Marginal probability distributions of random sets in R with Markovian refinements
Marginal probability distributions describing statistically random sets in R losed with probability one are introduced. These distributions are calculated in the case of random sets with Markovian refinements.
Saved in:
| Date: | 2005 |
|---|---|
| Main Authors: | , |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2005
|
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/4433 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Marginal probability distributions of random sets in R with Markovian refinements / Yu.P. Virchenko, O.L. Shpilinskaya // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 121–130. — Бібліогр.: 11 назв.— англ. |
Institution
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-4433 |
|---|---|
| record_format |
dspace |
| spelling |
Virchenko, Yu.P. Shpilinskaya, O.L. 2009-11-09T15:39:37Z 2009-11-09T15:39:37Z 2005 Marginal probability distributions of random sets in R with Markovian refinements / Yu.P. Virchenko, O.L. Shpilinskaya // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 121–130. — Бібліогр.: 11 назв.— англ. 0321-3900 https://nasplib.isofts.kiev.ua/handle/123456789/4433 519.21 Marginal probability distributions describing statistically random sets in R losed with probability one are introduced. These distributions are calculated in the case of random sets with Markovian refinements. This research has been partially supported by RFFI and Belgorod State University. en Інститут математики НАН України Marginal probability distributions of random sets in R with Markovian refinements Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Marginal probability distributions of random sets in R with Markovian refinements |
| spellingShingle |
Marginal probability distributions of random sets in R with Markovian refinements Virchenko, Yu.P. Shpilinskaya, O.L. |
| title_short |
Marginal probability distributions of random sets in R with Markovian refinements |
| title_full |
Marginal probability distributions of random sets in R with Markovian refinements |
| title_fullStr |
Marginal probability distributions of random sets in R with Markovian refinements |
| title_full_unstemmed |
Marginal probability distributions of random sets in R with Markovian refinements |
| title_sort |
marginal probability distributions of random sets in r with markovian refinements |
| author |
Virchenko, Yu.P. Shpilinskaya, O.L. |
| author_facet |
Virchenko, Yu.P. Shpilinskaya, O.L. |
| publishDate |
2005 |
| language |
English |
| publisher |
Інститут математики НАН України |
| format |
Article |
| description |
Marginal probability distributions describing statistically random sets in R losed with probability one are introduced. These distributions are calculated in the case of random sets with Markovian refinements.
|
| issn |
0321-3900 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/4433 |
| citation_txt |
Marginal probability distributions of random sets in R with Markovian refinements / Yu.P. Virchenko, O.L. Shpilinskaya // Theory of Stochastic Processes. — 2005. — Т. 11 (27), № 3-4. — С. 121–130. — Бібліогр.: 11 назв.— англ. |
| work_keys_str_mv |
AT virchenkoyup marginalprobabilitydistributionsofrandomsetsinrwithmarkovianrefinements AT shpilinskayaol marginalprobabilitydistributionsofrandomsetsinrwithmarkovianrefinements |
| first_indexed |
2025-11-26T12:29:28Z |
| last_indexed |
2025-11-26T12:29:28Z |
| _version_ |
1850621392602529792 |
| fulltext |
Theory of Stochastic Processes
Vol. 11 (27), no. 3–4, 2005, pp. 121–130
UDC 519.21
YU. P. VIRCHENKO AND O. L. SHPILINSKAYA
MARGINAL PROBABILITY DISTRIBUTIONS OF
RANDOM SETS IN R WITH MARKOVIAN REFINEMENTS
Marginal probability distributions describing statistically random sets in closed
with probability one are introduced. These distributions are calculated in the case
of random sets with Markovian refinements.
1. Introduction
The construction of probability distributions in sample spaces A ⊂ Ω = 2R
d
with
elements which are some sets in R
d, d ∈ N, due to the difficulty of the description of each
separate random element in Ω, is usually realized in applications by the following way.
It is implied that there exists the natural coordinatization of elements in A by means of
elements of the other space B, so that the latter is described using analytical tools by
an essentially simpler way, i.e. there is the definite image B �→ A. At the same time, it
is natural that the space A is found essentially poorer in comparison with the space Ω.
Further, the probability distribution is defined on B, i.e. the probability space 〈B, T,
Q〉 is constructed. Then the mapping pointed out generates the measurability structure
on A by the natural way, i.e. T → Σ = {A : B �→ A, B ∈ T }, where the mapping of
random events B �→ A is constructed by the rule A = {X̃ : Ỹ �→ X̃, Ỹ ∈ B}. Under
such a definition of random events in A, the probability distribution P on A is defined
by means of the equality P(A) = Q(B).
In statistical physics [1], [2], the construction of probability distributions is based
namely on the described scheme, and such an approach is found sufficient in that case
where the description of each typical random element is realized by a rather simple way.
