On singularity of distribution of random variables with independent   symbols of Oppenheim expansions

Liliia Sinelnyk; Grygoriy Torbin

arXiv:1706.00953·math.PR·June 20, 2017

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

Liliia Sinelnyk, Grygoriy Torbin

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TL;DR

This paper investigates the distribution properties of real numbers' expansions in the restricted Oppenheim framework, establishing conditions for singularity and finiteness of digit appearances in almost all cases.

Contribution

It introduces new conditions for the singularity of distributions of random variables with independent symbols in restricted Oppenheim expansions, extending to non-i.i.d. cases.

Findings

01

Almost all real numbers have expansions with finitely many occurrences of any digit.

02

Distributions of i.i.d. symbol-based random variables are singular relative to Lebesgue measure.

03

Sufficient conditions for singularity are provided for non-i.i.d. cases.

Abstract

The paper is devoted to restricted Oppenheim expansion of real numbers ( $R O E$ ),which includes as partial cases already known Engel, Silvester and L\"uroth expansions. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their $R O E$ -expansion contain arbitrary digit $i$ only finitely many times. Main results of the paper states the singularity (w.r.t. Lebesgue measure) of the distribution of random variable with i.i.d increments of symbols of restricted Oppenheim expansion. General non-i.i.d. case are also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

Equations140

ξ = Δ_{ξ_{1} ξ_{2} ... ξ_{n} ...}^{F},

ξ = Δ_{ξ_{1} ξ_{2} ... ξ_{n} ...}^{F},

x = x_{1};

x = x_{1};

d_{n} = [\frac{1}{x _{n}}] + 1;

d_{n} = [\frac{1}{x _{n}}] + 1;

d_{n + 1} > \frac{a _{n}}{b _{n}} d_{n} (d_{n} - 1) .

d_{n + 1} > \frac{a _{n}}{b _{n}} d_{n} (d_{n} - 1) .

d_{n + 1} \geq \frac{a _{n}}{b _{n}} d_{n} (d_{n} - 1) + 1.

d_{n + 1} \geq \frac{a _{n}}{b _{n}} d_{n} (d_{n} - 1) + 1.

x = \frac{1}{d _{1}} + \frac{1}{d _{1} d _{2}} + \dots + \frac{1}{d _{1} d _{2} \dots d _{n}} + \dots,

x = \frac{1}{d _{1}} + \frac{1}{d _{1} d _{2}} + \dots + \frac{1}{d _{1} d _{2} \dots d _{n}} + \dots,

x = \frac{1}{d _{1}} + \frac{1}{d _{2}} + \dots + \frac{1}{d _{n}} + \dots,

x = \frac{1}{d _{1}} + \frac{1}{d _{2}} + \dots + \frac{1}{d _{n}} + \dots,

x = \frac{1}{d _{1}} + \frac{1}{d _{1} ( d _{1} - 1 ) d _{2}} + \dots + \frac{1}{d _{1} ( d _{1} - 1 ) \dots d _{n} ( d _{n} - 1 ) d _{n + 1}} + \dots,

x = \frac{1}{d _{1}} + \frac{1}{d _{1} ( d _{1} - 1 ) d _{2}} + \dots + \frac{1}{d _{1} ( d _{1} - 1 ) \dots d _{n} ( d _{n} - 1 ) d _{n + 1}} + \dots,

P (d_{1} = j_{1}, \dots, d_{n} = j_{n}) = \frac{a _{1} a _{2} \cdot \dots \cdot a _{n - 1}}{b _{1} b _{2} \cdot \dots \cdot b _{n - 1}} \frac{1}{j _{n} ( j _{n} - 1 )}

P (d_{1} = j_{1}, \dots, d_{n} = j_{n}) = \frac{a _{1} a _{2} \cdot \dots \cdot a _{n - 1}}{b _{1} b _{2} \cdot \dots \cdot b _{n - 1}} \frac{1}{j _{n} ( j _{n} - 1 )}

P (d_{1} = j_{1}) = P {x : d_{1} (x) = j_{1}} = ∣ △_{j_{1}}^{R O E} ∣ = (\frac{1}{j _{1}}, \frac{1}{j _{1} - 1}) = \frac{1}{j _{1} ( j _{1} - 1 )} .

