Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension
The paper is devoted to the study of connections between fractal properties of one-dimensional singularly continuous probability measures and the preservation of the Hausdorf dimension of any subset of the unit interval under the corresponding distribution function. Conditions for the distribution f...
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irk-123456789-44972009-11-20T12:00:40Z Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension Torbin, G. The paper is devoted to the study of connections between fractal properties of one-dimensional singularly continuous probability measures and the preservation of the Hausdorf dimension of any subset of the unit interval under the corresponding distribution function. Conditions for the distribution function of a random variable with independent Q-digits to be a transformation preserving the Hausdorf dimension (DP-transformation) are studied in details. It is shown that for a large class of probability measures the distribution function is a DP-transformation if and only if the corresponding probability measure is of full Hausdorf dimension. 2007 Article Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension/ G. Torbin // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 281-293. — Бібліогр.: 12 назв.— англ. 0321-3900 http://dspace.nbuv.gov.ua/handle/123456789/4497 en Інститут математики НАН України |
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The paper is devoted to the study of connections between fractal properties of one-dimensional singularly continuous probability measures and the preservation of the Hausdorf dimension of any subset of the unit interval under the corresponding distribution function. Conditions for the distribution function of a random variable with independent Q-digits to be a transformation preserving the Hausdorf dimension (DP-transformation) are studied in details. It is shown that for a large class of probability measures the distribution function is a DP-transformation if and only if the corresponding probability measure is of full Hausdorf dimension. |
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Torbin, G. |
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Torbin, G. Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension |
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Torbin, G. |
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Torbin, G. |
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Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension |
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Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension |
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Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension |
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Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension |
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Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension |
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probability distributions with independent q-symbols and transformations preserving the hausdorff dimension |
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Інститут математики НАН України |
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Probability distributions with independent Q-symbols and transformations preserving the Hausdorff dimension/ G. Torbin // Theory of Stochastic Processes. — 2007. — Т. 13 (29), № 1-2. — С. 281-293. — Бібліогр.: 12 назв.— англ. |
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Theory of Stochastic Processes
Vol.13 (29), no.1-2, 2007, pp.281-293
GRYGORIY TORBIN
PROBABILITY DISTRIBUTIONS WITH
INDEPENDENT Q-SYMBOLS AND
TRANSFORMATIONS PRESERVING THE
HAUSDORFF DIMENSION
The paper is devoted to the study of connections between fractal
properties of one-dimensional singularly continuous probability mea-
sures and the preservation of the Hausdorff dimension of any subset
of the unit interval under the corresponding distribution function.
Conditions for the distribution function of a random variable with
independent Q− digits to be a transformation preserving the Haus-
dorff dimension (DP-transformation) are studied in details. It is
shown that for a large class of probability measures the distribu-
tion function is a DP-transformation if and only if the corresponding
probability measure is of full Hausdorff dimension.
1. Introduction
It is well known (see, e.g., [9]) that fractal analysis of singularly contin-
uous probability distributions allows us to study many essential properties
of such distributions. The first stage of such an analysis is the study of met-
ric, topological and fractal properties of the spectrum (topological support)
of the distribution, which leads to the metric-topological classification of
singular measures (see, e.g., [4], [9]). It should be mentioned here that the
determination of the Hausdorff-Besocovitch dimension even for the topolog-
ical support is often a very non-trivial problem (see, e.g., [1],[8], [9],[10]). On
the other hand, the topological support is a rather ”rough” characteristic
for a measure with a complicated local structure. For instance, for any pair
of real numbers p1 ∈ (0, 1
2
) and p2 ∈ (0, 1
2
), p1 �= p2 the distributions of the
following random variables ξ(1) and ξ(2) are mutually singular and they are
singular w.r.t. Lebesgue measure, where ξ(1) =
∞∑
k=1
ξ
(1)
k
2k , ξ(2) =
∞∑
k=1
ξ
(2)
k
2k ; ξ
(1)
k
2000 Mathematics Subject Classifications 60G30, 28A80, 11K55.
Key words and phrases. Singularly continuous probability distributions, Haus-
dorff dimension of probability measures, Hausdorff-Billingsley dimension, fractals, DP-
transformations.
