Internat. J. Mh. & Ma. Si.
Vol. 5 No. 4 (I982) 721-728
721
ON THE SPECTRUM OF A DISTRIBUTION FUNCTION
AND ON UNIQUE FACTORIZATION
T. PHAM.GIA
Departement de Physique-Mathematiques
Universite de Moncton
Monc ton, N.-B., EIA 3E9 CANADA
(Received November 13, 1980 and in revised form June 30, 1982)
ABSTRACT.
The spectrum of a distribution function is related to the quasi-analytlclty
of a class of functions
{C
M(j)I
where
M(j)
is a multisequence of positive numbers.
For a regular multlsequence, a result on the uniqueness of characteristic function
decomposition is given.
KEY WORDS AND PHPES.
analytic, regular,
Distribution, chractertic function, spectr, quasi-
factorization.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
i
Primary 26A93, 60E05.
INTRODUCTION.
A general principle formulated by N. Wiener, that a function and its Fourier
transform cannot both be very small at infinity, has been the origin of important but
quite different results in the fields of Complex Analysis and Probability Theory.
In
this note, we give some results connecting these two fields and an example to illustrate this connection.
Two fundamental theorems on the above principle are recalled below.
rely on the classical Paley-Wiener Theorems.
They both
Theorem i.i by Ingham and Levlnson [i]
has been used by various authors to establish results in Probability Theory, concern-
ing the spectrum of a distribution, unique factorization of characteristic functions,
etc. (see e.g. [2], [3]).
Theorem 1.2 by S. Mandelbrojt [4] has seen many applicat-
ions in Analysis, especially in the theory of quasi-analytic and non-quasi-analytic
functions (see e.g. [4], [5], [6]).
T. PHAM-GIA
722
THEOREM i.i.
Let G(u) e L(-, ) and suppose that
F(x)
l-i2
O(e
Let G(u)
(u)
G(u)-iUxdu"
where e(u) is a non-decreasing positive function such that
du
2
u
-e’u))
I
Then F(x) cannot be equivalent to zero in any interval unless it is equivalent to
zero over
If C(u) is a non-decreasing function on [0,
THEOREM 1 2
u
l’d, an
then
2
entire function F(z) not identically zero such that
lF(z)
-<
6lyl
c(Izl)
A number of deinltions and other results to be used later are recalled below.
A point x is called a point of the spectrum of a probability distribution F if
V6 > 0, F(x + 6) > F(x- 6).
_=
f(t)
eitxdF(x)
The characteristic function f of F is defined by
and the following theorem can be found in [2], under a slightly
different form.
THEOREM 1.3.
Let @(t) be a non-negative non-decreasing function for t > 0 and
f(t)=(e -@(Itl))
@.(t) dt
integral
let f(t) be the characteristic function of a distribution F such that
Itl
for large
Then F has (_oo,) as spectrum if and only if the
I-
t
2
diverges.
The proof of the above result is essentially based on Theorem i.i and various
properties of the Rademacher Series (see [2]).
2.
MAIN RESULTS.
Let
{M(j)}
(Jl’
be a multisequence of positive numbers where (j)
m
being an integer > 0 1 -< k < m. We set
jk and denote by
k=I7"
class of C
complex-valued functions defined on
fJl
8x Jl x Jm
I
m
where
f
and
Bf
1
C{M(j)}
the
such that
<
are positive constants which depend only on f.
we define the class
C{M(j))
Jm )’ Jk
Following Lelong [7],
to be quasi-analytic if it does not contain functions with
SPECTRUM OF A DISTRIBUTION FUNCTION
723
[7],[8],
Various properties of these classes have been obtained (see
compact support.
and [9]) and differ from the classical one-dlmensional case.
Let F be a probability distribution function with characteristic
THEOREM 2.1.
{M(j)I a multisequence
{llt;(J)l f(t) lloo/M(j < oo.
function f and
sup
of positive numbers such that.
Then F has (_oo,=) as spectrum if and only if
PROOF.
C{M(j)}
is quasi-analytic.
We begin by extending the ideas of Valiron points and half-planes to
several dimensions.
{M(j)}
Let
being an integer > 0.
p
We set
rapidly than any function of the form
m+l
In R
y
-log
m+l.
O; there
rP,r
{p}
grows more
> 0.
