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A random von Neumann theorem for uniformly distributed sequences of partitions

A random von Neumann theorem for uniformly distributed sequences of partitions 1IntroductionThe general study of uniformly distributed sequences of partitions was initiated in [1], inspired by a beautiful construction and result from the study by Kakutani [2]. The subject is closely related to the theory of uniformly distributed sequences, initiated in [3]. There are two classical references for the subject: [4] and [5].Kakutani took the interval I=[0,1]I=\left[0,1], a number α∈]0,1[\alpha \in ]0,1{[}and divided the interval in proportion α:1−α\alpha :1-\alpha . Then, he divided the longest interval of this partition in the same proportion and iterated the procedure dividing always the longest interval of the nth partition, so as to obtain a sequence of partitions of ]0,1[]0,1{[}. If at a certain step there were two or more intervals of maximal length, they were divided simultaneously.Kakutani proved that this sequence of partitions (denote it by {αnI}\left\{{\alpha }^{n}I\right\}) is uniformly distributed, which means that if αkI={0<t1k<t2k<⋯<tNkn<1}{\alpha }^{k}I=\left\{0\lt {t}_{1}^{k}\lt {t}_{2}^{k}\hspace{0.33em}\lt \cdots \lt {t}_{{N}_{k}}^{n}\lt 1\right\}is the kth partition, then limk→∞1Nk∑i=1Nkf(tik)=∫01f(t)dt,\mathop{\mathrm{lim}}\limits_{k\to \infty }\frac{1}{{N}_{k}}\mathop{\sum }\limits_{i=1}^{{N}_{k}}f\left({t}_{i}^{k})=\underset{0}{\overset{1}{\int }}f\left(t){\rm{d}}t,for every continuous function ff.In other words, the discrete measure concentrated in the points tik{t}_{i}^{k}converges weakly to the Lebesgue measure on [0,1]\left[0,1].The construction has been generalized in [1]. Let ρ\rho be any non trivial finite partition of II.In the first step, the longest interval(s) of ρ\rho is subdivided positively homothetically to ρ\rho . The partition obtained in this manner is denoted by ρ2I{\rho }^{2}I. In the second step, the same procedure is repeated on the longest interval(s), operating with ρ\rho on ρ2I{\rho }^{2}I. Iteration of this procedure leads to a sequence of partitions {ρnI}\left\{{\rho }^{n}I\right\}.If ρ={[0,α[,[α,1[}\rho =\left\{{[}0,\alpha {[},{[}\alpha ,1{[}\right\}, one gets Kakutani’s sequence.The following theorem includes the results of the study by Kakutani ([1], Theorem 2.7]).Theorem 1The sequence {ρnI}\left\{{\rho }^{n}I\right\}is uniformly distributed.There are interesting connections between the theory of u.d. sequences of partitions and u.d. sequences of points. This connection is far reaching in the construction of a significant subclass of ρ\rho -refinements, the so-called LSLS-sequences. The subject was initiated by the present author in [6], and it is connected with the van der Corput sequences of points.LSLS-sequences are constructed starting from the partition ρLS{\rho }_{LS}made of L+SL+Sintervals (LLand SSare positive integers) of length β\beta and β2{\beta }^{2}, respectively, where β\beta is the positive solution of the equation Lβ+Sβ2=1L\beta +S{\beta }^{2}=1.The present paper is concerned with a result in the domain of uniformly distributed sequences of partitions, related to a proposition by von Neumann for uniformly sequences of points [7].Theorem 2If {xn}\left\{{x}_{n}\right\}is a dense sequence of points in [0,1]\left[0,1], then there exists a rearrangement of these points, {xnk}\left\{{x}_{{n}_{k}}\right\}, which is uniformly distributed.One of the consequences of von Neumann’s result is that there are many u.d. sequences of points.We will now introduce the definitions we need.DefinitionsGiven a partition π={[ti−1,ti],1≤i≤N}\pi =\left\{\left[{t}_{i-1},{t}_{i}],1\le i\le N\right\}, we denote by li=ti−ti−1{l}_{i}={t}_{i}-{t}_{i-1}the length of its ith interval.The diameter of π\pi , denoted by LL, is equal to max1≤i≤Nli{\max }_{1\le i\le N}{l}_{i}.Given a sequence of partitions {πk}\left\{{\pi }_{k}\right\}, we say that it is dense if, denoted by Lk{L}_{k}the diameter of πk{\pi }_{k}, limk→∞Lk=0{\mathrm{lim}}_{k\to \infty }{L}_{k}=0.If π={[ti−1,ti],1≤i≤N}\pi =\left\{\left[{t}_{i-1},{t}_{i}],1\le i\le N\right\}is a partition, its random permutation is a partition π′={[sh−1,sh],1≤h≤N}\pi ^{\prime} =\left\{\left[{s}_{h-1},{s}_{h}],1\le h\le N\right\}defined by the points sh=∑j=0hlij,{s}_{h}={\sum }_{j=0}^{h}{l}_{{i}_{j}},for 0≤h≤N0\le h\le N, where s0=0{s}_{0}=0and the indices {ij}\left\{{i}_{j}\right\}, are successively taken at random, with probability 1N\frac{1}{N}, from the set {is:1≤s≤N}\left\{{i}_{s}:1\le s\le N\right\}.We will denote by π!\pi \hspace{0.1em}\text{&#x0021;}\hspace{0.1em}the set of all the N!N\&#x0021;permutations of π\pi .2Main resultsIn a previous paper [8], we proved the following result.Proposition 3If {πn}\left\{{\pi }_{n}\right\}is a dense sequence of partitions, then there exists a sequence of partitions {σn}\left\{{\sigma }_{n}\right\}, with σn∈πn!{\sigma }_{n}\in {\pi }_{n}\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}, which is uniformly distributed.In the same paper, we made the following conjecture.ConjectureIf {πn}\left\{{\pi }_{n}\right\}is a dense sequence of partitions and we select at random a partition σk∈πn!{\sigma }_{k}\in {\pi }_{n}\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}, then {σk}\left\{{\sigma }_{k}\right\}is uniformly distributed with probability 1.We need some preliminary calculations.Let q∈]0,1[q\in ]0,1{[}and denote by Nk(q){N}_{k}\left(q)the integer such that Nk(q)Nk≤q<Nk(q)+1Nk.\frac{{N}_{k}\left(q)}{{N}_{k}}\le q\lt \frac{{N}_{k}\left(q)+1}{{N}_{k}}.Consider a sequence {qm}\left\{{q}_{m}\right\}of points, which is dense in [0,1]\left[0,1]. For later convenience, we will denote by Nk(m){N}_{k}\left(m)the integer Nk(qm){N}_{k}\left({q}_{m}).Select at random from the Nk{N}_{k}intervals of πk{\pi }_{k}, with probability 1Nk\frac{1}{{N}_{k}}, Nk(m){N}_{k}\left(m)intervals. Denote by ξik{\xi }_{i}^{k}the length of the interval selected in the ith draw (1≤i≤Nk(m)1\le i\le {N}_{k}\left(m)) and consider the random variable ηkm=∑i=1Nk(m)ξik.{\eta }_{k}^{m}=\mathop{\sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}.Obviously, E(ηkm)=∑i=1Nk(m)E(ξik)=∑i=1Nk(m)1Nk=Nk(m)Nk,E\left({\eta }_{k}^{m})=\mathop{\sum }\limits_{i=1}^{{N}_{k}\left(m)}E\left({\xi }_{i}^{k})=\mathop{\sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}}=\frac{{N}_{k}\left(m)}{{N}_{k}},hence, ∣E(ηkm)−qm∣≤1Nk.| E\left({\eta }_{k}^{m})-{q}_{m}| \le \frac{1}{{N}_{k}}.It is easy to see that the second moment of ηkm{\eta }_{k}^{m}is uniformly bounded for any sequence of partitions.This, together with the independence of the ηkm{\eta }_{k}^{m}’s (for k∈Nk\in {\mathbb{N}}), would allow us to apply the strong law of large numbers and to conclude that, when kktends to infinity, the sequence ηkm{\eta }_{k}^{m}tends to qm{q}_{m}in the Cesàro mean (and nothing more, at least following this line of thought).But this is not what we were looking for.If we want to identify a class for which the conjecture is true, we have to make some assumptions on the sequence {πk}\left\{{\pi }_{k}\right\}.A simple sufficient assumption is expressed as follows: ∑i=1∞Lk2<∞.