At the same time, the physicists are interesting in the constructions of random sets when
the typical realizations have no any simple coordinatization. It is connected with the
creation of mathematical models of the so-called fractal structures, in particular, of frac-
tally disordered media. The above-described scheme of the construction of such random
sets does not longer act. In this case, it should be natural to define the distribution by
the ”statistical description” of each realization and to introduce the probability distribu-
tion of an infinite set of marginal distributions similarly to the case of separable random
processes [3]. But, the natural approach [4] to the construction of marginal probability
distributions in Ω based on some cylindrical events {X̃ ⊂ R
d : χ(xi|X̃) = αi, i = 1, ..., n},
αi ∈ {0, 1}, n ∈ N, where χ(x|X) = 1 for x ∈ X and χ(x|X) = 0 for x �∈ X , which is
equivalent to the introducing of the random field {θ̃(x) = χ(x|X̃); x ∈ Rd}, leads only
to the so-called separable random sets which have the topological dimension equal with
probability one to the dimension d of the submersion space. At the same time, random
geometric structures used in physics for the description of the fractal property should
be ones possessing arbitrary dimensions, in particular, they may have any value of the
2000 AMS Mathematics Subject Classification. Primary 60D05.
Key words and phrases. random sets, marginal distributions, Markovian refinements, c-systems.
This research has been partially supported by RFFI and Belgorod State University.
121
122 YU. P. VIRCHENKO AND O. L. SHPILINSKAYA
Hausdorff dimension. Therefore, the above way of the introducing of the measurability
structure is inadequate. Apparently, to embrace some possible cases which may appear
in applications, it is necessary to study the construction of probability distributions of
arbitrary random sets closed with probability one [4].
It is necessary to note that some specific constructions of ”fractal” random sets were
proposed in different works [5—8]. They are based on the above-described idea of con-
structing the probabilistic space by means of some special maps. In these cases, as is
mentioned above, each proposed construction of random sets has inevitably a rather
particular character.
In connection with the described situation, it is natural to pose the problem about
the method of the probabilistic description of random sets by means of some marginal
distributions such that it is useful for arbitrary random sets closed with probability one.
Such a method has been proposed in the work [9] for the case d = 1. In addition, we have
introduced the class of random sets which have been named random sets with Markovian
refinements [10]. They possess the fractal property, i.e., with probability one, each their
realization has the continuum cardinality together with zero Lebesgue’s measure defined
on the submersion space R
d. It has succeeded to prove that the Hausdorff dimension is
not random for all typical realizations of such random sets [11]. In this connection, there
is the problem concerning the possibility of the probabilistic description of these random
sets on the basis of some marginal distributions introduced in [9]. This work is devoted
to the solving of this problem for sets in R.
In Sections 2 and 3, we describe briefly the approach to the construction of probability
distributions on the basis of marginal distributions. In Section 4, the random sets with
Markovian refinements [10] are introduced, and the reduction formula (7) for them is
proved in Section 5. This formula expresses multi-interval marginal distributions using
the one-interval one. In Section 6, the one-interval probability distribution of random
sets with Markovian refinements in [0, 1] ⊂ R with the parameter N = 2 is calculated.
2. Marginal distributions of random sets in R
In this section, we describe briefly the statistical construction of the probability dis-
tribution of random sets in R.
Definition 1. The class of sets F ⊂ 2R
d
is called a c-system if it contains the empty set
and satisfies the following conditions:
(a) if A and B belong to the system F, there exists the finite disjoint collection {Ci ∈
F \ {∅}; i = 1, ..., m}, Ci �= ∅, Ci ∩Cj = ∅ as i �= j, i, j = 1, ..., m, such that the set A∩B
may be represented in the form of the union
A ∩ B =
m⋃
i=1
Ci ;
(b) for any A ∈ F, there exists the finite disjoint collection {Dj ∈ F; j = 1, ..., n},
Dj �= ∅, Di ∩ Dj = ∅ at i �= j, i, j = 1, ..., n, such that the complement A may be
represented in the form of the union
A =
n⋃
j=1
Dj .
It was proved in [9] that each finite measure defined on an arbitrary c-system is
uniquely extendable to the minimal σ-algebra containing this c-system.
Let Ω = {X̃} be the sample space of random sets in [0, 1]. We define the bundle
C(S) on the space S of all possible finite tuples Δ = 〈δ1, δ2, ..., δn〉, δi = [ai, bi) ⊂ [0, 1);
MARGINAL PROBABILITY DISTRIBUTIONS 123
bi ≤ ai+1, i = 1, ..., n− 1, n ∈ N. Each fiber C(Δ) at fixed Δ represents the set of tuples
Θ = 〈θ1, θ2, ..., θn〉, θi ∈ {0, 1}, i = 1, ..., n. We regard two pairs 〈Δ, Θ〉 and 〈Δ′, Θ′〉 in
the fiber space as equivalent ones if
⋃
i:θi=0
δi =
⋃
i:θ′
i=0
δ′i
and the subtuples 〈δi; θi = 1〉, 〈δ′i; θ′i = 1〉 coincide with each other.