281
282 GRYGORIY TORBIN
(ξ
(2)
k ) is a sequence of independent random variables taking the values 0 and
1 with probabilities p1(p2) and 1−p1(1−p2) correspondingly. Nevertheless,
the topological supports of the above distributions coincide with the unit
interval.
The second level of the fractal analysis of singular measures is the
determination of fractal properties of the density supports and essential
density supports, i.e., of the following sets Nμ = {x : F ′(x) �= 0} and
N∞
μ = {x : F ′(x) = +∞}. These sets describe properties of singularly
continuous measures essentially better than the topological support. In
particular, it is known (see, e.g., [6]) that a probability measure μ is sin-
gular if and only if μ(N∞
μ ) = 1. In [6] it has been constructed an example
of absolutely continuous probability distribution function such that the set
N∞
μ is everywhere dense of full Hausdorf dimension, which show that usual
derivative is ”too sensitive” to describe only essential properties of singular
measures.
The next level in the fractal analysis of a singularly continuous measure
is the investigation of dimensionally minimal supports of the measure. For
a probability measure μ one can define the Hausdorff dimension α0(μ) of
the measure as follows: α0(μ) = inf
E:μ(E)=1
{α0(E)}. If μ is discrete, then
α0(μ) = 0. If μ is absolutely continuous, then α0(μ) = 1; and α0(μ) ∈ [0, 1] if
μ is singularly continuous. Main problems of this level are the determination
of the Hausdorff dimension of the measure itself and the study of supports
whose Hausdorff dimension coincides with the Hausdorff dimension of the
measure (see, e.g., [1]).
In the present paper we show how results on the latter level of fractal
analysis of singularly continuous probability measures can be applied to the
study of continuous transformations preserving the Hausdorff dimensions
(DP-transformations) of any subset of real line. The development of a
general theory of DP-transformations is important from theoretical as well
as from applied point of view (see, e.g., [2], [3]). This paper is devoted to
the development of ideas and methods for the investigation of continuous
transformations which were proposed in [2], [3] and [5]. We show that the
problem of the study of continuous DP-transformations of the real line is
equivalent to the the investigation of the DP-properties of strictly increasing
probability distribution functions. Moreover we show that for a large class
of probability measures of the Jessen-Wintner type (see, e.g., [9]) the DP-
property is equivalent to the superfractality of the corresponding probability
measure (i.e., α0(μ) = 1). The main results of the paper are Theorem 1 and
Theorem 2 which give necessary conditions and sufficient conditions for the
distribution function of a random variable with independent Q−symbols to
be a DP-transformation. Corollaries after Theorem 2 show that the above
theorems are essential generalizations of results from [2], [3] and [5].
PROBABILITY DISTRIBUTIONS 283
2. Q-representation of real numbers and random variables
with independent Q-symbols.
Let us recall shortly the notion of the Q-representation of real numbers.
For any s > 1, s ∈ N let N0
s−1 = {0, 1, ..., s−1}, and let Q = (q0, q1, ..., qs−1)
be a stochastic vector with strictly positive coordinates. By using the vector
Q, we define the Q- partition of the unit interval [0, 1].
I step. We divide [0, 1] from the left to the right into s closed intervals
ΔQ
0 , ΔQ
1 , ..., ΔQ
s−1, with |ΔQ
i | = qi. We call the ΔQ∗
i ”first rank intervals”.
II step.We divide (from the left to the right) each of the closed first
rank interval ΔQ
i1 into s closed intervals, called second rank intervals, whose
lengths are in the following proportion : q0 : q1 : ... : q(s−1).
n-th step. We divide (from the left to the right) each of the closed (n-
1)-th rank interval ΔQ
i1i2...in−1
into s closed intervals of the n-th rank, whose
lengths are in the same proportion : q0 : q1 : ... : q(s−1).
