VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA87 10.15559/17-VMSTA87 Research Article On singularity of distribution of random variables with independent symbols of Oppenheim expansions SydorukLiliiasinelnyklilia@ukr.net TorbinGrygoriytorbin7@gmail.com∗ National Pedagogical Dragomanov University Corresponding author. 2017 2610201743273283 1382017 3102017 3102017 © 2017 The Author(s). Published by VTeX2017 Open access article under the CC BY license.

The paper is devoted to the restricted Oppenheim expansion of real numbers ( ROE ), which includes already known Engel, Sylvester and Lüroth expansions as partial cases. 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 state the singularity (w.r.t. the Lebesgue measure) of the distribution of a random variable with i.i.d. increments of symbols of the restricted Oppenheim expansion. General non-i.i.d. case is also studied and sufficient conditions for the singularity of the corresponding probability distributions are found.

 Restricted Oppenheim expansion singular probability distributions metric theory of ROE Sylvester expansion 11K55 60G30 Introduction

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., [7, 12, 39] and references therein). The study of fractal properties of different families of singularly continuous probability measures (see, e.g., [7, 16, 26, 28, 27, 29, 31, 38, 1, 42] and references therein) can be used to solve non-trivial problems in the metric number theory ([8, 9, 5, 4, 10, 30, 21]), in the theory of dynamical systems and DP-transformations and in fractal analysis ([6, 3, 11, 19, 18, 17, 22, 41, 43]).

On the other hand, for many families of probability measures the problem “singularity vs absolute continuity” is 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 (such probability distributions are of pure type [23]). The Lévy theorem [25] gives necessary and sufficient conditions for such measures to be discrete resp. continuous, and the main problem is to find sharp conditions for absolute resp. singular continuity. Infinite Bernoulli convolutions make an important subclass of such measures (see, e.g., [2, 32, 34, 36, 35, 37] and references therein). Another wide family of probability distributions where the problem “singularity vs absolute continuity” is still open consists of probability distributions of the following form: ξ=Δξ1ξ2…ξn…F, where ξn are independent symbols of some generalized F-expansion over some alphabet A. Random variables with independent symbols of s-adic expansions, continued fraction expansions, the Lüroth expansion, the Sylvester 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. Let us mention that many authors studied normal properties of real numbers in terms of digits of their Oppenheim expansion and the Hausdorff dimension of corresponding exceptional sets (see, e.g., [13, 15, 44, 45, 24]). In Section 2 we develop approach which has been invented by G. Torbin to study normal properties of the Ostrogradsky–Sierpinski–Pierce expansion [40] and get general results on normal properties of Oppenheim expansions. Based on these results in the last section of the paper we show that singularity is typical for the family of probability measures with independent symbols of ROE expansions.

 On metric theory of the restricted Oppenheim expansion

It is known ([14]) that any real number x∈(0,1) can be represented in the form of the Oppenheim expansion x∼1d1+a1b11d2+⋯+a1a2·…·anb1b2·…·bn1dn+1+⋯ where an=an(d1,…,dn) , bn=bn(d1,…,dn) are positive integer valued functions and the denominators dn are determined by the following procedure: for a given x we define the sequences {xn} and {dn} via x1:=x;dn=[1xn]+1;xn+1:=bnan(xn−1dn).

A sufficient condition for a series on the right-hand side in (1) to be the expansion of its sum is: dn+1≥anbndn(dn−1)+1.

We call the expansion (1) the restricted Oppenheim expansion (ROE) of x if an and bn depend only on the last denominator dn and if the function hn(j):=an(j)bn(j)j(j−1) is integer valued.

Let us consider some examples of the restricted Oppenheim expansions.

Let an=1 , bn=dn ( n=1,2,… ). Then the expansion (1) obtained by the algorithm (2) is the well-known Engel expansion of x: x=1d1+1d1d2+⋯+1d1d2…dn+⋯, where dn+1≥dn .

Let an=bn=1 (or an=bn=const ) ( n=1,2,… ). Then the expansion (1) obtained by the algorithm (2) is the well-known Sylvester expansion of x: x=1d1+1d2+⋯+1dn+⋯, where dn+1≥dn(dn−1)+1 .

Let an=1 , bn=dn(dn−1) . In this case we obtain the Lüroth series for a number x: x=1d1+1d1(d1−1)d2+⋯+1d1(d1−1)…dn(dn−1)dn+1+⋯, where dn+1≥2 .

Let us mention that metric, dimensional and probabilistic theories of Oppenheim series are not sufficiently developed. In fact, as evidenced by recent works and thesis in the field ([46, 20, 33]), even such partial cases of Oppenheim expansions as the Lüroth series, the Engel and Sylvester series generate a number of challenges for the 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 and to show their effectiveness in the study of Lebesgue structures of distributions of random variables with independent symbols of Oppenheim expansions.

Choose the probability space (Ω,A,P) , with Ω=(0,1) , A the set of Lebesgue measurable subsets of (0,1) and the Lebesgue measure as P.

