Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series
We study the structure, topological, metric and fractal properties of the distribution of random incomplete sum of the convergent positive series with independent terms under certain conditions on the rate of convergence of series and on the distributions of its terms. We also study the behaviour of...
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irk-123456789-44902009-11-20T12:00:43Z Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series Pratsiovytyi, M.V. Feshchenko, O.Yu. We study the structure, topological, metric and fractal properties of the distribution of random incomplete sum of the convergent positive series with independent terms under certain conditions on the rate of convergence of series and on the distributions of its terms. We also study the behaviour of the absolute value of the characteristic function of this random variable at infinity and the fractal dimension preservation by its distribution function. 2007 Article Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series / M.V. Pratsiovytyi, O.Yu. Feshchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 205-224. — Бібліогр.: 27 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4490 en Інститут математики НАН України |
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We study the structure, topological, metric and fractal properties of the distribution of random incomplete sum of the convergent positive series with independent terms under certain conditions on the rate of convergence of series and on the distributions of its terms. We also study the behaviour of the absolute value of the characteristic function of this random variable at infinity and the fractal dimension preservation by its distribution function. |
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Pratsiovytyi, M.V. Feshchenko, O.Yu. |
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Pratsiovytyi, M.V. Feshchenko, O.Yu. Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series |
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Pratsiovytyi, M.V. Feshchenko, O.Yu. |
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Pratsiovytyi, M.V. |
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Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series |
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Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series |
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Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series |
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Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series |
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Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series |
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topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series |
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Інститут математики НАН України |
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2007 |
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http://dspace.nbuv.gov.ua/handle/123456789/4490 |
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Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series / M.V. Pratsiovytyi, O.Yu. Feshchenko // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 205-224. — Бібліогр.: 27 назв.— англ. |
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2025-07-02T07:43:21Z |
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2025-07-02T07:43:21Z |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.205-224
M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
TOPOLOGICAL, METRIC AND FRACTAL
PROPERTIES OF PROBABILITY
DISTRIBUTIONS ON THE SET OF INCOMPLETE
SUMS OF POSITIVE SERIES
We study the structure, topological, metric and fractal properties of
the distribution of random incomplete sum of the convergent positive
series with independent terms under certain conditions on the rate
of convergence of series and on the distributions of its terms. We
also study the behaviour of the absolute value of the characteristic
function of this random variable at infinity and the fractal dimension
preservation by its distribution function.
1. Introduction
Let
a1 + a2 + . . . + ak + . . . =
∞∑
k=1
ak = r (1)
be a convergent positive series with ak+1 < ak for any k ∈ N , let Sm =
m∑
n=1
an
be a sequence of its partial sums, and let rm =
∞∑
n=m+1
an be a sequence of
its remainders, m = 1, 2, . . . .
Expression
∑
n∈M
an, where M is a finite or an infinite subset of set N of
positive integers, is called the subseries of series (1). Sum of any subseries
of series (1) is called the incomplete sum of series (1). In another words,
the sum of any series
∞∑
n=1
εnan, where εn ∈ {0, 1},
2000 Mathematics Subject Classifications. 11K55, 26A30, 28A80, 60E10.
Key words and phrases. Set of incomplete sums of series, singularly continu-
ous probability distributions, absolutely continuous probability distributions, Hausdorff-
Besicovitch dimension, Hausdorff-Billingsley dimension, fractals, characteristic function
of random variable, transformations preserving fractal dimensions.
205
206 M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
is called the incomplete sum of series (1). We denote the set of all in-
complete sums of series (1) by A. It is clear that all partial sums Sm and
remainders rm of series (1) belong to set A. Moreover, set A′ of incomplete
sums of any subseries of series (1) belongs to A.
For example, the set of incomplete sums of the series
∞∑
k=1
2 · 3−k is a
classical Cantor set, and the set of incomplete sums of the series
∞∑
k=1
3 · 4−k
is a self-similar set of the Cantor type.
The set of incomplete sums of given series is a Lebesgue measurable set.
As far as we know, necessary and sufficient conditions for this set to be
of zero Lebesgue measure in terms of sequences {an}, {rn} are not known
today. Some results and interesting examples are found in the paper [9].
The fractal properties of sets of incomplete sums of some series were studied
in [26,20]. However, in general case the problem does not have solution yet.
In this paper we study properties of the distribution of the random
variable
ξ =
∞∑
k=1
ηkak, (2)
where ηk is a sequence of independent random variables taking the values 0
and 1 with probabilities p0k and p1k correspondingly (pik ≥ 0, p0k +p1k = 1).
Generally speaking, properties of distribution of ξ depend both on series
∞∑
n=1
an and infinite “matrix” ||pik||.