It is proved in [9] that the class of all possible events
{X : χ(X |δi) = θi; i = 1, ..., n}where χ(X |δ) = {1, X ∩ δ �= ∅ ; 0, X ∩ δ = ∅}
, constructed on the basis of the fiber space 〈S, C(S)〉 of tuples 〈Δ, Θ〉 is a c-system. We
associate the probability
P (Δ, Θ) = Pr{X : χ(X |δi) = θi; i = 1, ..., n}
to each pair 〈Δ, Θ〉 in the fiber space 〈S, C(S)〉. It is done by such a way that all
equivalent pairs have equal probabilities. We assign the term of the marginal n-interval
probability distribution to the collection of all mentioned probabilities at a fixed n which
is the length of Δ and Θ. All marginal distributions satisfy, firstly, the normalization
condition and, secondly, the consistency condition
∑
Θ∈{0,1}n
P (Δ, Θ) = 1 , P (Δ, Θ) =
∑
Θ′:π(Θ′)=Θ
P (Δ′, Θ′)
, where Δ = π(Δ′) in the last equality and the symbol π denotes the projection operation.
It is applied to the tuples Δ′, Θ′ and consists in the deleting of all those δ′ from Δ′ which
are absent in the tuple Δ.
It has been proved in [9] that the collection of all probabilities P (Δ, Θ) defines the
probabilistic measure on Ω. This probabilistic measure is concentrated on the class of
left-closed random sets. Each left-closed realization (and, in particular, the closed one)
is uniquely restored by this measure with probability one.
3. Marginal distributions
We introduce a functional notation for the above-introduced marginal distributions.
Let Δ be an arbitrary disjoint tuple of semi-intervals 〈δi = [ai, bi) , i = 1, ..., n〉, and let
Θ be the corresponding tuple of ”filling numbers” 〈θ1, θ2, ..., θn〉, where ai < bi ≤ ai+1,
θi ∈ {0, 1}, i = 1, 2, ..., n, and an+1 ≡ 1, [a1, bn) ⊂ [0, 1). At the fixed number of
intervals, the collection of probabilities P (Δ, Θ) with the length of Δ (and, respectively,
of Θ) equal to n is obviously the vector function of the variables ai, bi (i = 1, ..., n) for
each Θ. It has the definite domain which is cut out by the above inequalities, and it has
2n components enumerated by the values of the parameters θi ∈ {0, 1}, i = 1, 2, ..., n.
We shall call this function the marginal n-interval probability distribution. The number
n is called the order of this distribution. We denote this vector function as
Fθ1,θ2,...,θn(a1, b1; a2, b2; ...; an, bn) = P (Δ, Θ) .
The consistency conditions show that some components of this vector function are ex-
pressed by other components. We introduce the special notation for components with
Θ = 0,
F (a1, b1; a2, b2; ...; an, bn) = F0,0,...,0(a1, b1; a2, b2; ...; an, bn) .
Let us prove now that all functions Fθ1,θ2,...,θn(a1, b1; a2, b2; ...; an, bn) at any n ∈ N and at
arbitrary n-component collection Θ ∈ {0, 1}n may be expressed using only the collection
of functions F (a1, b1; a2, b2; ...; an, bn), n ∈ N.
124 YU. P. VIRCHENKO AND O. L. SHPILINSKAYA
Theorem 1. For each n ∈ N and for each tuple 〈θ1, ..., θn〉 ∈ {0, 1}n, the function
Fθ1,θ2,...,θn(a1, b1; a2, b2; ...; an, bn) belongs to the linear manifold constructed on the basis
of the function collection F (ai1 , bi1 ; ai2 , bi2 ; ...; aik
, bik
) where 〈i1, i2, ..., ik〉 are all projec-
tions of the tuple 〈1, 2, ..., n〉 having the length k = 0, 1, ..., n (it is supposed that F (∅) = 1).
Proof. We prove the theorem by induction on n and on l = 0, 1, ..., n which is the number
of nonzero components in the tuple 〈θ1, ..., θn〉. The number l named the filling index
characterizes the property of being filled up. If l = 0, then, for any n ∈ N, each function
Fθ1,θ2,...,θn(·) is equal to the corresponding function F (·). If n = 1, then we have, at
l = 1, F1(a1, b1) = 1− F (a1, b1) from the normalization condition. Let the statement be
correct for all filling indices and for all functions with orders up to n inclusively. Further,
let the assertion be correct for the order n + 1 at filling index values up to l inclusively,
0 ≤ l < n + 1. Consider the function Fθ1,θ2,...,θn+1(a1, b1; a2, b2; ...; an+1, bn+1) with the
filling index l + 1. Then, without loss of generality, we may suppose that θn+1 = 1. The
consistency conditions yield
Fθ1,θ2,...,θn,1(a1, b1; a2, b2; ...; an+1, bn+1) =
= Fθ1,θ2,...,θn(a1, b1; a2, b2; ...; an, bn) − Fθ1,θ2,...,θn,0(a1, b1; a2, b2; ...; an+1, bn+1) .
The first term has order n and the second one has n + 1, but its filling index is l. Hence,
the induction step has been performed.
4. Random sets with Markovian refinements
We consider now the random sets with Markovian refinements in [0, 1] introduced in
[10]. Each such set is completely defined by the value of the natural subdivision parameter
N ≥ 2 and by the probability distribution q[·] on the set 2K1 , K1 = {[i/N, (i+1)/N); i =
0, 1, ..., N − 1}. Since the empty realization is not under consideration, we must exclude
it. For this, the distribution should satisfy the property q[∅] = 0.