It is easy to see that |ΔQ
i1i2...in| = qi1 · qi2 · ... · qin → 0 (n → ∞), and
the sequence of imbedded closed intervals ΔQ
i1 ⊃ ΔQ
i1i2 ⊃ ... ⊃ ΔQ
i1i2...in ⊃ ...
has a unique common point x.
Conversely, if a point x is not an end-point of any closed interval of the
above partition, then for the point x there is a unique sequence of imbedded
intervals ΔQ
α1(x) ⊃ ΔQ
α1(x)α2(x) ⊃ ... ⊃ ΔQ
α1(x)...αn(x) ⊃ ... containing the point
x.
Symbolically we write
x = ΔQ
α1(x)α2(x)...αn(x).... (1)
(1) is called the Q-representation of x. If a point x is an end-point of some
closed interval of the above partition, then x has two Q-representations.
Actually, this representation can also be thought as the representation,
which generates by the dynamical system on the unit interval with the
following transformation T (see, e.g., [11] for details):
T (x) =
1
qk
x −
k−1∑
i=0
qi
qk
,
for x ∈ [
k−1∑
i=0
qi, qk +
k−1∑
i=0
qi), with
−1∑
i=0
qi := 0.
For the convenience of a reader we give one more explanation of the
Q-representation: the cylinders of this representation are images of usual
s-adic cylinders under the distribution function Fψ of the random variable
ψ with independent identically distributed s-adic digits, i.e.,
ψ =
∞∑
k=1
ψk
sk
,
284 GRYGORIY TORBIN
where ψk are independent identically distributed random variables taking
the values 0, 1, ..., s − 1 with probabilities q0, q1, ..., qs−1 correspondingly.
Let {ηk} be a sequence of independent random variables taking the val-
ues 0, 1, ..., s − 1 with probabilities p0k, p1k, ..., p(s−1)k correspondingly, and
let us consider the random variable ξ:
ξ = ΔQ
η1η2...ηk.... (2)
ξ is said to be a random variable with independent Q-digits. The cor-
responding probability measure μξ can be obtained in the following way.
Let Ωk = {0, 1, ..., s − 1}, Fk = 2Ωk . We define a discrete measure νk on
the Fk by νk(i) = pik, i ∈ N0
s−1, and consider the infinite product of prob-
ability spaces: (Ω,F , ν) =
∏∞
k=1(Ωk,Fk, νk) and the bimeasurable mapping
f : Ω → [0, 1], defined for any element ω = (ω1, ω2, ..., ωk, ...) ∈ Ω as follows:
f(ω) = ΔQ
ω1ω2...ωk.... (3)
For any Borel subset E ⊂ [0, 1] we define the image measure ν∗ by the follow-
ing relation: ν∗(E) = ν(f−1(E)), where f−1(E) = {ω : ω ∈ Ω, f(ω) ∈ E}.
The measure ν∗ coincides with the probability measure μξ.
If pik = pi ∀k ∈ N , i ∈ N0
s−1 (i.e., ξ is a random variable with indepen-
dent identically distributed Q-digits), then the measure μξ is the self-similar
measure associated with the list (S0, ..., Ss−1, p0, ..., ps−1), where Si is a sim-
ilarity with the ratio qi, and the list (S0, ..., Ss−1) satisfies the open set
condition. More precisely, μξ is the unique Borel probability measure on
[0, 1] such that μξ =
∑s−1
i=0 pi ·μξ ◦S−1
i , (see, e.g., [8] for details). If we define
discrete measures νk in the following way: νk(i) = qi, i ∈ N0
s−1, k ∈ N, then
the measure ν∗ coincides with Lebesgue measure on [0, 1].
The distribution of the random variable ξ is of pure type. In [10] it has
been proved that it is of the discrete type iff
∞∏
k=1
max
i
pik > 0; (4)
it is of absolutely continuous type iff
∞∑
k=1
(
s−1∑
i=0
(1 − pik
qi
)2
)
< ∞; (5)
it is of singularly continuous type iff the infinite product (4) and series
(5) diverge. It is not hard to see that in this situation the singularity plays
a ”generic” role.