We are interested in the structure (i.e., the content of discrete, singular
and absolutely continuous component) of the distribution of ξ, and topolog-
ical, metric and fractal properties of essential sets of the distribution. We
also study the behaviour of the absolute value of the characteristic func-
tion fξ(t) of the random variable ξ, namely we study Lξ = lim sup
|t|→∞
|fξ(t)|.
The problem of the Hausdorff-Besicovitch dimension preservation by the
distribution function Fξ of the random variable ξ is also studied [3,21].
According to the Jessen-Wintner Theorem [11] (see also [10,20]) the ran-
dom variable ξ has a pure distribution, i.e., it is either pure discrete or pure
absolutely continuous or pure singular (with respect to the Lebesgue mea-
sure). So, the problem about structure of ξ is a problem about type of
distribution of ξ. Criterion for the discreteness (and so, for the continu-
ity) of ξ follows from P. Lévy Theorem [12] (see also [10,20]). Further we
consider only continuous distributions. Hence, the problem about type of
distribution of ξ is a problem about necessary and sufficient conditions for
singularity (and so, for absolute continuity) of ξ.
In this paper we understand topological, metric and fractal properties
of distribution of the random variable ξ as topological, metric and fractal
properties of its spectrum.
THE SET OF INCOMPLETE SUMS OF POSITIVE SERIES 207
Note that the random variable ξ is a generalisation of infinite symmet-
ric Bernoulli convolution [14,20,19,27] that has received attention of many
mathematicians from the thirties of last century [22,23,17,18,25]. But even
in simplest case an = λn, p0n = 1
2
= p1n the problem about type of distri-
bution (the structure) is not solved yet [17].
Distributions in cases an = 2−n (see [24,16,5,27]) and an = c ·λ−n, where
λ < 1
2
(see [7,20,27]), are studied in details. Note that a generalisation of
such distributions were studied in [2,1,9].
The random variable ξ if an ≥ rn for any n ∈ N , were studied in [20].
In this paper we continue to study its properties. The case if an < rn for
infinitely many n, is more complicated. Let us consider the case an = rn+1
for any n ∈ N here.
2. Cylindrical sets and their properties
The set Δ′
c1...cm
of all incomplete sums
m∑
n=1
cnan +
∞∑
n=m+1
εnan, where εn ∈ {0, 1},
of series (1) is called the cylinder of rank m with base c1 . . . cm (ci ∈ {0, 1}).
The segment
Δc1...cm =
[
inf Δ′
c1...cm
, sup Δ′
c1...cm
]
=
[
m∑
n=1
cnan, rm +
m∑
n=1
cnan
]
is called the cylindrical segment of rank m with base c1 . . . cm (ci ∈ {0, 1}).
We denote the interval with the same ends as in Δc1...cm by ∇c1...cm and call
it the cylindrical interval of rank m with base c1 . . . cm.
Depending on {an} and sequence (c1, . . . , cm) it is possible that Δ′
c1...cm
and Δc1...cm coincide or not, but always
Δ′
c1...cm
⊂ Δc1...cm.
Definitions directly imply the following properties of cylindrical sets:
1. inf Δc1...cm = inf Δ′
c1...cm
, sup Δc1...cm = sup Δ′
c1...cm
.
2. Δc1...cm = Δc1...cm0 ∪ Δc1...cm1, Δ′
c1...cm
= Δ′
c1...cm0 ∪ Δ′
c1...cm1.
3. inf Δc1...cm = inf Δc1...cm0 < inf Δc1...cm1,
sup Δc1...cm = sup Δc1...cm1.
4. |Δc1...cm | = rm → 0 (m → ∞).
208 M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
5.
∞⋂
m=1
Δc1...cm =
∞⋂
m=1
Δ′
c1...cm
≡ Δc1...cm... = x ∈ [0, r].
It is easy to see that
6.
|Δc1...cmc|
|Δc1...cm | = rm+1
am+1+rm+1
= 1
δm+1+1
, where δm+1 = am+1
rm+1
.
7. Δc1...cm
= Δs1...sk
if m
= k.
8. Δc1...cm = Δs1...sk
⇔
⎧⎨⎩ m = k,
m∑
n=1
(cn − sn)an = 0.
9. Δc1...cm0∩Δc1...cm1 =
⎧⎪⎨⎪⎩
[b + am+1, b + rm+1] if am+1 < rm+1;
Δc1...cm10...0... if am+1 = rm+1;
∅ if am+1 > rm+1;
where
b =
m∑
n=1
cnan.
Corollary. |Δc1...cm0 ∩ Δc1...cm1| = rm+1 − am+1.
10. The equality |Δc1...cm0 ∩ Δc1...cm1| = 1
2
|Δc1...cm0| is equivalent to
δm+1 =
am+1
rm+1
=
1
2
.
11. |Δc1...cm0 ∩ Δc1...cm1| < 1
2
|Δc1...cm0| < 1
2
rm+1.