We introduce the following notation. Let Km be the collection of all semi-intervals
being elements of a refinement of order m with the subdivision parameter N , Km =
{[i/Nm, (i + 1)/Nm); i = 0, 1, ..., Nm − 1}. We associate the subset
Dm(Γ) =
⋃
δ∈Γ
δ ⊂ [0, 1)
to each set Γ ⊂ Km. Denote the class of all such subsets by Km, i.e.
Km = {Dm(Γ); ∅ �= Γ ⊂ Km} .
Finally, we introduce the projection operation Km : 2[0,1) → Km. For any subset X from
[0, 1), we denote
Km(X) =
⋃
δ∈Km:δ∩X �=∅
δ ∈ Km .
Under the fixed value of the subdivision parameter N , each random realization X̃ of
the random set with Markovian refinements is defined as the limit in the set theory sense,
X̃ = lim
m→∞ X̃m, where {X̃m; m = 1, 2, ...}, X̃1 ⊃ X̃2 ⊃ ... is the decreasing sequence of
random sets, X̃m ∈ Km, m ∈ N. Components X̃m, m ∈ N are constructed in the form of
the union
X̃m =
⋃
δ∈Δ̃m
δ ,
using some random collections Δ̃m ⊂ Km, m = 1, 2, ... such that the embedding condition
X̃m+1 ⊃ X̃m is satisfied. At the same time, X̃m = Km(X̃), m = 1, 2, ... , is fulfilled.
MARGINAL PROBABILITY DISTRIBUTIONS 125
The probability distribution of a random set with Markovian refinements is defined
by means of the introduction of probability distributions of each sequence component
X̃m, m ∈ N. These distributions Pm(Z) = Pr{X̃m = Z}, Z ∈ Km, are united in the
uniform Markovian chain. Namely, for any pair X ∈ Km+1, Y ∈ Km such that X ⊂ Y ,
the equality
Pm+1(X) = Pr{X̃m+1 = X |X̃m = Y }Pm(Y ) (1)
takes place where the conditional probability is defined by the Markovian ”branching
condition”
Pr{X̃m+1 = X |X̃m = Y } =
∏
δ:δ∈Sm(Y )
q (Tm(X ∩ δ)) . (2)
The operation Tm shifts the set so that the initial point of the semi-interval δ ∈ Km
becomes zero and extends the shifted semi-interval by Nm times so that it becomes
[0, 1). In addition, in (2), the semi-interval tuple belonging to Km and forming the set
Y is denoted by Sm(Y ), Sm(Y ) = {δ ∈ Km : δ ∩ Y �= ∅}, and q(·) is the probability
distribution on K1 defined by the formula q(Z) = q[S1(Z)], Z ∈ K1, where q(∅) = 0.
On the basis of probabilities Pm(Z), Z ∈ Km, m ∈ N, the probability distribution
P (Δ, Θ) of random sets with Markovian refinements for arbitrary pairs 〈Δ, Θ〉, Θ ≡
〈θ(δ); δ ∈ Δ〉 is calculated as follows:
P (Δ, Θ) = lim
m→∞
∑
Z∈Km: χ(Z|δ)=θ(δ), δ∈Δ
Pm(Z) . (3)
To study probabilities P (Δ, Θ), we prove a remarkable formula.
Theorem 2. For any pair X ∈ Km+l, Y ∈ Km with m, l ∈ N such that Km(X) = Y , the
probabilities Pm(Y ), Pm+l(X) corresponding to random sets with Markovian refinements
satisfy the identity
Pm+l(X) =
⎡
⎣ ∏
δ∈Sm(Y )
Pl (Tm(X ∩ δ))
⎤
⎦ Pm(Y ) . (4)
Proof. We use the induction on l = 1, 2, ... . For l = 1 and m ∈ N, formula (4) coincides
with (1) since, by definition, P1(Z) = q(Z) for any Z ∈ K1. Supposing that formula (4)
is valid for any m and for the given l, we perform the induction step to the value l + 1.
For this, using the induction assumption, we represent the function value Pm+l+1(X) at
any X ∈ Km+l+1 in the form
Pm+l+1(X) =
⎡
⎣ ∏
δ∈Sm+1(Z)
Pl (Tm+1(X ∩ δ))
⎤
⎦ Pm+1(Z) (5)
where Km+1(X) = Z. Further, using formulas (1), (2), we represent each probability
Pm+1(Z) in the form
Pm+1(Z) =
⎡
⎣ ∏
σ∈Sm(Y )
q (Tm(Z ∩ σ))
⎤
⎦Pm(Y )
, where Km(Z) = Y . Substituting this representation to (5), we transform the product
of two expressions
⎡
⎣ ∏
δ∈Sm+1(Z)
(·)
⎤
⎦
⎡
⎣ ∏
σ∈Sm(Y )
(·)
⎤
⎦ to the iterated product
Pm+l+1(X) =
⎡
⎣ ∏
σ∈Sm(Y )
R(σ)
⎤
⎦Pm(Y ) , (6)
126 YU. P. VIRCHENKO AND O. L. SHPILINSKAYA
R(σ) ≡ q (Tm(Z ∩ σ))
∏
δ∈Sm+1(Z): δ⊂σ
Pl (Tm+1(X ∩ δ))
, where we used the fact that Km(Z) = Y when each semi-interval δ ∈ Sm+1(Z) is
contained in some semi-interval σ ∈ Sm(Y ). We associate the set γ = Tm(δ) to each
δ ∈ Sm+1(Z), δ ⊂ σ at fixed σ ∈ Sm(Y ). In this case, γ ∈ K1 and, in addition,
γ ∈ S1(Tm(Z ∩ σ)) where Tm(Z ∩ σ) ∈ K1 since Z ∈ Km+1 and σ ∈ Km. Conversely,
each definite δ ∈ Sm+1(Z), δ ⊂ σ, corresponds uniquely to each γ ∈ S1(Tm(Z ∩ σ)).