PROBABILITY DISTRIBUTIONS 285
3. DP-transformations with independent Q-symbols
A transformation f of Rn (in the sense of a bijective mapping of Rn into
itself) is said to be transformation preserving the Hausdorff dimension (DP-
transformation for short), if for any subset E ⊂ Rn and its image E ′ = f(E)
the following condition holds
α0(E) = α0(E
′).
From the countable stability of the Hausdorff dimension it follows that
a transformation f is a DP-transformation on R1 if and only if f preserves
the Hausdorff dimension of all subsets of any interval (a, b). So, to study
the DP-transformations of R1 it is sufficient to study DP-transformations
of intervals. Without loss of generality we shall consider the unit segment.
It is easy to see that a continuous function f is a transformation of [0, 1]
if and only if it is either a strictly increasing distribution function (in the
sense of probability theory) F on [0, 1] or it is of the form 1 − F .
Our main aim in this Section is to find conditions for the distribu-
tion functions of random variables with independent Q-symbols to be DP-
transformations.
The following theorems are the main results of this paper and they are
essential generalizations of results from [2], [3] and [5].
Let hk = − s−1∑
i=0
pik ln pik, and let bk = − s−1∑
i=0
pik ln qi.
Theorem 1. Let inf
i,j
pij > 0. If
lim
n→∞
h1 + h2 + ... + hn
b1 + b2 + ... + bn
= 1, (6)
then the distribution function Fξ of a random variable ξ with independent Q-
symbols preserves the Hausdorff dimension of any subset of the unit interval.
Proof. It is not hard to prove (see, e.g., [12]) that for any two probability
vectors −→p = (p0, p1, ..., ps−1) and −→q = (q0, q1, ..., qs−1) with qi > 0, ∀i ∈
N0
s−1 the following condition holds:
pp0
0 · pp1
1 · ... · pps−1
s−1 ≥ qp0
0 · qp1
1 · ... · qps−1
s−1 , (7)
and the equality holds if and only if pi = qi, ∀i ∈ N0
s−1. Therefore,
hk = − ln(pp0k
0k · pp1k
1k · ... · pp(s−1)k
(s−1)k ) ≤ bk = − ln(qp0k
0 · qp1k
1 · ... · qp(s−1)k
s−1 ), (8)
and condition (6) is equivalent to the existence of the following limit:
lim
n→∞
h1 + h2 + ... + hn
b1 + b2 + ... + bn
= 1. (9)
286 GRYGORIY TORBIN
Let ε be an arbitrary positive number such that ε < min
i
qi and let us
consider the following sets:
T+
ε,k =
{
j : j ∈ N, j ≤ k, |pij − qi| ≤ ε, ∀i ∈ N0
s−1
}
,
T−
ε,k =
{
j : j ∈ N, j ≤ k, |pij − qi| > ε for some i ∈ N0
s−1
}
.
Now we need the following lemma, which describes ”how dense” the sets
T+
ε,k is.
Lemma. If condition (6) holds, then lim
k→∞
|T+
ε,k
|
k
= 1.
Proof. Suppose, contrary to our claim, that lim
k→∞
|T+
ε,k
|
k
�= 1. Since | T+
ε,k |≤ k,
the latter assumption is equivalent to the existence of a sequence {kn} such
that lim
kn→∞
|T+
ε,kn
|
kn
= a0 < 1. From inequalities (8) and (7) it follows that for
any ε > 0 there exists a positive constant δ0 = δ0(ε) such that hj ≤ (1−δ0)bj
for any j ∈ T−
ε,k. Therefore,
k∑
j=1
hj
k∑
j=1
bj
=
∑
j∈T+
ε,k
hj +
∑
j∈T−
ε,k
hj
k∑
j=1
bj
≤
∑
j∈T+
ε,k
bj + (1 − δ0)
∑
j∈T−
ε,k
bj
k∑
j=1
bj
= 1 − δ0
∑
j∈T−
ε,k
bj
k∑
j=1
bj
.
