VMSTA Modern Stochastics: Theory and Applications 2351-6054 2351-6046 2351-6046 VTeXMokslininkų g. 2A, 08412 Vilnius, Lithuania VMSTA61 10.15559/16-VMSTA61 Research Article On spectra of probability measures generated by GLS-expansions LupainMarinamarinalupain@npu.edu.ua National Pedagogical Dragomanov University, Pyrogova Str. 9, Kyiv, Ukraine 2016 2610201633213221 582016 2792016 2792016 © 2016 The Author(s). Published by VTeX2016 Open access article under the CC BY license.

We study properties of distributions of random variables with independent identically distributed symbols of generalized Lüroth series (GLS) expansions (the family of GLS-expansions contains Lüroth expansion and Q∞ - and G∞2 -expansions). To this end, we explore fractal properties of the family of Cantor-like sets C[GLS,V] consisting of real numbers whose GLS-expansions contain only symbols from some countable set V⊂N∪{0} , and derive exact formulae for the determination of the Hausdorff–Besicovitch dimension of C[GLS,V] . Based on these results, we get general formulae for the Hausdorff–Besicovitch dimension of the spectra of random variables with independent identically distributed GLS-symbols for the case where all but countably many points from the unit interval belong to the basis cylinders of GLS-expansions.

 Random variables with independent GLS-symbols Q∞-expansion N-self-similar sets Hausdorff–Besicovitch dimension 11K55 28A80 Introduction

During the last 20 years, many authors studied singularly continuous probability measures generated by different expansions of real numbers (see, e.g., [2, 9, 10, 12–15]). All these measures are the distributions of random variables of the form ξ=Δξ1ξ2…ξk…F, where {ξk} are independent or Markovian, and F stands for some expansion of real numbers. For the case of expansions over finite alphabets, fractal properties of the spectra of the corresponding measures are relatively well studied. For the case of infinite alphabets, the situation is essentially more complicated. In [8] and [9], it has been shown that even for self-similar Q∞ -expansion and for i.i.d. case, the Hausdorff–Besicovitch dimension of the corresponding spectra cannot be calculated in a traditional way (as a root of the corresponding equation), and formulae for the Hausdorff dimension of the measure μξ are also unknown.

In this paper, we generalize results from [8] and [9] for the case of distributions of random variables with independent identically distributed GLS digits ξ=Δξ1ξ2…ξk…GLS and get general formulae for the determination of the Hausdorff–Besicovitch dimension of spectra of ξ for the case where all but countably many points from the unit interval belong to the basis cylinders of GLS-expansion.

 On GLS-expansion and fractal properties of related probability measures

Let Q∞=(q0,q1,…,qn,…) be an infinite stochastic vector with positive coordinates. Let us consider a countable sequence Δi=[ai,bi] of intervals such that Int(Δi)∩Int(Δj)=∅(i≠j) and |Δi|=qi . The sets Δi are said to be cylinders of GLS-expansion (generalized Lüroth series).

Let us remark that the placement of cylinders of rank 1 is completely determined by the preselected procedure.

For every cylinder Δi1 of rank 1, we consider a sequence of nonoverlapping closed intervals Δi1i2⊂Δi1 such that |Δi1i2||Δi1|=qi2 and the placement of Δi1i2 in Δi1 is the same as Δi1 in [0;1] . The closed intervals Δi1i2 are said to be cylinders of rank 2 of the GLS-expansion.

Similarly, for every cylinder of rank (n−1) Δi1i2…in−1 , we consider the sequence of nonoverlapping closed intervals Δi1i2…in⊂Δi1i2…in−1 such that |Δi1i2…in||Δi1i2…in−1|=qin,i∈N∪{0}, and the placement of Δi1i2…in in Δi1i2…in−1 is the same as Δi1 in [0;1] .

The closed intervals Δi1i2…in are said to be cylinders of rank n of the GLS-expansion. From the construction it follows that |Δi1i2...in|=qi1·qi2·…·qin≤(qmax)n→0(n→∞), where qmax:=maxiqi .

So, for any sequence of indices {ik} ( ik∈N∪{0}) , there exists the sequence of embedded closed intervals Δi1⊂Δi1i2⊂Δi1i2i2⊂⋯⊂Δi1i2…ik⊂⋯ with |Δi1i2…ik|→0,k→∞ . Therefore, there exists a unique point x∈[0,1] that belongs to all these cylinders.