Therefore, it is possible to enumerate the multipliers in the product with all possible γ:
R(σ) ≡ q (Tm(Z ∩ σ))
∏
γ∈S1(Tm(Z∩σ))
Pl (Tm+1(X ∩ δ)) .
We denote Tm(Z ∩ σ) = V . Since σ ∈ Km and Z ∈ Km+1, Z ∩ σ ∈ Km+1 and, hence,
V ∈ K1. Herein, q(V ) = P1(V ) is fulfilled. So
R(σ) ≡ P1 (V )
∏
γ∈S1(V )
Pl (Tm+1(X ∩ δ)) .
We notice that it is possible to represent the extension of a set by Nm+1 times in any
interval δ ∈ Km+1 as two extensions, firstly, by Nm times and, secondly, by N times.
That is, for any δ having a nonempty intersection with σ ∈ Km, it may be possible to
write the variable in the product in the form
T1 (Tm(X ∩ δ)) = T1 ([Tm(X ∩ σ)] ∩ δ) =
= T1 ([Tm(X ∩ σ)] ∩ [Tm(δ)]) = T1 ([Tm(X ∩ σ)] ∩ γ)
, where Tm(X ∩ σ) ∈ Kl+1. Then R(σ) is represented in the form
R(σ) ≡ P1 (V )
∏
γ∈S1(V )
Pl (T1 ([Tm(X ∩ σ)] ∩ γ)) = Pl+1(Tm(X ∩ σ))
on the basis of the induction assumption. The substitution of this expression in (6)
completes the induction step.
5. Reduction of multi-interval distributions
We now consider the problem of calculating the probabilities P (Δ, Θ) for random sets
with Markovian refinements. On the basis of Theorem 2, we shall obtain the general
formula which expresses all multi-interval marginal distributions via the one-interval
distribution.
A tuple Δ of semi-intervals is called the subordinated tuple to the subdivision of order
m (subordinated to Km) if, for each δ ∈ Δ, there exists a semi-interval σ ∈ Km such
that δ ⊂ σ. If a tuple Δ is subordinated to Km in a pair 〈Δ, Θ〉 where Θ = 〈θ(δ); δ ∈ Δ〉,
then, for each σ ∈ Km containing, at least, one component of Δ, it is possible to define
the pair 〈Δσ, Θσ〉 by the rule Δσ = 〈δ ∈ Δ; δ ⊂ σ〉, Θσ = 〈θ(δ); δ ∈ Δσ〉.
For each fixed m ∈ N and for each pair 〈Δ, Θ〉 with Δ subordinated to Km, we define
two sets to formulate below the reduction theorem. At first, we put
JΔ,m = Km
⎛
⎝ ⋃
δ∈Δ;θ(δ)=1
δ
⎞
⎠ .
The second set NΔ,m plays an important role in the degenerate case where semi-intervals
δ ∈ Δ are some elements of the subdivision with parameter N . Consider the class
NΔ = {δ ∈ Δ : θ(δ) = 0}. Define NΔ(Γ) =
⋃
δ∈Γ
δ for each Γ ⊂ NΔ. For fixed m and
each semi-interval σ ∈ Km, we denote, by Γσ, the subset of NΔ such that NΔ(Γσ) = σ.
MARGINAL PROBABILITY DISTRIBUTIONS 127
It is clear that the set Γσ is unique. But if such a set is absent for a given σ, then we
put Γσ = ∅. Further, we define
NΔ,m =
⋃
σ∈Km
Γσ .
The following assertion gives the main analytic tool, i.e. the reduction formula (7).
Theorem 3. For each pair 〈Δ, Θ〉 and for any m ∈ N such that the tuple Δ is subordi-
nated to Km, the expansion
P (Δ, Θ) =
∑
Y ∈Km: [0,1)\NΔ,m⊃Y ⊃JΔ,m
Pm(Y )
⎡
⎣ ∏
σ∈Sm(Y )
P (Tm(Δσ), Θσ)
⎤
⎦ (7)
takes place with semi-interval tuples Tm(Δσ) consisting of semi-intervals Tm(δ),
⋃
δ∈Δσ
δ ⊂
σ.
Proof. Since the semi-interval tuple Δ is subordinated to Km, the pair 〈Δσ, Θσ〉 is
defined, and two sets JΔ,m, NΔ,m are determined at each σ ∈ Km and for each pair
〈Δ, Θ〉.