Let −→pj = (p0j, p1j , ..., p(s−1)j) and −→r = (ln 1
q0
, ln 1
q1
, ..., ln 1
qs−1
). Since bj =
−→pj · −→r , we conclude that
bj ≤| −→pj | · | −→r |≤ 1 · (
s−1∑
i=0
ln2 qi)
1
2 ≤ d1 = d1(s, q0, ..., qs−1).
Since | −→r |= const, all coordinates of the vector −→r are strictly positive
and all coordinates of the vector −→pj are non-negative, from bj = −→pj · −→r it
follows that there exists a positive constant d0 = d0(s, q0, ..., qs−1) such that
bj ≥ d0, ∀j ∈ N. So, 0 < d0 ≤ bj ≤ d1 < ∞, ∀j ∈ N, and, therefore, there
exist constants Cε,k ∈ [d0, d1] and Dε,k ∈ [d0, d1] such that
∑
j∈T−
ε,k
bj =| T−
ε,k | ·Cε,k;
k∑
j=1
bj = k · Dε,k.
Hence,
kn∑
j=1
hj
kn∑
j=1
bj
≤ 1 − δ0
∑
j∈T−
ε,kn
bj
kn∑
j=1
bj
= 1 − δ0
| T−
ε,kn
| ·Cε,kn
kn · Dε,kn
≤ 1 − δ0d0
d1
| T−
ε,kn
|
kn
.
PROBABILITY DISTRIBUTIONS 287
Therefore,
1 = lim
n→∞
h1 + h2 + ... + hkn
b1 + b2 + ... + bkn
≤ lim
n→∞(1− δ0d0
d1
| T−
ε,kn
|
kn
) = 1− δ0d0
d1
(1−a0) < 1,
which is impossible. Let ΔQ
α1(x)...αk(x) be the cylinder of the Q-representation
containing the point x, let μ = μξ, and let λ be the Lebesgue measure.
Let pmin = inf
ij
pij, qmin = min
i
qi, qmax = max
i
qi, and let
Ni(x, k) = #{j : j ≤ k, αj(x) = i};
Ni(ε, x, k) = #{j : j ≤ k, αj(x) = i, j ∈ T+
ε,k}.
Then for any x ∈ [0, 1], for any k ∈ N , and for any ε < 1
2
qmin we have:
− ln μ(ΔQ
α1(x)...αk(x)) = −(ln[
k∏
j=1
pαj(x)j ]) = −(
∑
j∈T−
ε,k
ln pαj(x)j +
∑
j∈T+
ε,k
ln pαj(x)j)
=
∑
j∈T−
ε,k
ln
1
pαj(x)j
+
s−1∑
i=0
(
∑
αj(x)=i, j∈T+
ε,k
ln
1
pαj(x)j
) ≤
≤ ∑
j∈T−
ε,k
ln
1
pmin
+
s−1∑
i=0
(
∑
αj(x)=i, j∈T+
ε,k
ln
1
qi − ε
) =
=| T−
ε,k | ln
1
pmin
+
s−1∑
i=0
Ni(ε, x, k) ln
1
qi − ε
=
=| T−
ε,k | ln
1
pmin
+
s−1∑
i=0
Ni(ε, x, k)
(
ln
1
qi
+ ln(1 +
ε
qi − ε
)
)
≤
≤| T−
ε,k | ln
1
pmin
+
s−1∑
i=0
Ni(ε, x, k)
(
ln
1
qi
+
ε
qi − ε
)
≤
≤| T−
ε,k | ln
1
pmin
+
s−1∑
i=0
Ni(ε, x, k) ln
1
qi
+
s−1∑
i=0
Ni(ε, x, k)
2ε
qi
≤
≤| T−
ε,k | ln
1
pmin
+
s−1∑
i=0
Ni(ε, x, k) ln
1
qi
+
s−1∑
i=0
Ni(ε, x, k)
2ε
qmin
=
=| T−
ε,k | ln
1
pmin
+
s−1∑
i=0
Ni(ε, x, k) ln
1
qi
+ | T+
ε,k | 2ε
qmin
.