12. Let inf Δc1...cm < inf Δs1...sk
. Then
Δc1...cm∩Δs1...sk
=
⎧⎪⎪⎪⎨⎪⎪⎪⎩
[
k∑
i=1
siai,
m∑
i=1
ciai + rm
]
if
k∑
i=1
siai ≤
m∑
i=1
ciai + rm,
∅ if
k∑
i=1
siai >
m∑
i=1
ciai + rm.
13. Sets Δ′
c1...cm
and Δ′
(1−c1)...(1−cm) are symmetrical with respect to the
middle of the segment [0, r], since x ∈ Δ′
c1...cm
implies
x′ = r − x =
∞∑
k=1
an − m∑
n=1
cnan − ∞∑
n=m+1
εnan =
=
m∑
n=1
(1 − cn)an +
∞∑
n=m+1
(1 − εn)an ∈ Δ′
(1−c1)...(1−cm).
14. Sets Δ′
c1...cm0 and Δ′
c1...cm1 are symmetrical with respect to the middle
of cylindrical segment Δc1...cm.
THE SET OF INCOMPLETE SUMS OF POSITIVE SERIES 209
15. If ∇c1...cm0 ∩∇c1...cm1
= ∅, i.e., rm+1 > am+1, then
Δc1...cm0 ∩ Δc1...cm1 ⊂ Δc1...cm01
and
Δc1...cm0 ∩ Δc1...cm1 ⊂ Δc1...cm10,
hence,
Δc1...cm0 ∩ Δc1...cm1 = Δc1...cm01 ∩ Δc1...cm10.
16. The following equalities
Δc1...cm0 ∩ Δc1...cm1 = Δc1...cm011 = Δc1...cm100
hold.
3. Cylindrical representation of points of the set of
incomplete sums of given series
Definitions of cylindrical sets (cylinders and segments), properties 2, 4
and 5 imply that
Δc1 ⊃ Δc1c2 ⊃ . . . ⊃ Δc1...ck
⊃ . . . and Δ′
c1
⊃ Δ′
c1c2
⊃ . . . ⊃ Δ′
c1...ck
⊃ . . .
for any sequence {ck}, ck ∈ {0, 1}. Moreover, there exists unique number
x ∈ [0, r] such that
x =
∞⋂
m=1
Δc1...cm =
∞⋂
m=1
Δ′
c1...cm
=
∞∑
k=1
ckak. (3)
We denote expression (3) symbolically by x = Δc1...cm... and call the
cylindrical representation of number (point) x.
Set of all points x ∈ [0, r] having the cylindrical representation coincides
with the set of incomplete sums of series (1).
Directly from the cylindrical representation definition we obtain that
numbers u = Δc1...cm... and v = Δs1...sm... coincide if and only if
∞∑
i=1
(ci − si)ai = 0.
Lemma 1 ([20]). If ak ≤ rk that is equivalent to rk ≥ 2rk+1 then any point
of segment [0, r] has not more than two cylindrical representations.
210 M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
4. Topological, metric and fractal properties of the set
of incomplete sums of the series
The following proposition describes the structure and topological prop-
erties of the set of incomplete sums of series (1).
Theorem 1 ([9]). The set of incomplete sums A has the following proper-
ties
1. it is a perfect set (closed set without isolated points);
2. A =
⋃
(c1...cm)
Δ′
c1...cm
for any m ∈ N and all Δ′
c1...cm
are isometric;
3. it is a union of segments if inequality rj < aj holds only for finitely
many j;
4. it is a nowhere dense set otherwise.
If the set of incomplete sums of series (1) is of zero Lebesgue measure,
then the Hausdorff measure and the Hausdorff-Besicovitch dimension give
us more detailed information about its “massivity”. Let us recall these
notions.
Let E be a bounded subset of the space R1. The α-dimensional Haus-
dorff measure of E is defined as follows
Hα(E) = lim
ε→0
[
inf
|ui|≤ε
{∑
i
|ui|α :
⋃
i
ui ⊃ E
}]
,
where the infimum is taken over all coverings {ui} of the set E by segments
ui with |ui| ≤ ε, where |ui| is a diameter of ui. Generally speaking, the
measure Hα(E) may be equal to zero, infinity or positive integer. The
number
α0(E) = sup {α : Hα(E)
= 0} = inf {α : Hα(E) = 0}
is called the Hausdorff-Besicovitch dimension of the set E. This notion
characterise the massivity of a set and “compactness” of its points, since
it has the following properties: 1. If E1 ⊂ E2, then α0(E1) ≤ α0(E2).
2. α0
(⋃
i
Ei
)
= sup
i
α0(Ei).