P (d_{1} = j_{1}) = P {x : d_{1} (x) = j_{1}} = ∣ △_{j_{1}}^{R O E} ∣ = (\frac{1}{j _{1}}, \frac{1}{j _{1} - 1}) = \frac{1}{j _{1} ( j _{1} - 1 )} .

P (d_{1} = j_{1}, d_{2} = j_{2}) = P {x : x \in △_{j_{1} j_{2}}^{R O E}} = P {x : x = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}} + \dots};

P (d_{1} = j_{1}, d_{2} = j_{2}) = P {x : x \in △_{j_{1} j_{2}}^{R O E}} = P {x : x = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}} + \dots};

in f △_{j_{1} j_{2}}^{R O E} = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}};

in f △_{j_{1} j_{2}}^{R O E} = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}};

s u p △_{j_{1} j_{2}}^{R O E} = in f △_{j_{1} (j_{2} - 1)}^{R O E} = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2} - 1};

s u p △_{j_{1} j_{2}}^{R O E} = in f △_{j_{1} (j_{2} - 1)}^{R O E} = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2} - 1};

△_{j_{1} j_{2}}^{R O E} = (\frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}}, \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2} - 1}) .

△_{j_{1} j_{2}}^{R O E} = (\frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}}, \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2} - 1}) .

P (d_{1} = j_{1}, d_{2} = j_{2}) = ∣ △_{j_{1} j_{2}}^{R O E} ∣ = \frac{a _{1}}{b _{1}} \frac{1}{j _{2} ( j _{2} - 1 )} .

P (d_{1} = j_{1}, d_{2} = j_{2}) = ∣ △_{j_{1} j_{2}}^{R O E} ∣ = \frac{a _{1}}{b _{1}} \frac{1}{j _{2} ( j _{2} - 1 )} .

P (d_{1} = j_{1}, d_{2} = j_{2}, \dots, d_{n} = j_{n}) = P {x \in △_{j_{1} j_{2} \dots j_{n}}^{R O E}};

P (d_{1} = j_{1}, d_{2} = j_{2}, \dots, d_{n} = j_{n}) = P {x \in △_{j_{1} j_{2} \dots j_{n}}^{R O E}};

in f △_{j_{1} j_{2} \dots j_{n}}^{R O E} = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}} + \dots + \frac{a _{1} a _{2} \dots a _{n}}{b _{1} b _{2} \dots b _{n}} \frac{1}{j _{n}};

in f △_{j_{1} j_{2} \dots j_{n}}^{R O E} = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}} + \dots + \frac{a _{1} a _{2} \dots a _{n}}{b _{1} b _{2} \dots b _{n}} \frac{1}{j _{n}};

s u p △_{j_{1} j_{2} \dots j_{n}}^{R O E} = in f △_{j_{1} j_{2} \dots j_{n - 1} (j_{n} - 1)}^{R O E} = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}} + \dots + \frac{a _{1} a _{2} \dots a _{n - 1}}{b _{1} b _{2} \dots b _{n - 1}} \frac{1}{j _{n} - 1};

s u p △_{j_{1} j_{2} \dots j_{n}}^{R O E} = in f △_{j_{1} j_{2} \dots j_{n - 1} (j_{n} - 1)}^{R O E} = \frac{1}{j _{1}} + \frac{a _{1}}{b _{1}} \frac{1}{j _{2}} + \dots + \frac{a _{1} a _{2} \dots a _{n - 1}}{b _{1} b _{2} \dots b _{n - 1}} \frac{1}{j _{n} - 1};

∣ △_{j_{1} j_{2} \dots j_{n}}^{R O E} ∣ = \frac{a _{1} a _{2} \dots a _{n - 1}}{b _{1} b _{2} \dots b _{n - 1}} \frac{1}{j _{n} ( j _{n} - 1 )} .