288 GRYGORIY TORBIN
Therefore, for any x ∈ [0, 1] and for any ε < 1
2
qmin we have:
lim
k→∞
ln μ(ΔQ
α1(x)...αk(x))
ln λ(ΔQ
α1(x)...αk(x))
≤
≤ lim
k→∞
| T−
ε,k | ln 1
pmin
+
s−1∑
i=0
Ni(ε, x, k) ln 1
qi
+ | T+
ε,k | 2ε
qmin
− ln[
k∏
j=1
qαj(x)]
=
= lim
k→∞
| T−
ε,k | ln 1
pmin
+
s−1∑
i=0
Ni(ε, x, k) ln 1
qi
+ | T+
ε,k | 2ε
qmin
s−1∑
i=0
Ni(x, k) ln 1
qi
≤
≤ 1 + lim
k→∞
| T−
ε,k | ln 1
pmin
+ | T+
ε,k | 2ε
qmin
s−1∑
i=0
Ni(x, k) ln 1
qi
≤
≤ 1 + lim
k→∞
| T−
ε,k | ln 1
pmin
+ | T+
ε,k | 2ε
qmin
k ln 1
qmax
= 1 +
2ε
qmin · ln 1
qmax
.
On the other hand we have: − ln μ(ΔQ
α1(x)...αk(x)) =
= −(ln[
k∏
j=1
pαj(x)j ]) = −(
∑
j∈T−
ε,k
ln pαj(x)j +
∑
j∈T+
ε,k
ln pαj(x)j) =
=
∑
j∈T−
ε,k
ln
1
pαj(x)j
+
s−1∑
i=0
⎛
⎜⎝ ∑
αj(x)=i, j∈T+
ε,k
ln
1
pαj(x)j
⎞
⎟⎠ ≥
≥ ∑
j∈T−
ε,k
ln
1
pmax
+
s−1∑
i=0
⎛
⎜⎝ ∑
αj(x)=i, j∈T+
ε,k
ln
1
qi + ε
⎞
⎟⎠ =
=| T−
ε,k | ln
1
pmax
+
s−1∑
i=0
Ni(ε, x, k) ln
1
qi + ε
=
=| T−
ε,k | ln
1
pmax
+
s−1∑
i=0
Ni(ε, x, k)
(
ln
1
qi
+ ln(1 − ε
qi + ε
)
)
≥
≥| T−
ε,k | ln
1
pmax
+
s−1∑
i=0
Ni(ε, x, k)
(
ln
1
qi
− ε
qi + ε
)
≥
PROBABILITY DISTRIBUTIONS 289
≥| T−
ε,k | ln
1
pmax
+
s−1∑
i=0
Ni(ε, x, k) ln
1
qi
−
s−1∑
i=0
Ni(ε, x, k)
ε
qi
≥
≥| T−
ε,k | ln
1
pmax
+
s−1∑
i=0
Ni(ε, x, k) ln
1
qi
−
s−1∑
i=0
Ni(ε, x, k)
ε
qmin
=
=| T−
ε,k | ln
1
pmax
+
s−1∑
i=0
Ni(ε, x, k) ln
1
qi
− | T+
ε,k | ε
qmin
.