Theorem 2 ([20]). If series (1) satisfies the condition ak ≤ rk for any
k ∈ N (it is equivalent to δk = ak
rk
≥ 1), then the Hausdorff-Besicovitch
dimension of the set of its incomplete sums is equal to
α0(A) =
[
lim
k→∞
(
1
k
k∑
i=1
log2(δi + 1)
)]−1
. (4)
THE SET OF INCOMPLETE SUMS OF POSITIVE SERIES 211
Corollary 1. If conditions of Theorem 2 hold and lim
k→∞
δk = δ, the the
Hausdorff-Besicovitch dimension of the set of incomplete sums of series (1)
is equal to
α0(A) = log−1
2 (δ + 1).
Corollary 2. If conditions of Theorem 2 hold and lim
k→∞
δk = 1, then
α0(A) = 1.
Corollary 3. If conditions of Theorem 2 hold and lim
k→∞
δk = ∞, then
α0(A) = 0.
5. About one special series
There exists the unique positive series a1 + a2 + . . . + an + . . . = 1 such
that an = rn for any n ∈ N . This is the series
∞∑
k=1
2−k. The set of its
incomplete sums is [0, 1]. Any irrational point from [0, 1] has the unique
cylindrical representation corresponding to this series, and some rational
numbers have two cylindrical representations.
Let us consider the series with an = rn+1 for any positive integer n.
Lemma 2. If series a1 + a2 + . . . + an + . . . = 1 has property
an = rn+1 ⇔ (an+2 = an − an+1) (5)
for any n ∈ N , then
an = (−1)n−1(una1 − un−2) for n ≥ 2, (6)
where {un} is a classical Fibonacci sequence, i.e.,
u0 = 1, u1 = 1, un+1 = un + un−1.
Proof. 1. Since {
a1 + a2 + r2 = 1,
r2 = a1,
we have {
a2 = 1 − 2a1,
s2 = a1 + a2 = 1 − a1.
Analogously, since {
a1 + a2 + a3 + r3 = 1,
r3 = a2,
212 M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
we have {
a3 = 1 − s2 − a2 = a1 − a2 = 3a1 − 1,
s3 = s2 + a3 = 2a1.
So, equality (6) holds for n = 2 and 3.
2. Suppose that (6) holds for n = k.
3. Prove that 1 and 2 implies that equality (6) holds for n = k + 1.
From equality an = rn+1 we obtain
an = an+2 + rn+2, an = an+2 + an+1.
Hence, an+2 = an − an+1. This implies
sn = a1 + (1 − 2a1) + (a1 − a2) + . . . + (an−2 − an−1) = 1 − an−1.
From sn + an+1 + rn+1 = 1 and rn+1 = an it follows that
an+1 = 1 − sn − an = 1 − (1 − an−1) − (−1)n−1(una1 − un−2) =
= (−1)n[(un + un−1)a1 − (un−2 + un−3)] =
= (−1)n(un+1a1 − un−1).
By induction, equality (6) holds for any positive integer n.
Lemma 3. Let {un} be a classical Fibonacci sequence. Then sequence
xm = u2m−1
u2m+1
is increasing and sequence ym = u2m
u2m+2
is decreasing. Moreover,
lim
m→∞xm = lim
m→∞ ym =
1
ϕ2
, where ϕ =
1 +
√
5
2
.
Proof. It is known that u2
2m+1 − u2m−1u2m+3 = 1. Hence,
xm+1 − xm =
u2m+1
u2m+3
− u2m−1
u2m+1
=
1
u2m+3u2m+1
> 0,
and sequence {xm} is increasing.
Taking into account that u2
2m+2 − u2mu2m+4 = −1, we have
ym+1 − ym =
u2m+2
u2m+4
− u2m
u2m+2
=
−1
u2m+4u2m+2
< 0.
So, sequence {ym} is decreasing.
It is known that
lim
n→∞
un
un+1
=
1
ϕ
.
THE SET OF INCOMPLETE SUMS OF POSITIVE SERIES 213
Therefore,
lim
m→∞xm = lim
m→∞ ym = lim
n→∞
(
un
un+1
· un+1
un+2
)
=
1
ϕ2
,
which proves the Lemma.
Theorem 3. There exists the unique positive series with property (5). It
has a1 = 1
ϕ2 and
an = rn+1 = (−1)n−1
(
un
ϕ2
− un−2
)
= −βan−1 = (−1)n−1βn−1
ϕ2
, (7)
where ϕ = 1+
√
5
2
, β = 1−√
5
2
, and {un} is a classical Fibonacci sequence.
Proof. If series has property (5), then according to Lemma 2 the n-th term
of the series has a form (6). Then condition an > 0 is equivalent to the
following conditions {
a2m+1 = u2m+1a1 − u2m−1 > 0,
a2m = u2m−2 − u2ma1 > 0,
i.e.,
u2m−1
u2m+1
< a1 <
u2m−2
u2m
.
According to Lemma 3, there exists unique a1 such that the latter double
inequality holds for any positive integer m, namely a1 = 1
ϕ2 .