Let △j1j2…jnROE:={x:d1(x)=j1,d2(x)=j2,…,dn(x)=jn} be the cylinder of rank n with base (j1,j2,…,jn) .

([14]) Let x be the random variable, which is uniformly distributed on the unit interval and let dj:=dj(x) . Then P(d1=j1,…,dn=jn)=a1a2·…·an−1b1b2·…·bn−11jn(jn−1) where ai=ai(ji) , bi=bi(ji) (i=1,2,…,n−1) .

([14]) The sequence dn ( n=1,2,… ) forms the Markov chain P(d1=j)=1j(j−1);P(dn=k|dn−1=j)=hn−1(j)k(k−1),k>hn−1(j); h_{n-1}(j);\end{array}\]]]> and 0 otherwise.

Therefore, we get the following properties of cylinders:

△j1j2…jn−1ROE= ⋃i=1∞△j1j2…jn−1iROE .

sup△j1j2…jnROE=inf△j1j2…jn−1(jn−1)ROE .

inf△j1j2…jnROE=1j1+a1b11j2+⋯+a1a2…anb1b2…bn1jn ,

sup△j1j2…jnROE=1j1+a1b11j2+⋯+a1a2…an−1b1b2…bn−11jn−1 .

|△j1j2…jnROE|=a1a2…an−1b1b2…bn−11jn(jn−1) .

If the first k symbols of ROE are fixed, then (k+1) -st symbol of ROE cannot take values 2,3,…,akbkdk(dk−1) , ∀k∈N .

Each of the cylinders of ROE can be uniquely rewritten in terms of the difference restricted Oppenheim expansion ( ROE‾ ): α1=d1−1;αk+1=dk+1−akbkdk(dk−1).

Then series (1) can be rewritten as follows: x=1α1+1+a1b11a1b1d1(d1−1)+α2+a1a2b1b21a2b2d2(d2−1)+α3+⋯=:Δα1α2…αn…ROE‾. where αk∈{1,2,3,…} .

If there exists a sequence lk , such that ∀x∈[0,1] : bk−1ak−11dk−1(dk−1−1)<lk and series ∑k=1∞lk<+∞, then for any digit i0 almost all (with respect to the Lebesgue measure) real numbers x∈[0,1] contain symbol i0 only finitely many times in ROE‾ .

Let Ni(x) be a number of symbols “i” in ROE‾ of number x. Let us prove that the Lebesgue measure of the set Ai={x:Ni(x)=∞} is equal to 0 for all i∈N .

Consider the set △¯ik={x:x=△α1α2…αk−1iαk+1…ROE‾,αj∈N,j≠k}.

From the definition of the set △¯ik and properties of cylindrical sets it follows that △¯ik= ⋃α1=1∞⋯ ⋃αk−1=1∞△α1…αk−1iROE‾

Let us consider the following ratio: |△α1…αk−1iROE‾||△α1…αk−1ROE‾|=|△d1…dk−1(ak−1bk−1dk−1(dk−1−1)+i)ROE||△d1…dk−1ROE|=a1…ak−1b1…bk−1·1(ak−1bk−1dk−1(dk−1−1)+i)(ak−1bk−1dk−1(dk−1−1)+i−1):a1…ak−2b1…bk−2·1dk−1(dk−1−1)≤ak−1bk−1·dk−1(dk−1−1)ak−1bk−1dk−1(dk−1−1)·1ak−1bk−1dk−1(dk−1−1)=bk−1ak−1·1dk−1(dk−1−1)<lk Then λ(△¯ik)= ∑α1=1∞⋯ ∑αk=1∞|△α1…αk−1iROE‾|≤lk.

It is clear, that the set Ai is the upper limit of the sequence of sets {△¯ik} , i.e., Ai=lim supk→∞△¯ik= ⋂m=1∞( ⋃k=m∞△¯ik).

Since ∑k=1∞λ(△¯ik)≤ ∑k=1∞bk−1ak−11dk−1(dk−1−1)≤ ∑k=1∞lk<+∞, from the Borel–Cantelli Lemma it follows that λ(Ai)=0,∀i∈N.

Therefore, λ(A¯i)=1,∀i∈N.

Let A¯= ⋂i=1∞A¯i. It is clear that λ(A¯)=1 , which proves the theorem.  □

Consider the Sylvester series: d1∈{2,3,…},dk+1=dk(dk−1)+i,i∈{1,2,3,…}. If d1=2 , then the minimal admissible value of d2 is 3. Therefore dk+1≥dk(dk−1)+1≥(dk−1(dk−1−1)+1)(dk−1(dk−1−1)+1)≥(dk−1(dk−1−1))2+1≥((dk−2(dk−2−1)+1)(dk−2(dk−2−1)+1−1))2+1≥(dk−2(dk−2−1))4≥(dk−3(dk−3−1))23≥⋯≥(dk−(k−2)(dk−(k−2)−1))2k−2=(d2(d2−1))2k−2≥3·22k−2.

So for the Sylvester series: 1dk−1(dk−1−1)<13·22k−4·(3·22k−4)=:lk. It is clear that ∑k=1∞lk<∞.