According to (3), the probability P (Δ, Θ) is presented in the form
P (Δ, Θ) = lim
l→∞
∑
Z∈Km+l:χ(Z|δ)=θ(δ),δ∈Δ
Pm+l(Z) . (8)
For each σ ∈ Km and Z ∈ Km+l, we define the set σ∩Z. This set is empty in a trivial
way if σ �∈ Sm(K(Z)). It also can be empty for the other reason. If, for the given σ, there
is δ ⊂ NΔ,m, δ ∈ Δ, then, due to the definition NΔ,m, it follows that there exists such
Γσ ⊂ KΔ that the equality NΔ(Γσ) = σ holds. Then θ(δ′) = 0 takes place for all semi-
intervals δ′ ∈ Γσ and, therefore, χ(Z|σ) = 0, Z ∩σ = ∅. The converse statement is valid,
too. If the set Z corresponds to the term in (8), then σ ∩NΔ,m = ∅. In this connection,
the property restricting the set of configurations {Z ∈ Km+l : Z ⊂ [0, 1)\NΔ,m} included
in the sum in (8) can be written as follows:
[∀σ ∈ Sm(Km(Z))] [Z ∩ σ �= ∅, [χ(Z ∩ σ|δ) = θ(δ), δ ∈ Δσ]] . (9)
According to (4) and this requirement, all terms Pm+l(Z) in (8) are represented in the
form
Pm+l(Z) = Pm(Km(Z))
⎡
⎣ ∏
σ∈Sm(Km(Z))
Pl(Tm(Z ∩ σ))
⎤
⎦ . (10)
In this case, Km(Z) ⊂ [0, 1) \NΔ,m. In addition, if there exists δ ∈ Δσ with θ(δ) = 1 for
anyone σ ∈ Km, then necessarily σ ⊂ Km(Z). Therefore, Km(Z) ⊃ JΔ,m.
Taking into account formula (10) and requirement (9) for configurations Z ∈ Km+l,
we represent the sum in (8) in the form of iterated sums
∑
Z∈Km+l:χ(Z|δ)=θ(δ),δ∈Δ
Pm+l(Z) =
=
∑
Y ∈Km:
JΔ,m⊂Y ⊂[0,1)\NΔ,m
Pm(Y )
⎛
⎜⎝
∏
σ∈S(Y )
∑
V ∈Km+l:V ⊂σ
χ(V |δ)=θ(δ),δ∈Δ,δ⊂σ
Pl(Tm(V ))
⎞
⎟⎠
, where changes of the summation variables Y = Km(Z) and V = Z ∩ σ have been done.
Let us consider the multiplied sums on the right-hand side of this formula. Producing
128 YU. P. VIRCHENKO AND O. L. SHPILINSKAYA
the replacement of the summation variables in each of them U = Tm(V ) ∈ Kl and taking
into account that Tm transforms V ∩ δ = (V ∩ σ) ∩ δ to U ∩ Tm(δ), we obtain
∑
V ∈Km+l:V ⊂σ
χ(V |δ)=θ(δ),δ∈Δ,δ∈Δσ
Pl(Tm(V )) =
∑
U∈Kl:
χ(U|Tm(δ))=θ(δ),δ∈Δσ
Pl(U) =
=
∑
U∈Kl:
χ(U|δ′)=θ(δ′),δ′∈Tm(Δσ)
Pl(U) = Pr{X̃ : χ(Kl(X̃)|δ) = θ(δ), δ ∈ Tm(Δσ)} .
Here, it is taken into account that the condition δ ∈ Δ, δ ⊂ σ in the initial sum can be
changed by δ ∈ Δσ and, in this case, Tm(δ) ∈ Tm(Δσ). Passing to the limit l → ∞ in
the last expression, we obtain (7) on the basis of definition (3) since
lim
l→∞
Pr{X̃ : χ(Kl(X̃)|δ) = θ(δ), δ ∈ Tm(Δσ)} =
= Pr{X̃ : χ(X̃|δ) = θ(δ), δ ∈ Tm(Δσ)} = P (Tm(Δσ), Θσ) .
Remark 1. For factors corresponding to those σ when Δσ = ∅ on the right-hand side of
formula (7), it is necessary to put P (∅, Θ) = 1.
Remark 2. The tuple of semi-intervals may be such that it is not subordinated to Km
for all m ∈ N. But, in some cases, this difficulty can be passed over, namely, it may be if
it is possible to change the pair 〈Δ, Θ〉 by the equivalent one obtained by a subdivision of
semi-intervals δ ∈ Δ having θ(δ) = 0. For the equivalent pair consisting of a semi-interval
collection obtained by the subdivision, it may be already found such m ∈ N that it is
subordinated to Km. In particular, if Θ = 0, then all m ∈ N will be such that there is an
equivalent pair 〈Σ, 0〉 for the pair 〈Δ, 0〉, where Σ is subordinated to Km.
The following theorem allows us to express all multi-interval marginal distributions
by means of the one-interval one.