By using the known Binet formula
un =
(
1+
√
5
2
)n −
(
1−√
5
2
)n
√
5
,
one can to express the ratio
an+1
an
= −un+1 − ϕ2un−1
un − ϕ2un−2
= −β =
√
5 − 1
2
.
Hence, equality (7) holds.
6. Type and properties of distribution of random variable ξ
As it is known, the random variable ξ has a pure distribution. The
following statement follows from P. Lévy Theorem [12].
214 M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
Theorem 4. The random variable ξ has a discrete distribution if and only
if
M =
∞∏
k=1
max{p0k, p1k} > 0.
Corollary. The random variable ξ has a continuous distribution if and only
if M = 0.
The spectrum Sζ of the distribution of random variable ζ is the minimal
closed support of ζ , i.e.,
Sζ = {x : P{ζ ∈ (x − ε, x + ε)} > 0 ∀ ε > 0} =
= {x : Fζ(x + ε) − Fζ(x − ε) > 0 ∀ ε > 0}.
It is easy to prove the following proposition.
Lemma 4. The spectrum of ξ is the set
Sξ = {x : x = Δc1c2...ck..., pckk > 0 ∀k ∈ N},
which is a subset of the set of incomplete sums of series (1).
Let us focus our attention on the distribution of the random variable ξ
if the corresponding series has property an = rn+1.
Theorem 5. Let an is given by (7), let M = 0 and let
p0(2m)p1(2m) = 0, p0(2m−1)p1(2m−1)
= 0.
Then the random variable ξ has a singular distribution of the Cantor type
with a self-similar spectrum. The Hausdorff-Besicovitch dimension of the
spectrum of ξ is equal to
α0(Sξ) = − logβ2 2.
Proof. Taking into account that an = rn+1, we have
sup Δc1...cm00 = inf Δc1...cm10,
sup Δc1...cm01 = inf Δc1...cm11.
Let Δ∗
c1...cm
= Δc1...cm ∩ Sξ. From p01p11
= 0, p0(2m) = 1 it follows that
p1(2m) = 0 and
Sξ = Δ∗
00 ∪ Δ∗
10,
THE SET OF INCOMPLETE SUMS OF POSITIVE SERIES 215
where the sets Δ∗
00 and Δ∗
10 are isometric (congruent) and self-similar sets.
That is
Δ∗
c0 = Δ∗
c000 ∪ Δ∗
c010, c = 0, 1,
Sξ
k∼ Δ∗
c010 = r4 ⊕ Δ∗
c000,
where
k =
|Δc000|
|Δc0| =
r4
r2
=
a3
a1
= β2.
The set Δ∗
c0 is a perfect set. Its self-similar dimension αs coincides with
its Hausdorff-Besicovitch dimension [6] and it is a solution of the Moran
equation
β2x + β2x = 1, i.e., x = − logβ2 2.
By the Corollary after Theorem 4, the random variable ξ has a contin-
uous distribution, since M = 0. Since αs(Sξ) < 1, the Lebesgue measure
λ(Sξ) = 0. So, the random variable ξ has a singular distribution of the
Cantor type by definition.
Theorem 6. Let an is given by (7), let M = 0 and let p0(2m)p1(2m) = 0.
Then the random variable ξ has a singular distribution of the Cantor type.
The Hausdorff-Besicovitch dimension of the spectrum of ξ is equal to
α0(Sξ) = − lim
j→∞
logdj
Aj,
where
Aj = 2
j∑
k=1
[bk−1]
, bk = #{i : pi(2k+1)
= 0}, dj =
β2j+1
−ϕ2
.
Proof. The random variable ξ has a singular distribution, since M = 0. The
spectrum of ξ is a subset of the set
B = {x : x = Δc1...cm..., where c2m−1 ∈ {0; 1}, c2m are fixed} .
The Lebesgue measure of B is equal to 0 (see the proof of the previous
theorem). So, ξ has a singular distribution of the Cantor type.
If p01p11
= 0, then sets Δ∗
00 = Δ00∩Sξ and Δ∗
10 = Δ10∩Sξ are isometric.
If p(1−c)1 = 0, then Sξ = Δ∗
c1. Therefore, the Hausdorff-Besicovitch dimen-
sion of the spectrum Sξ coincides with the Hausdorff-Besicovitch dimension
of the set Δ∗
c0, where pc1
= 0.
The set Δ∗
c0 is a union of Aj isometric closed sets of diameter dj. The
α-volume of such a covering is given by
lαj ≡ Ajd
α
j ,
216 M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
where
dj = r2j+2 −
∞∑
i=j+1
a2i = a2j+1 − a2j+4
1 − β2
= a2j+1
(
1 +
β3
1 − β2
)
=
=
a2j+1
1 − β2
(1 − β2 + β3) = −βa2j+1 = −β2j+1a1 =
β2j+1
−ϕ2
.