Jl
I
Xm
Jm and
{M(j)},
called the Valiron points of the sequence
in the lower halfp
r
Since-- 0, r > 0 and a > 0, there exist at most a finite number
p
of indices p such that p log r- log
a >
and suppose that
let s consider the points with co-ordinates x
M(j),
space of R
{M(j)}
inf
I(J)I=P
Jm)’ Jk
(Jl
be a multisequence of positive numbers where (j)
p
> a.
A fortiori, f(r)
(rl,
rm)
and
are at most a finite number of Valiron points such that
Jk
Now fix (r) and fix a > 0.
Let’s
log r k
log
M(j)
(2.1)
a.
m+l
determined by the
consider the hyperplane in R
equation:
m
Y
k-- xk
log rk
+ a.
It intersects the y-axis at the point with ordinate a and the
point
a
log
r
with i < % < m.
By (2.1), b
point is above y =-7 x log r
k
k
+
b(rl,
rm)
Valiron point is on the hyperplane itself.
k=l
b(r
rm)
I
and, at the same time, at least one
such that no Valiron
rm)
_>
-log
M(j)
or
b(rl,
m)
r
>-sup
(j)
at the
Hence, for Valiron points, we have
log rk + b(r I,
Jk
x-axis
{_ Jk
k:l--
log r k
log
M(j)};
T. PHAM-G IA
724
in fact
r
rm
b(r I,
sup
(j)
{log
log
T(rl,
Jm
Jl
r
m
I
M(j)
rm
where
r
rm)
T(rl,
For (r)
m
r..
y
{M(j)}
rm)
(rl,
x log r
k
k
+
We set r
r^z
I
main diagonal of R
m
b(r)
r
M
Jm
m
(j)
given, the hyperplane H((r)) with equation
rm)
b(rl,
(r)
at the point
sup {
(j)
Jl
I
is called the Valiron
hyperPlane of the sequence
m.
rm
R
r and consider the Valiron hyperplanes defined on the
m
x + b(r). Here
whose equations become y
-log r
=E
k
kl
log (r) where (r)
T(r, r,
r).
We can see from the above arguments
that b(r) is a positive non-decreasing function of r > 0.
Suppose now that
C{M(j)}
is
b (r)
By [8] we know that
quasi-analYtic.
i
dr
r
diverges.
Let
sup
(J)
Then
If(t)l
<
Y
M(j)
and hence
If(t)
0(I/y(Itl)
as
t > 0%
Using Theorem 1.3, F then has (_oo,oo) as spectrum.
(_oo,) as spectrum.
Since
If(t)
0(e
-b(Itl))
t
,
Conversely, suppose that F has
b(t)
the integral
ges again by [8], this is a sufficient condition for the class
C{M(j)}
’t2be dt diver-
to
quasi-
analytic.
For one dimension, we have the following example and corollary.
EXAMPLE.
P(X)
[
02
If we associate
+ (x
) 2]
to the Cauchy distribution
< x <
, @, D
(with density
real, 8 > O) the sequence
we can easily verify that the above theorem is satisfied.
Mj
e
SPECTRUM OF A DISTRIBUTION FUNCTION
be a sequence of positive numbers such that M
Let {M }oo
0
J j=O
COROLLARY
M2n
<
Mn-i Mn+l
Y.
n=l
Mn_ l
I
1,2,3,..., and F is a distribution fuctlon with characteristic
with n
I ItJf(t) llo/M}
function f such that sup
j>_0
only if
725
< oo.
Then F has (_oo,oo) as spectrum if and
oo.
M
n
By the classical Denjoy-Carleman Theorem (see [9]), if M. is logarithml-
PROOF.
tally convex, the divergence of the integral
gence of the series
dt is equivalent to the diver-
Mn_ I
F.
n=l
1
t
M
n
UNIQUE FACTORIZATION.
3.
An interesting property of distribution functions is that the cancellation law
does not hold in general, i.e. F
F
I * 2
F
* F3
1
does not imply F
has given examples of characteristic function f such that f(t)
fl(t) f3(t)
with
f2(t)
#
F(x + a)
F(x
fl(t) f2(t)
F(-x), Vx and, for
associate the marginal multisequence
M
l(j)l, 0,
{Mj)} defined by
M0, I(J)l, 0,
M0,
A multisequence
DEFINITION.
i)
2)
M
0
0
M(j)
{C
>
Mj)}
0
V(J),
M*j
a
> 0,
a).