∣ △_{j_{1} j_{2} \dots j_{n}}^{R O E} ∣ = \frac{a _{1} a _{2} \dots a _{n - 1}}{b _{1} b _{2} \dots b _{n - 1}} \frac{1}{j _{n} ( j _{n} - 1 )} .

P (d_{1} = j) = \frac{1}{j ( j - 1 )};

P (d_{1} = j) = \frac{1}{j ( j - 1 )};

P (d_{n} = k ∣ d_{n - 1} = j) = \frac{h _{n - 1} ( j )}{k ( k - 1 )}, k > h_{n - 1} (j);

P (d_{n} = k ∣ d_{n - 1} = j) = \frac{h _{n - 1} ( j )}{k ( k - 1 )}, k > h_{n - 1} (j);

P (d_{n} = k ∣ d_{1} = j_{1}, \dots, d_{n - 1} = j) = \frac{P ( d _{n - 1} = j , \dots , d _{n} = k )}{P ( d _{n - 1} = j , \dots , d _{1} = j _{1} )} .

P (d_{n} = k ∣ d_{1} = j_{1}, \dots, d_{n - 1} = j) = \frac{P ( d _{n - 1} = j , \dots , d _{n} = k )}{P ( d _{n - 1} = j , \dots , d _{1} = j _{1} )} .

\frac{a _{1} a _{2} \dots a _{n - 1}}{b _{1} b _{2} \dots b _{n - 1}} \frac{1}{k ( k - 1 )} : \frac{a _{1} a _{2} \dots a _{n - 2}}{b _{1} b _{2} \dots b _{n - 2}} \frac{1}{j ( j - 1 )} = \frac{a _{n - 1} ( j )}{b _{n - 1} ( j )} \frac{j ( j - 1 )}{k ( k - 1 )} = \frac{h _{n - 1} ( j )}{k ( k - 1 )}, k > h_{n - 1} (j)

\frac{a _{1} a _{2} \dots a _{n - 1}}{b _{1} b _{2} \dots b _{n - 1}} \frac{1}{k ( k - 1 )} : \frac{a _{1} a _{2} \dots a _{n - 2}}{b _{1} b _{2} \dots b _{n - 2}} \frac{1}{j ( j - 1 )} = \frac{a _{n - 1} ( j )}{b _{n - 1} ( j )} \frac{j ( j - 1 )}{k ( k - 1 )} = \frac{h _{n - 1} ( j )}{k ( k - 1 )}, k > h_{n - 1} (j)

α_{1} = d_{1} - 1;

α_{1} = d_{1} - 1;

α_{k + 1} = d_{k + 1} - \frac{a _{k}}{b _{k}} d_{k} (d_{k} - 1) .

α_{k + 1} = d_{k + 1} - \frac{a _{k}}{b _{k}} d_{k} (d_{k} - 1) .

x = \frac{1}{α _{1} + 1} + \frac{a _{1}}{b _{1}} \frac{1}{\frac{a _{1}}{b _{1}} d _{1} ( d _{1} - 1 ) + α _{2}} + \frac{a _{1} a _{2}}{b _{1} b _{2}} \frac{1}{\frac{a _{2}}{b _{2}} d _{2} ( d _{2} - 1 ) + α _{3}} + \dots =: Δ_{α_{1} α_{2} ... α_{n} ...}^{\overline{R O E}} .

x = \frac{1}{α _{1} + 1} + \frac{a _{1}}{b _{1}} \frac{1}{\frac{a _{1}}{b _{1}} d _{1} ( d _{1} - 1 ) + α _{2}} + \frac{a _{1} a _{2}}{b _{1} b _{2}} \frac{1}{\frac{a _{2}}{b _{2}} d _{2} ( d _{2} - 1 ) + α _{3}} + \dots =: Δ_{α_{1} α_{2} ... α_{n} ...}^{\overline{R O E}} .