Therefore [13],
α0(Δ
∗
c0) = α0(Sξ) = − lim
j→∞
logdj
Aj ,
which proves the Theorem.
Let us return to the case an ≥ rn for any n ∈ N . It is sufficient to define
the distribution function Fξ(x) of the random variable ξ at the points of
the spectrum Sξ, since it is defined by the continuity and monotonicity in
remaining points.
Lemma 5. At the point x = Δc1c2...ck... = x ∈ Sξ the distribution function
Fξ of the random incomplete sum ξ is of the following form
Fξ(x) = βc11 +
∞∑
k=2
⎛⎝βckk
k−1∏
j=1
pcjj
⎞⎠ , (8)
where
βckk =
{
0, if ck = 0,
p0k, if ck = 1.
Proof. The event {ξ < x} can be represented in the following form
{ξ < x} = {η1 < c1} ∪ {η1 = c1, η2 < c2} ∪ {η1 = c1, η2 = c2, η3 < c3} ∪ . . .
. . . ∪ {η1 = c1, η2 = c2, . . . , ηk−1 = ck−1, ηk < ck} ∪ . . . .
Then P{ξ < x} =
= P{η1 < c1} + P{η1 = c1, η2 < c2} + P{η1 = c1, η2 = c2, η3 < c3}+
. . . + P{η1 = c1, η2 = c2, . . . , ηk−1 = ck−1, ηk < ck} + . . .
and
P{η1 = c1, η2 = c2, . . . , ηk−1 = ck−1, ηk < ck} =
=
⎛⎝k−1∏
j=1
P{ηj = cj}
⎞⎠ · P{ηk < ck} = βckk
k−1∏
j=1
pcjj .
Hence, Fξ(x) = P{ξ < x} is expressed as (8).
THE SET OF INCOMPLETE SUMS OF POSITIVE SERIES 217
Lemma 6 ([20]). If δk = ak
rk
≥ 1 (it is equivalent to rk ≥ 2rk+1) and
p0kp1k
= 0 for any k ∈ N , then the spectrum Sξ of the random variable ξ is
a perfect nowhere dense set. Moreover, its Lebesgue measure is equal to
λ(Sξ) = 2 lim
k→∞
2krk = 2r
∞∏
k=1
1
δk + 1
=
⎧⎪⎪⎨⎪⎪⎩
0 if
∞∑
k=1
(δk − 1) = ∞,
A > 0 if
∞∑
k=1
(δk − 1) < ∞.
Theorem 7 ([20]). Let δk = ak
rk
for any k ∈ N and let the series
∞∑
k=1
(δk−1)
is convergent. Then the random variable ξ has an absolutely continuous
distribution if and only if
∞∑
k=1
ln[
√
p0kp1k(δk + 1)] < ∞.
Theorem 8 ([20]). Let δk ≥ 1 (it is equivalent to rk ≥ 2rk+1) and
p0kp1k
= 0 for any k ∈ N , let M = 0, and let the series
∞∑
k=1
(δk − 1) is
divergent. Then the random variable ξ has a singular distribution of the
Cantor type. Moreover, the Hausdorff-Besicovitch dimension of its spec-
trum is equal to (4).
7. Fractal dimension preservation
Let us recall that the α-dimensional Hausdorff-Billingsley measure of a
set E is defined as follows
Ĥα(E) = lim
ε↓0
(
inf
μ(ui)≤ε
∑
i
μα(ui),
⋃
i
ui ⊃ E
)
,
where the infimum is taken over all coverings {ui} of the set E ⊂ A by
segments ui with μ(ui) ≤ ε. The number
αμ = αμ(E) = inf{α : Ĥα(E) = 0} = sup{α : Ĥα(E)
= 0}
is called the Hausdorff-Billingsley dimension of the set E with respect to
measure μ.
We say that a distribution function Fξ(x) preserves the fractal dimension
if the Hausdorff-Billingsley dimension αμ(·) of any subset E ⊂ Sξ is equal to
the Hausdorff-Besicovitch dimension of its image E ′ = Fξ(E), i.e., αμ(E) =
αμξ
(E) ≡ α0(E
′), where μ is a probability measure which gives uniform
distribution on Sξ.
218 M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
Theorem 9. If an ≥ rn for any n ∈ N and
lim
k→∞
p0k =
1
2
, (9)
then the distribution function Fξ of the random variable ξ defined by (2)
preserves the fractal dimension.