Theorem 4. For random sets with Markovian refinements, all multi-interval marginal
distributions F (a1, b1; ...; an, bn), n ∈ N, are expressed by the one-interval distribution
due to the following formula
F (a1, b1; ...; an, bn) =
∑
Y ∈Km: [0,1)\NΔ,m⊃Y
Pm(Y )
∏
σ∈Sm(Y )
F (σ)(a′
i, b
′
i), (11)
where m is the smallest order of the subdivision of the interval [0, 1), when there exists
the tuple of semi-intervals Δ′ = 〈[a′
i, b
′
i); i = 1..., n′〉 subordinated to Km and such that
the pair 〈Δ′, 0〉 being equivalent to the pair 〈Δ, 0〉 possesses the following property. In
each semi-interval σ ∈ Km, there exists no more than one semi-interval of Δ′. In this
case, F (σ)(a′
i, b
′
i) = 1 if σ ∩ [a′
i, b
′
i) = ∅ and F (σ)(a′
i, b
′
i) = P (Tm([a′
i, b
′
i)), 0) otherwise.
Proof. In view of the above notice, there is a tuple pointed out in the theorem. After
that, we apply formula (7).
6. One-interval distribution
Let us calculate the one-interval distribution in the elementary special case where the
subdivision parameter N is equal to 2 and the subdivision is spatially uniform q[δ1] =
q[δ2], where K1 = {δ1, δ2}, δ1 = [0, 1/2), δ2 = [1/2, 1). Herewith, the random set
is defined completely by two probabilities q2 = Pr{X̃1 = [0, 1)}, q1 = Pr{X̃1 = δ},
δ ∈ K1, 2q1 + q2 = 1. Hereinafter, we use a binary presentation of numbers in [0, 1)
in the calculations. For any number ξ ∈ [0, 1) with the binary decomposition ξ =
0.ξ1ξ2..., ξi ∈ {0, 1}, i ∈ N, we introduce the notations of n-component restrictions of the
fraction ξ, namely, ξ|n = 0.ξ1ξ2...ξn with the lack and ξ|n = ξ|n + 2−n with the excess,
correspondingly. At the same time, ξ|0 = 0, ξ|0 = 1 if ξ ∈ (0, 1). Firstly, let us calculate
MARGINAL PROBABILITY DISTRIBUTIONS 129
the probability F (0, b) = Pr{X̃ : X̃ ∩ [0, b) = ∅}. To this end, we consider the probability
F1(0, b) and represent it as the limit
F1(0, b) = lim
m→∞F1(0, b|m) = lim
m→∞ Pr{X̃ : Km(X̃) ∩ [0, b|m) �= ∅} . (12)
Thus, it is necessary to calculate the probability F1(0, b|m), m ∈ N. Considering the
function F1(0, b|m+1) as the probability of the sum of two disjoint events, we represent
it in the form
F1(0, b|m+1) = F1(0, b|m) + F01(0, b|m; b|m, b|m+1) . (13)
The following relation is established similarly:
fm ≡ F01(0, b|m; b|m, b|m) = F01(0, b|m+1; b|m+1, b|m) + F01(0, b|m; b|m, b|m+1) .
At bm+1 = 0, we have b|m = b|m+1 and, therefore, the last summand is equal to zero. If
bm+1 = 1, then b|m+1 = b|m. Hence, in both cases,
F01(0, b|m; b|m, b|m+1) = bm+1(fm − fm+1) . (14)
Thus, it is sufficient to calculate the sequence fm, m ∈ N. We construct the recurrent
relation for fm. For this, we represent fm+1 as the value P (Δ, Θ) at Δ = 〈δ(m+1)
i ; i =
1, ..., 2m+1b|m+1 +1〉 and Θ = 〈0, ...., 0, 1〉, where δ
(m+1)
i = [(i−1)/2m+1, i/2m+1). Then,
on the basis of (7), we have
fm+1 =
∑
Y ∈Km :
[0,1)\NΔ,m⊃Y ⊃[b|m+1,b|m+1)
Pm(Y )
⎡
⎣ ∏
σ∈Sm(Y )
P (Tm(Δσ), Θσ)
⎤
⎦ . (15)
Notice that NΔ,m = [0, b|m) and only one semi-interval σ = [b|m, b|m) contained in
Sm(Y ) can have a nonempty intersection with semi-intervals of Δ. In this case, the
tuple Δσ consists of semi-intervals [b|m, b|m+1), [b|m+1, b|m+1), respectively (the first is
empty as bm+1 = 0), the tuple Tm(Δσ) consists of [0, 0.1) as bm+1 = 0 and of the pair
〈[0, 0.1), [0.1, 1)〉 as bm+1 = 1. Respectively, Θσ is either 〈1〉 or 〈0, 1〉. Therefore, in the
first case, P (Tm(Δσ), Θσ) equals
F10(0, 0.1; 0.1, 1) + F11(0, 0.1; 0.1, 1) = q1 + q2
and, in the second one, it equals F01(0, 0.1; 0.1, 1) = q1. Consequently, we rewrite (15)
in the form
fm+1 = (q1 + (1 − bm+1)q2)
∑
Y ∈Km :
[b|m,1)⊃Y ⊃[b|m,b|m)
Pm(Y ) = (q1 + (1 − bm+1)q2)fm.