Conversely, if x∈[0,1] belongs to some cylinder of rank k for any k∈N and x is not an end-point for any cylinder, then there exists a unique sequence of the cylinders Δi1(x)⊃Δi1(x)i2(x)⊃Δi1(x)i2(x)i3(x)⊃⋯⊃Δi1(x)i2(x)…ik(x)⊃⋯ containing x, and x= ⋂k=1∞Δi1(x)i2(x)…ik(x)=Δi1(x)i2(x)…ik(x)…. The latter expression is called the GLS-expansion of x (see, e.g., [1, 3, 4, 6, 7] for details).

Let us remark that the Lüroth expansion and Q∞ -expansion [8, 9] are particular cases of the GLS-expansion. For the case where the ratio of lengths of two embedded cylinders of successive ranks depends on the last index and it is a power of φ=1+52 , we get the G∞2 -expansion of x [11].

Let Q∞=(q0,q1,…,qn,…) be a stochastic vector with positive coordinates, and let x=Δi1(x)i2(x)…ik(x)…GLS be the GLS-expansion of x∈[0,1] .

Let {ξk} be a sequence of independent identically distributed random variables: P(ξk=i):=pi≥0, where ∑i=0∞pi=1.

Using the sequence {ξk} and a given GLS-expansion, let us consider the random variable ξ=Δξ1ξ2...ξk...GLS, which is said to be the random variable with independent identically distributed GLS-symbols. Let μξ be the corresponding probability measure.

To investigate metric, topological, and fractal properties of the spectrum of the random variable with independent identically distributed GLS-symbols, let us study properties of the following family of sets. Let V be a subset of N0:={0,1,2,…} , and let C[GLS,V]={x:x=Δα1(x)…αk(x)…GLS,αk∈V}.

If the set V is finite, then C[GLS,V] is a self-similar set satisfying the open set condition (see, e.g., [5]). So, its Hausdorff–Besicovitch dimension coincides with the root of the equation ∑i∈Vqix=1.

If the set V is countable, then the situation is essentially more complicated. In particular, there exist stochastic vectors Q∞ and subsets V such that equation (1) has no roots on the unit interval.

For example, if qi=A(i+2)ln2(i+2) and V=N , then the equation ∑i∈Vqix=1 has no roots on [0;1] .

If a stochastic vector Q∞ and a set V⊂N0 are such that the equation ∑i∈Vqix=1 has a root α0 on [0,1] , then dimH(C[GLS,V])=α0.

First, let us show that for any k∈N , the set C[GLS,V] can be covered by cylinders of rank k and that the α0 -volume of this covering is equal to 1.

For k=1 , the set C[GLS,V] can be covered by cylinders of rank 1. It easy to see that the α0 -volume is equal to 1: ∑i1∈V|Δi1|α0=∑i1∈Vqi1α0=1.

Suppose that for k=n−1 , the α0 -volume of the covering of C[GLS,V] by cylinders of rank n−1 is equal to 1. Let us show that for k=n , the α0 -volume of the covering of C[GLS,V] by cylinders of rank n will not change. We have ∑ij∈V|Δi1i2...in−1in|α0=∑ij∈V(qi1qi2…qn−1qn)α0=∑i1∈Vqi1α0·∑ij∈V(qi1qi2…qn−1)α0=1.

So, for any ε>0 0$]]>, we get Hεα0(C[GLS,V])≤1. Hence, Hα0(C[GLS,V])≤1.

By the definition of the Hausdorff–Besicovitch dimension we get dimH(C[GLS,V])≤α0.

Let us show that dimH(C[GLS,V])≥α0 . To this end, let us consider sets V={i1,…,ik,…} , Vk={i1,…,ik},k≥2,k∈N , and the sequence C[GLS,Vk] of subsets of C[GLS,V] . For all k≥2,k∈N , we have C[GLS,Vk]⊂C[GLS,Vk+1], and, therefore, dimH(C[GLS,Vk])≤dimH(C[GLS,Vk+1]).

Let dimH(C[GLS,Vk])=αk . The sets C[GLS,Vk] are self-similar and satisfy the open set condition (OSC). Hence, the Hausdorff–Besicovitch dimension αk of C[GLS,Vk] coincides with the solution of the equation ∑i∈Vkqix=1. It is clear that α2<α3<⋯<αk<⋯ and αk<α0 . So, the sequence {αk} is increasing and bounded. Therefore, there exists a limit limk→∞αk=α∗ .

It is clear that α∗≤α0 because αk<α0(∀k∈N) . Let us prove that α∗=α0 .