Proof. Without loss of generality, let us assume that pik > 0. Then the
expression of the distribution function Fξ(x) is a Q∗-representation [20] of
the number Fξ(x) and
μ(Δα1(x)...αk(x)) = 2−k,
μξ(Δα1(x)...αk(x)) = P{ξ ∈ Δα1(x)...αk(x)} =
k∏
j=1
pαj(x)j =
k∏
j=1
(
1
2
ταj(x)j
)
,
where ταj(x)j = 2pαj(x)j . Taking into account (9), we have
ταj(x)j → 1 (j → ∞) and lim
j→∞
ln(ταj (x)j) = 0. (10)
Then
lim
k→∞
ln μξ(Δα1(x)...αk(x))
lnμ(Δα1(x)...αk(x))
= lim
k→∞
ln
k∏
i=1
pαj(x)j
ln 2−k
=
= lim
k→∞
k∑
j=1
ln pαj(x)j
−k ln 2
= lim
k→∞
k ln 2−1 +
k∑
j=1
ln ταj (x)j
−k ln 2
= 1 + lim
k→∞
1
k
k∑
j=1
ln ταj(x)j
− ln 2
.
Taking into account (10), we have
lim
k→∞
1
k
k∑
j=1
ln ταj(x)j = 0.
Therefore,
lim
k→∞
ln μξ(Δα1(x)...αk(x))
ln μ(Δα1(x)...αk(x))
= 1 (11)
for any x ∈ Sξ.
For cylindrical representation and the Hausdorff-Billingsley dimension
the analogue of Billingsley Theorem for s-adic intervals [4] holds: if ν1 and
ν2 are continuous probability measures and
E ⊂ E0 =
{
x : lim
k→∞
ln ν1(Δα1(x)...αk(x))
ln ν2(Δα1(x)...αk(x))
= δ
}
,
then αν2(E) = δαν1(E). Hence, from equality (11) it follows that αμ(E) =
1 · αμξ
(E). So, Fξ preserves the fractal dimension.
THE SET OF INCOMPLETE SUMS OF POSITIVE SERIES 219
8. Characteristic function of the random incomplete sum
with independent terms
Characteristic function fξ(t) of random variable ξ is a mathematical
expectation of random variable eitξ, i.e.
fξ(t) = Meitξ.
Characteristic functions provide suitable tools for the investigation of
the structure and properties of real-valued random variables. In particular,
it is known that
Lξ = lim sup
|t|→∞
|fξ(t)|
is equal to
1) 1, if ξ has a discrete distribution;
2) 0, if ξ has an absolutely continuous distribution.
For singular distributions Lξ can be equal to any number from [0, 1].
Singular distributions with Lξ = 1 are close to discrete distributions, and
distributions with Lξ = 0 are close to absolutely continuous ones. Hence, the
value Lξ characterise how close are the properties of singular distribution to
the properties of discrete and absolutely continuous ones. Note that measure
μξ such that Lξ = 0 is called the Rajchman measure. Some important for
probability theory problems are related with such measures [15].
Lemma 7. The characteristic function of random variable ξ defined by (2)
is of the following form
fξ(t) =
∞∏
k=1
(
p0k + p1ke
itak
)
=
∞∏
k=1
(p0k + p1k cos(akt) + ip1k sin(akt)) ,
and its absolute value is f the following form
|fξ(t)| =
∞∏
k=1
|fk(t)|, where |fk(t)| =
√
1 − 4p0kp1k sin2 tak
2
.
Proof. From properties of characteristic functions and mathematical expec-
tation we obtain
fξ(t) = Meitξ = Me
it
∞∑
k=1
akηk
= M
∞∏
k=1
eitakηk =
=
∞∏
k=1
Meitakηk =
∞∏
k=1
(p0k + p1ke
itak) =
=
∞∏
k=1
(p0k + p1k cos(tak) + ip1k sin(tak)) =
∞∏
k=1
fk(t)
220 M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
and
|fk(t)| =
√
p2
0k + 2p0kp1k cos(tak) + p2
1k =
√
1 − 4p0kp1k sin2 tak
2
,
which proves the Lemma.
Let us study the behaviour of the absolute value of the characteristic
function of random variable ξ at infinity under extra conditions.
Theorem 10. If
1
gn
=
an+1
an
→ 0 (n → ∞), where 2 ≤ gn ∈ N, (12)
then
Lξ = lim sup
|t|→∞
|fξ(t)| = 1.
Proof. Let us consider the sequence tn = 2π
an
. And let us estimate
|fξ(t)| =
∞∏
k=1
√
1 − 4p0kp1k sin2 tak
2
≥
∞∏
k=1
√
1 − sin2 tak
2
=
∞∏
k=1
∣∣∣∣cos
tak
2
∣∣∣∣ .
Hence,
Lξ ≥ lim
n→∞ |fξ(tn)| = lim
n→∞
∞∏
k=1
|fk(tn)| ≥ lim
n→∞
∞∏
k=1
∣∣∣∣cos
tnak
2
∣∣∣∣ .
