Adv.
Appl. Probe 21, 708-710 (1989)
Printed in N. Ireland
© Applied Probability Trust 1989
SOME POWER SERIES WITH RANDOM GAPS
PHILIP
HOLGATE,*
Birkbeck College,
London
Abstract
Power series E zX
n
are studied, where
{X
n
}
is a strictly increasing
integer-valued stochastic process.
RANDOM
POWER
SERIES;
NATURAL
BOUNDARY
The
function theoretic properties of
random
functions are discussed in the monograph of
Kahane (1985),
and
random
power series in particular are surveyed by Lukacs (1975),
Chapter
5. Walk (1968) drew attention to
the
parallels between probability 1 properties of
random power series and those of certain deterministic lacunary series. Brief references to the
properties of fixed lacunary series can be found in, for example Titchmarsh (1939), §7.4.
Arnold (1966) studied series in which
the
sequence of powers for which
the
coefficient is
non-zero is specified deterministically,
but
the
values of
the
coefficients
are
random. This
note
is about lacunary series of
the
form 1(z) = E zX
n
in which the gap lengths are random
while the coefficients
are
all 1. 0
Corresponding to any
random
power series we define
the
mean
function
Il(z)
=
EI(z)
and
the real variance function
v (z) = E
(I
(z) - Il (z )), provided
the
relevant sums converge.
The
indices occurring in a realisation of
I(z)
form a catalogue of
the
states visited by the sample
sequence
{X
n
}
and
Il(z)
is a generating function for
the
expected numbers of visits of the
process to the states.
n
Case 1. Suppose
that
X
o
= 0, X; = E
lj,
and
the
{Y
n
}
are
independent,
positive
i=O
integer-valued
random
variables, with common p.g.f.
n(z).
The
corresponding
mean
series is
Il(z)
=
E~
nn(z) =
1/(1-
n(z)),
which is singular at z = 1.
Examples.
For
the
uniform distribution on
1,···,
k,
Il(z)
= k(1 -
z)/
{k
- (k +
l)z
+
Zk+l}.
For
the geometric distribution Pi = (1 - B)(Ji-t, j
~
1, we have
Il(z)
=
(1-
Bz)/(I-
z).
The
real variance is
00 00 00
L L
Ezxmzxn
-11l(Z)1
2
= L E
Izl
2X
m L (ZX
j
+ ZX
j
) - L
Izl
2X
n -11l(Z)1
2
m=O
n=O
m=O
i=O
n=O
= 1l(lz1
2
) {
Il
(z) +
ll(z)}
- 1l(lz 1
2
) -
III(x )1
2
•
With probability 1 the series diverges at z = 1, and by Pringsheim's
theorem,
I(z)
has a
singularity there.
For
any integer k we can write
k-l
k-l
I(z)
= L
zi~(j)(zk)(Xn-j)lk
= L
zit(z),
i=O
i=O
where the summation
~(j)
extends
over
those n for which X;
==
j
(mod
k).
Each
t(z)
has the
same probability distribution as
I(Zk),
and
hence with probability 1 has singularities at each
Received 7 April 1989; revision received 10 May 1989.
* Postal address: Department of Statistics, Birkbeck College, University of London, Malet S1.,
London
WCIE
7HX.
708
https://doi.org/10.2307/1427645 Published online by Cambridge University Press
Letters to the editor
709
kth
root of unity.
The
functions will only
add
up to cancel
out
the
singularities if a fixed
relationship holds between the
{fi(z)}. This is
not
so, since if k - 1 of
them
are
fixed,
the
kth
has positive conditional .variance. Thus
f(z)
has a natural boundary
produced
by
the
stochastic irregularity of
the
gap lengths.
The
examples show
that
a
random
series may have a
natural boundary with probability
1,
but
its
mean
series be analytically continuable.
Let
cp(z) =
f(z)/Jl(z)
=
(1-
:rc(x»f(z).
Then
Ecp(z) = 1, lim,
__
t
var cp(z) = 1. Multiplying
and
dividing by
1-
z we find lim,
__
t
cp(z) =
:rc'(1)
lim,
__
t
(1-
z)f(z).
Now
:rc'(1)
=
EXt.
Since
(1 -
z) can only cancel
out
a simple pole, cp(z) has with probability 1 an essential singularity
at z
= 1.
Case 2. Suppose now
that
{X
n
}
is a stochastic branching process (Harris (1963» whose
offspring p.g.f. is
:rc(z),
with expectation m > 1.