Therefore, for any x ∈ [0, 1] and for any ε < qmin we have:
lim
k→∞
ln μ(ΔQ
α1(x)...αk(x))
ln λ(ΔQ
α1(x)...αk(x))
≥
≥ lim
k→∞
| T−
ε,k | ln 1
pmax
+
s−1∑
i=0
Ni(ε, x, k) ln 1
qi
− | T+
ε,k | ε
qmin
− ln[
k∏
j=1
qαj(x)]
=
= lim
k→∞
| T−
ε,k | ln 1
pmax
+
s−1∑
i=0
Ni(ε, x, k) ln 1
qi
− | T+
ε,k | ε
qmin
s−1∑
i=0
Ni(x, k) ln 1
qi
=
= 1 + lim
k→∞
| T−
ε,k | ln 1
pmax
− | T+
ε,k | ε
qmin
s−1∑
i=0
Ni(x, k) ln 1
qi
≥
≥ 1 + lim
k→∞
| T−
ε,k | ln 1
pmin
− | T+
ε,k | ε
qmin
k ln 1
qmin
= 1 − ε
qmin · ln 1
qmin
.
So, for any x ∈ [0, 1] and for any ε < 1
2
qmin we have:
1 − ε
qmin · ln 1
qmin
≤ lim
k→∞
ln μ(ΔQ
α1(x)...αk(x))
ln λ(ΔQ
α1(x)...αk(x))
≤
≤ lim
k→∞
ln μ(ΔQ
α1(x)...αk(x))
ln λ(ΔQ
α1(x)...αk(x))
≤ 1 +
2ε
qmin · ln 1
qmax
.
Therefore, for any x ∈ [0, 1] the following condition holds:
lim
k→∞
ln μ(ΔQ
α1(x)...αk(x))
ln λ(ΔQ
α1(x)...αk(x))
= 1. (10)
290 GRYGORIY TORBIN
Finally, from formula (10) and from Billingsley’s theorem ([7], p.142),
we have for all E ⊂ [0, 1] :
αλ(E) = 1 · αμ(E),
where αλ(E) and αμ(E) are the Hausdorff-Billingsley dimensions with re-
spect to measures λ and μ correspondingly (see, e.g., [7] or [12] for details).
The Hausdorff-Billingsley dimension with respect to the Lebesgue mea-
sure coincides with the classical Hausdorff dimension: αλ(E) = α0(E), ∀E ⊂
[0, 1]. From inf
ij
pij = pmin > 0 and from Theorem 1 of the paper [1] it follows
that the Hausdorff-Billingsley dimension of an arbitrary subset E ⊆ [0, 1]
with respect to the measure μ coincides with the Hausdorff dimension of
the set E
′
= Fξ(E) : αμ(E) = α0(Fξ(E)).
So, Fξ is a DP-transformation on the unit interval.
Remark. The condition inf
i,j
pij > 0 plays an essential role in the theorem
1. The following example shows that there exists a random variable ξ with
independent Q-adic digits such that condition (11) holds but the distribu-
tion function Fξ does not preserve the Hausdorff dimension.
Example. Let s = 3 and q0 = q1 = q2 = 1
3
, i.e., ξ =
∞∑
k=1
ηk
3k . Let p1k =
1
3
, p2k = 2
3
− p0k, and let
p0k =
{
1
3
, if k �= 3n, n ∈ N ;
7−7n
if k = 3n, n ∈ N.
It is easy to see that hj = ln 3 for j �= 3k, 0 < hj < ln 3 for j = 3k, and
bj = ln 3 for any j ∈ N.
Let L(k) = {i : i = 3n, i ≤ k, n ∈ N}, and l(k) =| L(k) |. Then
lim
k→∞
h1 + h2 + ... + hk
b1 + b2 + ... + bk
= lim
k→∞
(k − l(k)) ln 3 +
∑
j∈L(k)
hj
k ln 3
= 1,
because l(k)
k
→ 0 as k → ∞.
Let us consider the set T (Q) =
=
{
x : x =
∞∑
k=1
αk
3k
; αk = 0 if k = 3n; αk ∈ {0, 1, 2} if k �= 3n, n ∈ N
}
.
The set T (Q) is the topological support of a specially constructed ran-
dom variable ξ∗ with independent Q-digits (q∗0 = q∗1 = q∗2 = 1
3
; p∗0k =
1 if k = 3n, p∗0k = p∗1k = p∗2k = 1
3
if k �= 3k). From Theorem 2
of the paper [1] it follows that the Hausdorff dimension of the set T (Q) is
equal to 1.
