·
.
UNIVERSITY OF NORTH CAROLINA
Department of Statistics
Chapel Hill, N. C.
A NOTE ON CHARACTERISTIC FUnCTIOllTS WHICH
VANISH IDENTICALLY IN AN DH'ERVAL
by
Halter L. Smith
Septenber, 1961
This research was supported by the Office of
Naval Research under Contract No. Nonr-855(09)
for research in probability and statistics at
Cha.pel Hill. Reproduction in whole or in part
is perQitted for a.ny purpose of the United States
Government.
Institute of Sta$istics
Mimeogral?h Series No. 306
A NOTE ON CHARACTF...RISTICS FunCTIONS 't'ffiICH
V.AHISH IDENTICALLY IN .All INTERVAL
by
Halter L. Smith
Some years ago, in connection with some unpublished work in the
theory of queues, the question arose as to whether the characteristic
function of a non-negative random variable could vanish identically in
an interval.
The purpose of this note is to sbow that such a thing is
impossible.
There are, of course, characteristic functions Which vanish identically in intervals.
f(x) =
In fact the probability density function
2
2 sin &X!2)
~
,
-co
<
x
<+
co
,
x
is associated with the characteristic function
¢ (Q) =
1 -
IQI
0
,
IQ I
,
JQ J
which vanishes in semi-infinite intervals.
-<
1,
> 1,
This characteristic function,
however, is associated with a distribution over both positive and negative values.
It is not clear what can be achieved for distributions
over positive values only, althouGh the follQ't>Ting easily proved theorem
gives some indications.
Theorem 1.
If 0 <
Q
l
< Q2 < Q < •..
angles, and if k , k , k , •..
l
2 3
and if
3
is an increasing sequence of
is a sequence of positive integers,
2
00
(1)
<
E
00
n=l
then there is a non-necative random variable whose characteristic function
vanishes at
Ql' Q2' Q3' ••• , the zero at
Proof.
~
Let
Qn being kn-fOld, for all
n.
be a random variable associated with the probability
density function
0< x<
for
=
Let
like
,
"
~,Xn ,
o
21C
~
,
otherwise.
... , Xn(Ien )
be k
independent random variables distributed
n
'"
(k )
X, and write Z = X + X +. •• + X n
Then
n
n
n
n
n
,
and
1C
(3)
k
n
1
kn
( 4)
1C
2
= 3 Q2
Furthermore, in view of
bounded.
(1),
n
the randOQ variables
Thus if Zl' Z2' Z3' ••• are
independ~nt,
1Zn 1are uniformly
the series
00
S
=
E
j=l
is almost certainly convergent, in virtue of the three series theorem and
3
(3),
(1),
(4).
and
The characteristic function of S is evidently
co
=
where
* (Q)
,
IT
n=l
is given in (2).
n
viously has a
Since
S
1s non-negative and ¢(Q) ob-
zero at Qn' for a.ll n, the theorem is proved.
l~n-fold
We do not for a moment suppose that Theorem 1
possible result of its kind that can be proved.
is the strongest
It merely provides, as
we ha.ve said, some indication of the distribution of zeros which can occur.
It would naturally be of interest to obtain critical results, but we have
been unable to obtain such.
Nevertheless the following theorem shC't'1's tbat
the characteristic fun.ction of a non-negative random variable has some
restriction on the location of its zeros.
Theorem 2.
If
¢(Q) is the characteristic function of a non-negative
random variable then ¢( Q) cannot vanish identically in an interval.
Proof:-
Let
F(X)
be the distribution function of the non-negative
random variable and write
J
co
(6)
i
(u + iV) =
ei(U+iV)X dF(x)
,
o
~
(u + iV)
Evidently
for v > 0,
@
=
iJr
(u + iV)
1 - i(u + iV)
(u + iV) is reaular for v >
° and, since fie u + iV) I ~ 1
J
4
+00
I
~
2
1
(u + i v)
du
v.
exists and is bounded for every positive
Thus, for v > 0, by Titchmarsh (1948, 1'- 125), we have, for I
Z
> 0,
+00
(8)
~
(z)
J
1
= 21C1
<l' (u)
"There, in the present situation,
of
cI>
du
u - z
,
(u) is the ordinary limit as
cI>
v -> 0+
(u + i v) • Thus
00
<l' (u)
=
'" (u)
1 - iu
1
= --....;~1 - iu
J
dF
°
Now suppose the cInI'acteristic function
in
t(u) vanishes identically
< b. Then (8) may be rewritten, for I z > 0,
(a,b), a
<l'(z)
(x)
1
=-21(i
j+j
-00
b
<P
(u)
du
u - z
The right side of (9) is a reBUJ-ar function of
plane.
z
in the upper half-
It is not hard to see that this regular function can be analytically
continued, by a well-knmm expansion method, through the segment (a,b) on
the real axis into the lower balf-plane.
Thus
domain which contains a limit point of zeros.
<i>(z)
is regular in a,n open
TIlis implies that
vanishes identically, and thus provides a contradiction.
conclude that
<l'(u)
cI>(z)
He:may therefore
cannot vanish identically in an interval.
'l'le point out tInt an alternative proof of Theorem 2 can be deduced from
a result of
a circle.
Bieberbach (19'1 , 1"
156)
concerning functions analytic in
5
REFERENCES
BIERBERBACH, L., (1931). 'Lehrbuch der Funlttionentheorie' Band II,
Ed.
B. G. Teubner, Leipzig and Berlin.
TITCm~H,
E. C., (1948).
'Theory of Fourier Integrals',
Oxford University Press.
2nd. Ed.
2nd.