\frac{a _{k - 1}}{b _{k - 1}} \frac{b _{k}^{2}}{a _{k}^{2}} \frac{1}{d _{k - 1} ( d _{k - 1} - 1 )} < l_{k}

\frac{a _{k - 1}}{b _{k - 1}} \frac{b _{k}^{2}}{a _{k}^{2}} \frac{1}{d _{k - 1} ( d _{k - 1} - 1 )} < l_{k}

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TopicsMathematical Dynamics and Fractals · Stochastic processes and financial applications · Stochastic processes and statistical mechanics

Full text

On singularity of distribution of random variables with independent symbols of Oppenheim expansions

Liliia Sinelnyk, Grygoriy Torbin

Abstract.

The paper is devoted to restricted Oppenheim expansion of real numbers ( $ROE$ ),which includes as partial cases already known Engel, Silvester and Lüroth expansions. We find conditions under which for almost all (with respect to Lebesgue measure) real numbers from the unit interval their $ROE$ -expansion contain arbitrary digit $i$ only finitely many times. Main results of the paper states the singularity (w.r.t. Lebesgue measure) of the distribution of random variable with i.i.d increments of symbols of restricted Oppenheim expansion. General non-i.i.d. case are also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

AMS Subject Classifications (2010): 11K55, 60G30.

**Key words: ** Restricted Oppenheim expansion, singular probability distributions, metric theory of ROE, Silvester expansion.

1. Introduction

It is well known that singularly continuous probability measures were studied during almost all XX century and there are a lot of open problems related to them. The fractal and multifractal approaches to the study of such measures are known to be extremely useful (see, e.g., [4, 13, 34] and references therein). Study of fractal properties of different families of singularly continuous probability measures (see, e.g., [4, 18, 22, 21, 23, 24, 25, 35] and references therein) can be used to solve non-trivial problems in metric number theory ([1, 2, 7, 8, 12, 26, 27]), in the theory of dynamical systems and DP-transformations and in fractal analysis ([3, 9, 10, 15, 16, 17, 20, 36, 38]).

On the other hand for many families of probability measures the problem "singularity vs absolute continuity" are extremely complicated even for the so-called probability distributions of the Jessen-Wintner type, i.e., distributions of random variables which are sums of almost surely convergent series of independent discretely distributed random variables. Infinite Bernoulli convolutions form one important subclass of such measures (see, e.g., [6, 28, 30, 31, 32, 33] and references therein). Another wide family of probability measures where the problem "singularity vs absolute continuity" are still open consists of probability distributions of the following form:

[TABLE]

where $\xi_{n}$ are independent symbols of some generalize $F$ -expansion over some alphabet $A$ . Random variables with independent symbols of $s-$ adic expansions, continued fraction expansions, Lüroth expansion, Silvester and Engel expansions are among them. This paper is devoted to the development of probabilistic theory of Oppenheim expansions of real numbers which contains many important expansions as rather special cases. Main result of the paper shows that in this family of probability measures the singularity are generic.

2. On metric theory of restricted Oppenheim expansion

It is known ([14]) that any real number $x\in(0,1)$ can be represented in the form of the Oppenheim expansion

[TABLE]

where $a_{n}=a_{n}(d_{1},\ldots,d_{n})$ , $b_{n}=b_{n}(d_{1},\ldots,d_{n})$ are positive integers and the denominators $d_{n}$ are determined by the algorithm:

[TABLE]

and satisfy inequalities:

[TABLE]

A sufficient condition for a series on the right-hand side in (1) to be the expansion of its sum by the algorithm (2) is:

[TABLE]

We call the expansion (1) (obtained by the algorithm (2)) the restricted Oppenheim expansion (ROE) of $x$ if $a_{n}$ and $b_{n}$ depend only on the last denominator $d_{n}$ and if the function

[TABLE]

is integer valued.