PROBABILITY DISTRIBUTIONS 291
If x ∈ T , then
lim
k→∞
ln λ(ΔQ
α1(x)α2(x)...αk(x))
ln μ(ΔQ
α1(x)α2(x)...αk(x))
= lim
k→∞
ln
(
1
3
)k
ln(pα1(x)1 · pα2(x)2 · ... · pαk(x)k)
=
= lim
k→∞
k ln 1
3
(k − l(k)) ln 1
3
+
l(k)∑
j=1
7j ln 1
7
= 0.
Therefore, αμ(T (Q)) = 0 · αλ(T (Q)) which is equivalent to the condition
α0(Fξ(T (Q))) = 0, and we conclude that Fξ does not preserve the Hausdorff
dimension. Moreover, Fξ transforms the superfractal set T (Q) into the
anomalously fractal set T
′
(Q) = Fξ(T (Q)).
The following theorem gives us general necessary conditions for the dis-
tribution function Fξ to be a DP-transformation.
Theorem 2. If the distribution function Fξ of a random variable ξ with
independent Q-symbols preserves the Hausdorff dimension of any subset of
the unit interval, then
lim
n→∞
h1 + h2 + ... + hn
b1 + b2 + ... + bn
= 1. (11)
Proof. Let Aξ be the set of all possible ”supports” of the distribution of
the random variable ξ, i.e. Aξ = {E : E ∈ B, μξ(E) = 1}. The
number α0(ξ) = inf
E∈Aξ
{α0(E)} is said to be the Hausdorff dimension of the
probability distribution ξ and a set M with α0(M) = α0(ξ) is said to be
the minimal dimensional support of the measure μξ. Generally speaking,
the Hausdorff dimension of a probability distribution is less or equal to the
Hausdorff dimension of the topological support (minimal closed support)
of the distribution. Usually, the fractal analysis of minimal dimensional
supports is a rather nontrivial problem.
In [1] an explicit formula for the determination of the Hausdorff dimen-
sion of probability distributions with independent Q∗−symbols has been
found (under the restriction inf
i,j
qij > 0). If we put qij = qi, ∀i ∈ N0
s−1, then
the above mentioned formula gives us the exact value for the Hausdorff
dimension of our probability distributions ξ with independent Q-symbols:
α0(ξ) = lim
n→∞
h1 + h2 + ... + hn
b1 + b2 + ... + bn
.
If α0(ξ) < 1, then there exists a support E such that such that α0(ξ) ≤
α0(E) < 1. Since μξ(E) = 1, we conclude that α0(Fξ(E)) = 1 �= α0(E),
which contradicts the assumption of the theorem.
292 GRYGORIY TORBIN
Corollary 1. Let inf
i,j
pij = p > 0. Then the distribution function Fξ of a
random variable ξ with independent Q-symbols is a DP-transformation of
[0,1] if and only if
lim
n→∞
h1 + h2 + ... + hn
b1 + b2 + ... + bn
= 1,
i.e., if and only if the Hausdorff dimension of the measure μξ is equal to 1.
Corollary 2. If qi = s−1, ∀i ∈ N0
s−1 (i.e. ξ is a random variable with
independent s-adic digits), and inf
i,j
pij = p > 0, then the distribution function
Fξ is a DP-transformation if and only if
lim
n→∞
h1 + h2 + ... + hn
n ln s
= 1.
Corollary 3. If lim
k→∞
pik = qi, (∀i ∈ N0
s−1), then the distribution function
Fξ is a DP-transformation of the unit interval.
Acknowledgment. This work was partly supported by Alexander von
Humboldt Foundation and by DFG 436 UKR 113/78,80 projects.
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Institut für Angewandte Mathematik, Universität Bonn, Bonn, Ger-
many; National Pedagogical University, Kyiv, Ukraine; Institute
for Mathematics of NASU, Kyiv.
E-mail: torbin@wiener.iam.uni-bonn.de
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