We now consider a more specific multisequence called regular.
Mj)
Gnedenko [I0]
3.
f3(t).
For a distribution function F, we define F*(x)
(A a F)(x)
F
2
..{M(j)I
with
{M(j) },
let’s
0
(J)l,
0
For
V(J)
is regular if:
sup
(j)
{(
M(j)
M
<
(j)
is quasi-analytic with each marginal sequence
0
of the form M
0
0
(!)
[()]
where
m
F.
i and (x) is a continuously differentiable function (depending on the mark=l k
1
ginal sequence) with (0)
(x) > 0 and x.’.(XL)
oo.
0(i) as x
’
REMARK.
(x)
The class of regular multisequence, which is obviously not empty, was
introduced first by Hirschman [ii] for one dimension, to generalize the m-th logarithmic sequence.
T. PHAM-GIA
726
THEOREM.
Let
C{M(j)}
be a regular class of functions and f the characteristic
Then there exists a continuous non-decreaslng function (t) for t > O,
function of F.
such that
0(e-ltl/(Itl)), then F has (_oo,o) as spectrum
(AaF*)(u) 0(e-Itl/([tl)) for some a > 0 and f admits
l)
If f(t)=
2)
If
fl(t) f2(t),
tion f(t)
PROOF.
f2
then
where
We first show that, for a regular multisequence
(r)
fl"
is uniquely determined by f and
b*(r) associated with the marginal multisequence
r +
a decomposi-
{Mj)}
{M(j)},
is such that
is a continuous, non-decreasing function with
the function
b*(r)
r
’(r)
as
9(0) > 0.
We have
(3.1)
with
P* linf(J)l:P {Mj)}
[m(p)]P
We also have %<0)
(x)
l,
>
[i (p) ]p
(p!)
x’ (x__)
O, and (x)
m
m
[(p)]P
p!
(p!)
:
where
.
o(1) as x
k1
.
Following Hirschman [ii], write
b*(r)
max
{p log
r- p!
p log
(p)}
(3.2)
p>0
and let p*(r) be the integer for which the maximum is attained.
% pP e-p,
dG(p)
we have
log p
log r
dp
dG(p)
), the solution of
attained as
p!
p log r- p!
G(p)
,riable in
a continuou
d
dp
p log (p), the maximum would be
Using Stirling’s formula
0.
(log F(p + i))
log (p)
If we now treat p as
log p
+ (I) by the
+ o(1) and, therefore,
properties of
.
Hen ce,
log r
and r
%
s(r)
",
log s(r)
b*(r)
%
+ (i)
(s(r)).
Putting (3.3) in (3.2) when p
and so b*(r)
log (s(r))
r
(s(r))
for r > 0.
p*(r), we can obtain
p*(r) [i + (i)]
%
s(r)
(3.3)
SPECTRUM OF A DISTRIBUTION FUNCTION
and s, there exists a continuous function
By the non-decreasing property of
(r), non-decreaslng with (0) > 0 such that b*(r)
Now let
g
sup
{(
<
Mj)
and
r
%
Now,
Mj)
M(j)
Mj)
I (J)l
n
> M
(j)
then afortiori f(t)
=O(1/X(Itl))
H(j)
>-
Mj)
implies l(r)_< l(r).
(j) implies %*(nr) < %(r) by (3 I).
being quasi-analytic, the integral
above inequalities, implies the divergence of
for r > O.
(r)
We have l*(rir) -< t*(r) by ’eorem 2 of [8] while
On the other hand,
727
I b()r
b(r) dr diverges and by the
2
dr.
If f(t)
and by Theorem 1.3, F has (_oo,) as its spectrum.
The second part of the theorem follows from a result by E. Lukacs
combined with the regularity of
C{M(j)}.
([3, Theorem
The proof, based on classical results of
Khintchine, the inversion formula, and Parseval’s Theorem, follows closely Lukacs
(see [3]).
Research partially supported by NSERC (Canada) grant A9249.
ACKNOWLEDGEMENTS.
The
author wishes to thank the referee for many helpful suggestions.
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728
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T. PHAM-G IA
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