Since
tnak
2
=
{
πgkgk+1 . . . gn−1 if k ≤ n,
π
gn+1gn+2...gk
if k > n,
we have ∣∣∣∣cos
tnak
2
∣∣∣∣ =
{
1 if k ≤ n,
cos π
gn+1gn+2...gk
if k > n.
So,
∞∏
k=1
|fk(tn)| ≥
∞∏
k=n+1
cos
π
gn+1gn+2 . . . gk
. (13)
Since g−1
n → 0 (n → ∞), there exists n0 such that gn ≥ 4 for all n > n0.
Then for k > n > n0 we have
cos
π
gn+1gn+2 . . . gk
≥ cos
π
4k−n
= 1 − 2 sin2 π
2 · 4k−n
> 1 − 2π2
4 · 16k−n
=
= 1 − π2
24k−4n+1
.
THE SET OF INCOMPLETE SUMS OF POSITIVE SERIES 221
Hence,
∞∏
k=n+1
cos
π
gn+1gn+2 . . . gk
≥
∞∏
k=n+1
(
1 − π2
24k−4n+1
)
.
Since series
∞∑
k=n+1
π2
24k−4n+1 converges, infinite product (13) also converges,
hence,
lim
n→∞
∞∏
k=n+1
cos
π
gn+1gn+2 . . . gk
= 1
and Lξ ≥ 1. Since always Lξ ≤ 1, we have Lξ = 1.
Corollary. If sequence {an} satisfies the condition (12) and condition
M = 0 holds for matrix ||pik||, then the random variable ξ has an anoma-
lously fractal singular distribution.
Lemma 8. If for any positive integer n condition (12) holds and Lξ = 0,
then
p0k → 1
2
(k → ∞)
and gn = 2 for any n > n0 for some n0.
Proof. Since Lξ = 0, the equality
lim
n→∞ |fξ(tn)| = 0 (14)
holds for any sequence {tn} such that tn → ∞. Let us consider tn =
2g1 . . . gnπ. Then
sin2 tnak
2
= sin2 πg1 . . . gk
g1 . . . gn
=
⎧⎪⎨⎪⎩
sin2 πgk+1 . . . gn if k < n,
sin2 π if k = n,
sin2 π
gn+1...gk
if k > n,
and fk(tn) = 1 if k ≤ n. Hence,
|fξ(tn)| = |fn+1(tn)| · Bn,
where
|fn+1(tn)| =
√
1 − 4p0(n+1)p1(n+1) sin2 π
gn+1
and
Bn =
∞∏
k=n+2
|fk(tn)| =
∞∏
k=n+2
√
1 − 4p0kp1k sin2 π
gn+1 . . . gk
.
The sequence {Bn} is convergent, moreover,
Bn ≥
∞∏
k=n+2
√
1 − sin2 π
gn+1 . . . gk
=
∞∏
k=n+2
cos
π
gn+1 . . . gk
= b > 0.
222 M.V.PRATSIOVYTYI AND O.YU.FESHCHENKO
Therefore, equality (14) holds if and only if |fn+1(tn)| → 0 (n → ∞). It is
possible if p0n → 1
2
(n → ∞) and gn = 2 for n > n0.
Corollary. Let M = 0, let condition (12) holds, and let lim
k→∞
p0k
= 1
2
. Then
the random variable ξ has a singular distribution.
Lemma 9. Let condition (12) holds, let gn = 2 for n > n0, and let p0k → 1
2
(k → ∞). Then Lξ = 0.
Proof. Let us consider the random variable
ζ =
∞∑
m=1
2−mηn0+m
with independent binary digits ηn0+m. The behaviour of Lζ was studied in
the paper [8]. It is known [8] that
Lζ = 0 ⇔ p0k → 1
2
(k → ∞).
This fact remains valid for the random variable ζ̂ = 1
g1...gn0
ζ . Since
ξ = ξ1 + ζ̂ , where ξ1 =
n0∑
n=1
anηn,
and
fξ(t) = fξ1(t) · fζ̂(t),
we have that Lξ = 0 if and only if Lζ̂ = 0, i.e., if condition p0k → 0 (k → ∞)
holds.
Lemmas 8 and 9 imply the following proposition.
Theorem 11. Let M = 0, and let for any k ∈ N
ak =
1
g1g2 . . . gk
, 2 ≤ gk ∈ N.
Then Lξ = 0 if and only if{
p0k → 1
2
(k → ∞),
gn = 2 for n > n0.
Acknowledgment. This work was partly supported by DFG 436 UKR
113/78, DFG 436 UKR 113/80 projects.
THE SET OF INCOMPLETE SUMS OF POSITIVE SERIES 223
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Dragomanov National Pedagogical University, Kyiv, Ukraine;
Institute for Mathematics of NAS of Ukraine, Kyiv, Ukraine
E-mail address: prats2@yandex.ru
Institute for Mathematics of NAS of Ukraine, Kyiv, Ukraine
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