Let us consider some examples of restricted Oppenheim expansions.

Example 1.

Let $a_{n}=1$ , $b_{n}=d_{n}$ , ( $n=1,2,\ldots$ ).Then the expansion (1) obtained by the algorithm (2) is the well known Engel expansion of $x$ :

[TABLE]

where $d_{n+1}\geq d_{n}.$

Example 2.

Let $a_{n}=b_{n}=1$ (or $a_{n}=b_{n}=const$ ) ( $n=1,2,\ldots$ ). Then the expansion (1) obtained by the algorithm (2) is the well known Silvester expansion of $x$ :

[TABLE]

where $d_{n+1}\geq d_{n}(d_{n}-1)+1.$

Example 3.

Let $a_{n}=1,$ $b_{n}=d_{n}(d_{n}-1)$ . In this case we obtain Lüroth series for a number $x:$

[TABLE]

where $d_{n+1}\geq 2.$

It is known [14] that metric, dimensional and probabilistic theories of Oppenheim series are underdeveloped (in fact, as evidenced by its recent work and dissertation ([39], [19], [29]) even such partial cases of Oppenheim expansions as Luroth series, Engel and Silvester series generate a number of challenges metric and probabilistic number theory). The main purpose of this article is to develop some general methods of the metric theory of numbers and Oppenheim expansions, show their effectiveness in the study of Lebesgue structures of distributions of random variables with independent symbols of Oppenheim expansions.

Choose the probability space $(\Omega,\mathcal{A},P),$ as $\Omega=(0,1)$ , $\mathcal{A}$ the set of Lebesgue measurable subsets of $(0,1)$ and Lebesgue measure as $P$ .

Let $\triangle_{j_{1}j_{2}\ldots j_{n}}^{ROE}:=\{x:d_{1}(x)=j_{1},d_{2}(x)=j_{2},\ldots,d_{n}(x)=j_{n}\}$ be the cylinder of rank $n$ with base $(j_{1},j_{2},\ldots,j_{n}).$

Lemma 1.

([14])

[TABLE]

where $a_{i}=a_{i}(j_{i})$ , $b_{i}=b_{i}(j_{i})$ $(i=1,2,\ldots,n-1)$ .

Proof.

At first calculate $P(d_{1}=j_{1}).$

[TABLE]

As follows

[TABLE]

Similarly

[TABLE]

∎

Theorem 1.

([14]) The sequence $d_{n}(n=1,2,\ldots)$ forms a Markov chain

[TABLE]

and 0 otherwise.

Proof.

Since $\{d_{1}=j_{1},\ldots,d_{n-1}=j\}>0,$ then

[TABLE]

Thus, by Lemma, this ratio is equal to:

[TABLE]

∎

Therefore, we get the following properties of cylinders:

$\triangle_{j_{1}j_{2}\ldots j_{n-1}}^{ROE}=\bigcup\limits_{i=1}^{\infty}\triangle_{j_{1}j_{2}\ldots j_{n-1}i}^{ROE}.$
$sup\triangle_{j_{1}j_{2}\ldots j_{n}}^{ROE}=inf\triangle_{j_{1}j_{2}\ldots j_{n-1}j_{n}+1}^{ROE}.$
$inf\triangle_{j_{1}j_{2}\ldots j_{n}}^{ROE}=\frac{1}{j_{1}}+\frac{a_{1}}{b_{1}}\frac{1}{j_{2}}+\ldots+\frac{a_{1}a_{2}\ldots a_{n}}{b_{1}b_{2}\ldots b_{n}}\frac{1}{j_{n}},$

$\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ sup\triangle_{j_{1}j_{2}\ldots j_{n}}^{ROE}=\frac{1}{j_{1}}+\frac{a_{1}}{b_{1}}\frac{1}{j_{2}}+\ldots+\frac{a_{1}a_{2}\ldots a_{n-1}}{b_{1}b_{2}\ldots b_{n-1}}\frac{1}{j_{n}-1}.$