Therefore, for λ-almost all x∈[0,1] their difference Sylvester expansion contain arbitrary digit i only finitely many times.

Consider the case where an=dn , bn=1 . Then dn+1≥dn·dn(dn−1)+1≥dn2≥(dn−12)2=dn−14≥dn−28≥dn−324≥⋯≥dn−(n−1)2n=d12n≥22n.

So for this case 1dk−1(dk−1−1)<122k=:lk.

Then, ∑k=1∞lk<∞.

So for λ-almost all x∈[0,1] the difference expansion contains arbitrary digit i only finitely many times.

 On singularity of distribution of random variables with independent symbols of the difference restricted Oppenheim expansion

A probability measure μξ of a random variable ξ is said to be singularly continuous (with respect to the Lebesgue measure) if μξ is a continuous probability measure and there exists a set E, such that λ(E)=0 and μξ(E)=1 .

Let x=Δα1(x)α2(x)…αn(x)…ROE‾ be ROE‾ of real numbers, let ξ1,ξ2,…,ξk,… be a sequence of independent random variables taking values 1,2,…,n,… with probabilities p1k,p2k,…,pnk,… correspondingly, and let ξ=Δξ1ξ2…ξn…ROE‾ be a random variables with independent ROE‾ -symbols.

Let assumptions of Theorem 2 hold. If there exists a digit i0 such that ∑k=1∞pi0k=+∞ , then the probability measure μξ is singular with respect to the Lebesgue measure.

Consider sets △¯i0n={x:αn(x)=i0} and Ai0={x:Ni0(x)=+∞}.

It is clear, that Ai0=lim‾n→∞△¯i0n .

From the definition of △¯i0n it follows that μξ(△¯i0n)=pi0n .

Since the random variables ξ1,ξ2,…,ξn,… are independent, we conclude that μξ(△¯i0k1∩△¯i0k2∩⋯∩△¯i0ks)=μξ({x:αk1(x)=i0,αk2(x)=i0,…,αks(x)=i0})=μξ({x:αk1(x)=i0})·μξ({x:αk2(x)=i0})·…·μξ({x:αks(x)=i0})=pi0k1·pi0k2·…·pi0ks.

So, events △¯i01,△¯i02,…,△¯i0n,… are independent with respect to measure μξ .

Since ∑k=1∞pi0k=+∞ and {△¯i0n} is a sequence of independent events, from the Borel–Cantelli Lemma it follows that μξ(Ai0)=1.

Let λ be the Lebesgue measure. Events △¯i01,△¯i02,…,△¯i0n,… , in general, are not independent w.r.t. the Lebesgue measure. We estimate the Lebesgue measure of the set △¯i0n : λ(△¯i0n)=λ({x:αn(x)=i0})= ∑α1(x)=1∞ ∑α2(x)=1∞⋯ ∑αn−1=1∞|Δα1(x)α2(x)…αn−1(x)i0ROE‾|= ∑α1(x)=1∞ ∑α2(x)=1∞⋯ ∑αn−1=1∞|Δα1(x)α2(x)…αn−1(x)i0ROE‾||Δα1(x)α2(x)…αn−1(x)ROE‾|·|Δα1(x)α2(x)…αn−1(x)ROE‾|≤ln(i0)· ∑α1(x)=1∞ ∑α2(x)=1∞⋯ ∑αn−1=1∞|Δα1(x)α2(x)…αn−1(x)ROE‾|=ln·1, where ln are defined in Theorem 2. Therefore, ∑n=1∞λ(△¯i0n)≤ ∑n=1∞ln<+∞.

So by the Borel–Cantelli Lemma, λ(Ai0)=0 , i.e. for λ-almost all x∈[0,1] their ROE‾ contains arbitrary digit i only finitely many times.

Hence λ(Ai0)=0 , and μξ(Ai0)=1 . So, probability measure μξ is singular with respect to the Lebesgue measure  □

Let assumptions of Theorem 2 hold. If ξk are independent and identically distributed random variables, then the probability measure μξ is singular with respect to the Lebesgue measure.

If ξ1,ξ2,…,ξn,… are independent and identically distributed random variables, then pik=pi .

Since ∑i=1∞pi=1 , it is clear that there exists a number i0 such, that: pi0>0 0$]]>. Therefore ∑k=1∞pi0k=+∞, and the singularity of μξ follows directly from Theorem 3.  □

Let x=Δα1(x)α2(x)…αn(x)…S‾ be the difference version of the Sylvester expansion (S‾ -expansion) and let ξ=Δξ1ξ2…ξn…S‾ be the random variable with independent symbols of S‾ -expansion.

If there exists a digit i0 such that ∑k=1∞pi0k=+∞ , then the probability measure μξ is singular with respect to the Lebesgue measure.

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

 Acknowledgments

This work was partly supported by the project STREVCOM FP-7-IRSES 612669 and by the Ministry of Education and Science of Ukraine.

Authors express their sincere gratitude to the referees for useful comments and remarks.

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