On
taking expectations of
both
sides of
the
series for the mean, we find Jl(z) =
E:=o
:rcn(z),
where
:rcn(z)
the
p.g.f. of X; is the
nth
functional iterate of
:rc(z).
We can write this as a functional equation, Jl(z) = z + Jl(:rc(z».
Write Jl(z) = E b.z' for [z]
~
roo
Suppose
that
:rc(0)
= 0,
then
b., =
o.
Substitution into
the
functional equation gives the relation b; = D
t n
+ E {n!
bEklk
t!
..
·k
n!)}(pt
1!)k
t
• • • (Pnn!)k
n
(summation over all positive integral solutions of
k,
+ 2k
z
+
...
+
nk;
=
n),
= D
t n
+
An(b;pt'
..
Pn), where D
t n
is a Kronecker symbol
and
An is
the
nth
Bell polynomial
(Riordan
(1958), p. 34).
The
first few coefficients
are
found to be: b
t
= 1/(1 - Pt), b
z
=
pz/«l
-
pt)(l
-
pi»,
b,
=
p3/«1-
Pt)(l-
p~»
+
2ptP~/«1-
Pt)(l-
pi)(l
-
p~».
The
mean
series cannot be
finite at z = 1, for
that
would imply Jl(l) = 1 + Jl(l).
An
expression can be derived for
v(z)
in
this case,
but
it is less simple
than
in Case 1.
It follows from the strong law of large numbers
that
Xn+t/Xn~m
with probability 1,
and
therefore the sequence has
Hadamard
gaps
and
a natural boundary. However, it is still
possible for
the
mean series to be analytically continuable.
Examples.
For
the
zero-modified geometric distribution it is possible to obtain an explicit
expression for
:rc
n
(z)
for all n, namely
:rcn(z)
=
1-
m
n(l-
q)(m
n
-
q)-t{l
-
(1-
q)z(m
n
- q -
im" -
l)z)-t}
where
q =
:rc(0),
m =
:rc'(1).
It can be seen
that
in this case Jl(z) has poles at z =
tm"
-
q)(m
n
-
n'
for n = 1, 2,
...
,
and
hence
that
z = 1 is a limit point of poles. However, Jl(z) is regular everywhere else on its
circle of convergence.
For
the
Petersburg distribution, Pr
(X
= Z") =
(~)n+t,
n = 0, 1,
...
,
the
p.g.f.
:rc(z)
= E
(~)n+tzzn
has a natural boundary at [z] = 1.
Then
:rc(z)
becomes
unbounded
as
z~zo
for any z, on the unit circle. Thus Jl(z) is
unbounded
as
z~zo,
and
has a natural
boundary.
The similarity in behaviour of
random
power series
and
deterministic lacunary series
investigated
here
reflects a wider parallelism between
the
latter
and
general sequences of
independent or nearly independent
random
variables. This is seen most strongly in
the
probability theoretic
and
function theoretic formulations of
the
law of
the
iterated
logarithm.
It is discussed in the references cited by Bingham «1986), Section 20.1), in particular in
the
papers of Salem
and
Zygmund (1950),
and
Makarov (1985).
References
ARNOLD,
L. (1966) Konvergenzproblemen bei zufalligen
Potenzreihen
mit Lucken. Math. Zeitschr.
92, 356-365.
BINGHAM,
N. H. (1986) Variants of the law of the
iterated
logarithm. Bull. London Math. Soc. 18,
433-467.
HARRIS,
T. E. (1963) Branching Processes. Springer-Verlag, Berlin.
KAHANE,
J.-P. (1985) Some Random Series
of
Functions, 2nd
edn.
Cambridge University Press.
LUKACS,
E. (1975) Stochastic Convergence, 2nd
edn.
Academic Press, New
York.
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710
Letters to the editor
MAKAROV,
N. G. (1985)
On
the
distortion
of
boundary
sets
under
conformal mapping. Proc. London
Math. Soc.
(3) 51, 368-384.
RIORDAN,
J. (1958) Combinatorial Analysis. Wiley, New
York.
SALEM,
R.
AND
ZYGMUND,
A. (1950) La loi du logarithme itere
pour
les series trigonometriques
lacunaires. Bull Sci. Math. 74, 209-224.
TITCHMARSH,
E. C. (1939) The Theory
of
Functions,
2nd
edn.
Oxford
University Press.
WALK,
H. (1968)
Uber
das
Randverhalten
zufalligen
Potenzreihen.
J. reine angew. Math. 230,
66-103.
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