Assume the opposite: let α∗<α0 . Then there exists α′ such that α∗<α′<α0 . Then ∑i∈Vkqiα′<1 for all k∈N . Since ∑i∈Vkqiαk=1 , we get ∑i∈Vkqiα′<1 for all k∈N . Let us consider the series ∑k=1∞qikα′ . It is clear that ∑k=1nqikα′<1 for all n∈N . So limn→∞∑k=1nqikα′≤1 . Therefore, ∑i∈Vqiα′≤1 .

Since qiα′>qiα0 {q_{i}^{\alpha _{0}}}$]]> for all i∈V and ∑i∈Vqiα0=1 , we get ∑i∈Vqiα′>∑i∈Vqiα0=1 \sum _{i\in V}{q_{i}^{\alpha _{0}}}=1$]]>, which contradicts the already proven inequality ∑i∈Vqiα′≤1 . This proves that α∗=α0 .

Since for any k≥2,k∈N , αk=dimH(C[GLS,Vk])≤dimH(C[GLS,V]), we get α0≤dimH(C[GLS,V]). Thus, α0=dimH([GLS,V]).  □

If the matrix Q∞ and the set V={i1,i2,…,ik,ik+1,…} are such that equation ∑i∈Vqix=1 has no roots on [0,1] , then dimH(C[GLS,V])=limk→∞dimH([GLS,Vk]), where Vk={i1,i2,…,ik},k∈N,k≥2 .

The sets C[GLS,Vk] are self-similar and satisfy the OSC. Thus the dimension αk can be obtained as a solution of the equation ∑i∈Vkqix=1 . It is easy to see that α2<α3<⋯<αk−1<αk<1.

Therefore, there exists the limit limk→∞αk=α∗. It is clear that α∗≤dimH(C[GLS,V]) because C[GLS,Vk]⊂C[GLS,V] for all k≥2 . Suppose that α∗<dimH(C[GLS,V]) . Then there exists a number α′ such that α∗<α′<dimH(C[GLS,V]) . It is clear that ∑i∈V2qiα′<∑i∈V3qiα′<⋯<∑i∈Vkqiα′<1.

So, ∑i∈Vqiα′≤1 .

On the other hand, Hα′(C[GLS,V])=∞ by the definition of the Hausdorff–Besicovitch dimension (because α′<dimH(C[GLS,V]) ). Then the α′ -dimensional Hausdorff measure of the set C[GLS,V] with respect to the family Φ of coverings that are generated by the GLS-expansion of the unit segment is equal to Hα′(C[GLS,Φ])=∞ (where the family Φ is a locally fine system of the coverings of the unit segment, i.e., for any ε>0 0$]]>, there exists such a covering of [0,1] by the subsets Ej∈Φ such that |Ej|<ε and [0,1]=⋃jEj ). Since the set C[GLS,V] can be covered by cylindrical segments of the GLS-expansion with indices from V, we deduce that for any M>0 0$]]>, there exists k(M) such that for all k>k(M) k(M)$]]>, we have the inequality ∑iq∈V,q∈{1,…,k}|Δi1i2…ik|α′>M,∑iq∈V,q∈{1,…,k}|Δi1i2…ik|α′=∑iq∈V,q∈{1,…,k−1}|Δi1i2…ik−1|α′·∑ik∈VΔikα′<∑iq∈V,q∈{1,…,k−1}|Δi1i2…ik−1|α′=∑iq∈V,q∈{1,…,k−2}|Δi1i2…ik−2|α′·∑ik−1∈Vqik−1α′<∑iq∈V,q∈{1,…,k−2}|Δi1i2…ik−2|α′<⋯<∑i1∈V|Δi1|α′=∑i1∈Vqi1α′<1. M,\\{} \displaystyle \sum \limits_{i_{q}\in V,q\in \{1,\dots ,k\}}|\varDelta _{i_{1}i_{2}\dots i_{k}}{|}^{{\alpha ^{\prime }}}& \displaystyle =\sum \limits_{i_{q}\in V,q\in \{1,\dots ,k-1\}}|\varDelta _{i_{1}i_{2}\dots i_{k-1}}{|}^{{\alpha ^{\prime }}}\cdot \sum \limits_{i_{k}\in V}{\varDelta _{i_{k}}^{{\alpha ^{\prime }}}}\\{} & \displaystyle <\sum \limits_{i_{q}\in V,q\in \{1,\dots ,k-1\}}|\varDelta _{i_{1}i_{2}\dots i_{k-1}}{|}^{{\alpha ^{\prime }}}\\{} & \displaystyle =\sum \limits_{i_{q}\in V,q\in \{1,\dots ,k-2\}}|\varDelta _{i_{1}i_{2}\dots i_{k-2}}{|}^{{\alpha ^{\prime }}}\cdot \sum \limits_{i_{k-1}\in V}{q_{i_{k-1}}^{{\alpha ^{\prime }}}}\\{} & \displaystyle <\sum \limits_{i_{q}\in V,q\in \{1,\dots ,k-2\}}|\varDelta _{i_{1}i_{2}\dots i_{k-2}}{|}^{{\alpha ^{\prime }}}<\cdots \\{} & \displaystyle <\sum \limits_{i_{1}\in V}|\varDelta _{i_{1}}{|}^{{\alpha ^{\prime }}}=\sum \limits_{i_{1}\in V}{q_{i_{1}}^{{\alpha ^{\prime }}}}<1.\end{array}\]]]>

