 0
such
that
1 + 2a < r ( X r ) / L ( X r )
r ! r o ~ r o ( a ).
interval
(1.29)
Now
for each
B ! Yr i Xr
such
such
r
, we
can
find
a number
Yr
in
that
sup
{ Y r Y L ( Y r ) / X r Y L ( X r )} + ~ 9
{tYL(t)}/xrYL(x r)
B ~ t ~ xr
Clearly
(1.28)
the
and
Yr
(1.30)
Since
fact
as
m a y be
(1.29)
chosen
it f o l l o w s
monotone
1 + ~ < YrYL(Yr)/xrYL(xr
tYL(t)
+ ~
as
the m o n o t o n e
r § -
, and
Representation
t § |
, by
sequence
L
{yr }
is b o u n d e d on
Theorem
non-decreasing
with
r
.
From
that
in
that
for the
I~
) , r _9 r o
it f o l l o w s
satisfies
finite
right-hand
Yr
from
§ "
intervals.
side
of
(I.30)
, since
xr
Invoking
the
(1.30),
§
we m a y w r i t e