At f0 = Pr{X̃ ∩ [0, 0) = ∅, X̃ ∩ [0, 1) �= ∅} = 1, the obtained difference equation gives
us the formula for fm,
fm =
m∏
j=1
(q1 + (1 − bj)q2) = qrm
1 (q1 + q2)m−rm
, where the sequence 〈rm; m = 1, 2...〉 determines the number of units in the binary
decomposition b|m. Substituting this expression to (14) and using (12), (13), we find
F1(0, b) = (q1 + q2)
∞∑
m=0
bm+1fm , F (0, b) = 1 − F1(0, b) . (16)
The probability F (b, 1) is calculated in the same way, i.e.
F1(b, 1) = (q1 + q2)
∞∑
m=0
(1 − bm+1)gm , F (b, 1) = 1 − F1(b, 1) ,
130 YU. P. VIRCHENKO AND O. L. SHPILINSKAYA
gm =
m∏
j=1
(q1 + bjq2) = qm−rm
1 (q1 + q2)rm .
At last, in the general case, the probability F (a, b) is expressed via the probabilities
F− = F (2m(a|m−a), 1), F+ = F (0, 2m(b−b|m)), b|m = a|m on the basis of the reduction
formula. Namely, we find the minimal subdivision order m, when there exists the unique
point i/2m ∈ [a, b]. Then, we apply the reduction formula to F (a, b) = P (Δ, 0), where the
equivalent pair 〈Δ, 0〉 with Δ = 〈δ−, δ+〉 = 〈[a, i/2m), [i/2m, b)〉 is used. Semi-intervals
of this pair may have a nonempty intersection with no more than two semi-intervals
σ ∈ Sm(Y ). Then it follows from (7) that
F (a, b) =
∑
Y ∈Km:Y ⊂[0,1)
Pm(Y )P (Tm(δ−), 0)P (Tm(δ+), 0) . (17)
Here, it is taken into account that NΔ,m = ∅ under the minimality condition of the
subdivision order m. The sum in (17) is split into four parts in accordance with the
following summand groups {Y : Y ∩ (δ+ ∪ δ−) = ∅}, {Y : Y ∩ δ+ = ∅, Y ∩ δ− �= ∅},
{Y : Y ∩ δ+ �= ∅, Y ∩ δ− = ∅}, {Y : Y ∩ δ+ �= ∅, Y ∩ δ− �= ∅}. Accordingly, in the first
group, both factors P (Tm(δ±), 0) are equal identically to unity (the first group is absent
as m = 1), in the second and in the third group, one of the summands differs from unity
and, in last, both of them possess this property. If these factors differ from zero, then
they are equal to P (Tm(δ±), 0) = F±. Therefore, (17) is represented in the form
F (a, b) = Pr{Km(X̃) ∩ Km(δ+ ∪ δ−) = ∅} +
+Pr{Km(δ−) ⊂ Km(X̃) ⊂ [0, 1) \ Km(δ+)}F+ +
+ Pr{Km(δ+) ⊂ Km(X̃) ⊂ [0, 1) \ Km(δ−)}F−+
+ Pr{Km(δ+ ∪ δ−) ⊂ Km(X̃)}F−F+ =
= 1 − (q1 + q2)m−1 [1 − q1(F− + F+) − q2F−F+] , m ∈ N . (18)
The simple calculation of each summand in (18) is based on formulas (1) and (2).
Acknowledgement: We are grateful to RFFI and Belgorod State University for the
financial support of this work
Bibliography
1. J.M. Ziman, Models of Disorder. The Theoretical Physics of Homogeneously Disordered Sys-
tems, Cambridge University Press, Cambridge, 1979.
2. A.A. Katsnelson, A.I. Olemskoi, Microscopic Theory of Inhomogeneous Structures, Moscow
State University, Moscow, 1987.
3. I.I. Gikhman, A.V. Skorokhod, Introduction to the Theory of Random Processes, Dover Publi-
cations, Inc., New York, 1996.
4. G. Matheron, Random Sets and Integral Geometry, John Wiley and Sons, New York, 1975.
5. K.J. Falconer, Random fractals, Math. Proc. Cambridge Phil. Soc. 100 (1986), 559–582.
6. S. Graf, Statistically self-similar fractals, Probab. Th. Rel. Fields 74 (1987), 357–392.
7. R.D. Mauldin, S.C. Williams, Random recursion constructions: asymptotic geometric and topo-
logical properties, Trans. Am. Math. Soc. 295 (1986), 325–346.
8. M. Barnsley, J. Hutchinson, O. Stenflo, A fractal valued random iteration algorithm and fractal
hierarchy, arXiv:math.PR/0312187 v1.
9. Yu.P. Virchenko, O.L. Shpilinskaya, Probabilistic space of stochastic fractals, Ukr. Mat. Zh. 56
(2004), no. 11, 1467–1483.
10. Yu.P. Virchenko, O.L. Shpilinskaya, Random fields with Markovian refinements and the geom-
etry of fractally disordered media, Theor. Math. Phys. 124 (2000), no. 3, 1273-1284.
11. Yu.P. Virchenko, O.L. Shpilinskaya, Stochastic fractals with Markovian refinements, Theor.
Math. Phys. 128 (2001), no. 2, 983-995.
E-mail : virch@isc.kharkov.ua
|