From the obtained contradiction it follows that limk→∞αk=dimH(C[GLS,V]), where αk=dimH(C[GLS,Vk]) .  □

Theorems 1 and 2 can be considered as natural generalizations of results from [8].

The Hausdorff–Besicovitch dimension can be calculated as follows: dimH(C[GLS,V])=sup{x:∑i∈Vqix≥1} for any Q∞ and V={i1,i2,…,ik,ik+1,…} .

Let α0=dimH(C[GLS,V]) . Show that sup{x:∑i∈Vqix≥1}≥α0 . Let us consider the function φ(x)=∑i∈Vqix and denote the set A+={x:φ(x)≥1}. Let αk and αk+1 be the solutions of the equations ∑i∈Vkqix=1 and ∑i∈Vk+1qix=1, respectively.

Let us show that αk<αk+1<α0 . If α0 is the solution of ∑i∈Vqix=1 , then it is easy to see that αk<αk+1<α0 . If ∑i∈Vqix=1 has no roots on [0,1] , then α0=limk→∞αk and αk<αk+1 , so that αk<αk+1<α0 .

Express the function φ(x) as follows: φ(x)=qi1x+⋯+qikx+qik+1x︸≥1+ ∑j=k+1∞qijx. It is easy to see that for all x∈[αk,αk+1] (x<α0,k∈N) , φ(x)≥1 . Then A+⊃(−∞;α0) and supA+≥α0 .

Let us show that supA+≤α0 . Suppose the opposite. If supA+>α0 \alpha _{0}$]]>, then there exists x1 such that x1∈(α0;supA+] , x1∈A+ , and φ(x1)≥1 . So ∑i∈Vqix1≥1.

Since αk is a solution of ∑i∈Vkqix=1 and αk<α0 , we get ∑i∈Vkqiαk=1 and ∑i∈Vkqiα0≤1. It is clear that ∑i∈Vqiα0<1 and ∑i∈Vqix1≤1.

So, from the obtained contradiction it follows that supA+=α0 .  □

Let Δ∞GLS be the set of those x∈[0;1] that do not belong to any cylinder of the first rank of the GLS-expansion. The set Δ∞GLS can be empty, countable, or of continuum cardinality.

Let us recall that the nonempty and bounded set E is called N-self-similar if it can be represented as a union of a countably many sets Ej(dimH(Ei∩Ej)<dimHE,i≠j) such that the set E is similar to the sets Ej with coefficient  kj .

Since the spectrum Sξ of the distribution of a random variable ξ with independent identically distributed GLS-symbols is a self-similar or N-self-similar set and Sξ=(C[GLS,V])cl , we can apply the above results to calculate the Hausdorff–Besicovitch dimension of the spectrum Sξ for the case where Δ∞GLS is an at most countable set.

So, we get the following theorem, which can be considered as a corollary of Theorems 1 and 2.

Let V:={i:pi>0} 0\}$]]>. If Δ∞GLS is at most countable, then the Hausdorff–Besicovitch dimension of the spectrum of the distribution of a random variable ξ with independent identically distributed GLS -symbols can be calculated in the following way.

1) If the equation ∑i∈Vqix=1 has one root α0 on [0,1] , then dimHSξ=α0.

2) If the equation ∑i∈Vqix=1 has no roots on [0,1] , then dimHSξ=limk→∞αk, where αk are the roots of the equations ∑i∈Vkqix=1,Vk={i1,i2,…,ik}⊂V .

 Acknowledgments

The author expresses her sincere gratitude to the referee for useful comments and remarks.

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