$|\triangle_{j_{1}j_{2}\ldots j_{n}}^{ROE}|=\frac{a_{1}a_{2}\ldots a_{n-1}}{b_{1}b_{2}\ldots b_{n-1}}\frac{1}{j_{n}(j_{n}-1)}.$

If first $k$ symbols of ROE are fixed, then $(k+1)-th$ symbol of ROE can not take values $2,3,\ldots,\frac{a_{k}}{b_{k}}d_{k}(d_{k}-1),$ $\forall k\in\mathbb{N}.$

Each of the cylinders of ROE can be uniquely rewritten in terms of the difference restricted Oppenheim expansion ( $\overline{ROE}$ ):

[TABLE]

Then series (1) can be rewritten as follows:

[TABLE]

where $\alpha_{k}\in\{1,2,3,\ldots\}.$

Theorem 2.

If there exists a sequence $l_{k}$ , such that $\forall x\in[0,1]:$

[TABLE]

and series

[TABLE]

then for any digit $i_{0}$ almost all (with respect to the Lebesgue measure) real numbers $x\in[0,1]$ contain symbol $i_{0}$ only finitely many times in $\overline{ROE}.$

Proof.

Let $N_{i}(x)$ be a number of symbols $"i"$ in $\overline{ROE}$ of number $x.$ Let us prove that Lebesgue measure of set $A_{i}=\{x:N_{i}(x)=\infty\}$ is equal to 0 for all $i\in\mathbb{N}.$

Consider the set

[TABLE]

From the definition of the set $\bar{\triangle}_{i}^{k}$ and properties of cylindrical sets it follows that

[TABLE]

Let us consider the following ratio:

[TABLE]

Then

[TABLE]

It is clear, that the set $A_{i}$ is the upper limit of the sequence of sets $\{\bar{\triangle}_{i}^{k}\},$ i.e.,

[TABLE]

Since

[TABLE]

from Borel-Cantelli Lemma it follows that

[TABLE]

Therefore,

[TABLE]

Let

[TABLE]

It is clear that $\lambda(\bar{A})=1,$ which proves the theorem. ∎

Example 4.

Consider the Silvester series:

[TABLE]

If $d_{1}=2,$ then $\min d_{2}=3.$ Therefore

[TABLE]

So for the Silvester series:

[TABLE]

It is clear that

[TABLE]

Therefore, for $\lambda$ -almost all $x\in[0,1]$ their Silvester series contain arbitrary digit $i$ only finitely many times.

Example 5.

Consider the case where $a_{n}=d_{n},$ $b_{n}=1.$ Then

[TABLE]

So for this case

[TABLE]

Then,

[TABLE]

So for $\lambda$ -almost all $x\in[0,1]$ the expansion contains arbitrary digit $i$ only finitely many times.

3. On singularity of distribution of random variables with independent symbols of $\overline{ROE}$

Definition.

A probability measure $\mu_{\xi}$ of a random variable $\xi$ is said to be singularly continuous (with respect to Lebesgue measure) if $\mu_{\xi}$ is a continuous probability measure and there exists a set $E$ , such that $\lambda(E)=0$ and $\mu_{\xi}(E)=1.$

Let $x=\Delta^{\overline{ROE}}_{\alpha_{1}(x)\alpha_{2}(x)...\alpha_{n}(x)...}$ be $\overline{ROE}$ of real numbers, let $\xi_{1},\xi_{2},\ldots,\xi_{k},\ldots$ be a sequence of independent random variables taking values $1,2,\ldots,n,\ldots$ with probabilities $p_{1k},p_{2k},\ldots,p_{nk},\ldots$ correspondingly, and let

[TABLE]

be a random variables with independent $\overline{ROE}$ - symbols.

Theorem 3.

Let assumptions of Theorem 2 hold. If there exists a digit $i_{0}$ such that $\sum\limits_{k=1}^{\infty}p_{i_{0}k}=+\infty,$ then the probability measure $\mu_{\xi}$ is singular with respect to Lebesgue measure.

Proof.

Consider sets

[TABLE]

and

[TABLE]

It is clear, that $B=\varlimsup\limits_{n\to\infty}B_{n}.$

From the definition of $B_{n}$ it follows that $\mu_{\xi}(B_{n})=p_{i_{0}n}.$

Since the random variables $\xi_{1},\xi_{2},\ldots,\xi_{n},\ldots$ are independent, we conclude that

[TABLE]

So, events $B_{1},B_{2},\ldots,B_{n},\ldots$ are independent with respect to measure $\mu_{\xi}.$

Since $\sum\limits_{k=1}^{\infty}p_{i_{0}k}=+\infty$ and $\{B_{n}\}$ is a sequence of independent events from Borel-Cantelli Lemma it follows that

[TABLE]

Let $\lambda$ be Lebesgue measure. Events $B_{1},B_{2},\ldots,B_{n},\ldots$ are generally speaking, not independent w.r.t. Lebesgue measure. We estimate the Lebesgue measure of the set $B_{n}$ :

[TABLE]

Therefore,

[TABLE]

So by Borel-Cantelli Lemma , $\lambda(B)=0,$ i.e. for $\lambda$ -almost all $x\in[0,1]$ their $\overline{ROE}$ contains arbitrary digit $i$ only finitely many times.

Hence $\lambda(B)=0$ , аnd $\mu_{\xi}(B)=1$ . So, probability measure $\mu_{\xi}$ is singular with respect to Lebesgue measure ∎

Theorem 4.

Let assumptions of Theorem 2 hold. If $\xi_{k}$ are independent and identically distributed random variables, then the probability measure $\mu_{\xi}$ is singular with respect to Lebesgue measure.

Proof.

If $\xi_{1},\xi_{2},\ldots,\xi_{n},\ldots$ are independent and identically distributed random variables, then the matrix $P=\|p_{ik}\|$ is of the following form

[TABLE]

Since $\sum\limits_{i=1}^{\infty}p_{i}=1,$ it is clear that there exists a number $i_{0}$ such, that: $p_{i_{0}}>0$ . Therefore

[TABLE]

∎

Corollary 1.

Let

[TABLE]

be the difference version of Silvester expansion $(\overline{S}$ -expansion) and let

[TABLE]

be the random variable with independent symbols of $\overline{S}$ -expansion.

If there exists a digit $i_{0}$ such that $\sum\limits_{k=1}^{\infty}p_{i_{0}k}=+\infty,$ then the probability measure $\mu_{\xi}$ is singular with respect to Lebesgue measure.

In particular, the distribution of the random variable with independent identically distributed symbols of $\overline{S}$ -expansion is singular w.r.t. Lebesgue measure.

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8[8] Albeverio S., Pratsiovytyi M., Torbin G. Singular probability distributions and fractal properties of sets of real numbers defined by the asymptotic frequencies of their s-adic digits, Ukr.Math.J. , 57 (2005), no.9, 1361-1370.

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On singularity of distribution of random variables with independent symbols of Oppenheim expansions

Abstract.

1. Introduction

2. On metric theory of restricted Oppenheim expansion

Example 1**.**

Example 2**.**

Example 3**.**

Lemma 1**.**

Proof.

Theorem 1**.**

Proof.

Theorem 2**.**

Proof.

Example 4**.**

Example 5**.**

3. On singularity of distribution of random variables with independent symbols of ROE‾\overline{ROE}ROE

Definition**.**

Theorem 3**.**

Proof.

Theorem 4**.**

Proof.

Corollary 1**.**

Example 1.

Example 2.

Example 3.

Lemma 1.

Theorem 1.

Theorem 2.

Example 4.

Example 5.

3. On singularity of distribution of random variables with independent symbols of $\overline{ROE}$

Definition.

Theorem 3.

Theorem 4.

Corollary 1.