\mathop{\sum }\limits_{i=1}^{\infty }{L}_{k}^{2}\lt \infty .Theorem 4If the series of squares of diameters of {πk}\left\{{\pi }_{k}\right\}is convergent, then its random permutations σk{\sigma }_{k}are uniformly distributed with probability 1.ProofWe have Var(ηkm)=E∑i=1Nk(m)ξik−∑i=1Nk(m)1Nk2=E∑i=1Nk(m)ξik−1Nk2=E∑i=1Nk(m)(ξik)2−2∑i=1Nk(m)ξik1Nk+∑i=1Nk(m)1Nk2=E∑i=1Nk(m)(ξik)2−21NkE∑i=1Nk(m)ξik+E∑i=1Nk(m)1Nk2≤E∑i=1Nk(ξik)2−2NkE∑i=1Nkξik+∑i=1Nk1Nk=E∑i=1Nk(ξik)2−∑i=1Nk1Nk≤∑i=1NkLk2−1<∑i=1NkLk2.\begin{array}{rcl}\hspace{0.1em}\text{Var}\hspace{0.1em}\left({\eta }_{k}^{m})& =& E\left({\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}-\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}}\right)}^{2}\right)\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\left({\xi }_{i}^{k}-\frac{1}{{N}_{k}}\right)}^{2}\right)\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\left({\xi }_{i}^{k})}^{2}-2\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}\frac{1}{{N}_{k}}+\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}^{2}}\right)\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\left({\xi }_{i}^{k})}^{2}\right)-2\frac{1}{{N}_{k}}E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}\right)+E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}^{2}}\right)\\ & \le & E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{\left({\xi }_{i}^{k})}^{2}\right)-\frac{2}{{N}_{k}}E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{\xi }_{i}^{k}\right)+\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}\frac{1}{{N}_{k}}\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{\left({\xi }_{i}^{k})}^{2}\right)-\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}\frac{1}{{N}_{k}}\le \mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{L}_{k}^{2}-1\lt \mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{L}_{k}^{2}.\end{array}Apply now the Čebišëv inequality. By our assumption, we have, for every ε>0\varepsilon \gt 0(and every mm), ∑k=1∞P(∣ηkm−E(ηkm)∣>ε)≤∑n=1∞Var(ηkm)ε2<∞.\mathop{\sum }\limits_{k=1}^{\infty }P\left(| {\eta }_{k}^{m}-E\left({\eta }_{k}^{m})| \gt \varepsilon )\le \mathop{\sum }\limits_{n=1}^{\infty }\frac{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({\eta }_{k}^{m})}{{\varepsilon }^{2}}\lt \infty .Recalling that E(ηkm)E\left({\eta }_{k}^{m})tends to qm{q}_{m}and applying the Borel-Cantelli lemma [9, Theorem 4.2.1], we obtain that limk→∞ηkm=qm\mathop{\mathrm{lim}}\limits_{k\to \infty }{\eta }_{k}^{m}={q}_{m}almost surely for every m∈Nm\in {\mathbb{N}}.The set {qm}\left\{{q}_{m}\right\}is countable; therefore, the aforementioned limit holds almost surely for all the values of mmsimultaneously.Observe now that limk→∞Nk(q)Nk{\mathrm{lim}}_{k\to \infty }\frac{{N}_{k}\left(q)}{{N}_{k}}is an increasing function of qq. Therefore, it follows that, almost surely, limk→∞Nk(q)Nk=q\mathop{\mathrm{lim}}\limits_{k\to \infty }\frac{{N}_{k}\left(q)}{{N}_{k}}=qfor every q∈[0,1]q\in \left[0,1].In other words, the empirical distribution function Fk{F}_{k}of σk{\sigma }_{k}tends almost surely to the distribution function of the random variable UUuniformly distributed on [0,1]\left[0,1].On the other hand, convergence in distribution is known to be equivalent to weak convergence, so the desired conclusion follows.□ http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Open Mathematics de Gruyter

A random von Neumann theorem for uniformly distributed sequences of partitions

Open Mathematics , Volume 20 (1): 4 – Jan 1, 2022

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© 2022 Ingrid Carbone, published by De Gruyter
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Abstract

1IntroductionThe general study of uniformly distributed sequences of partitions was initiated in [1], inspired by a beautiful construction and result from the study by Kakutani [2]. The subject is closely related to the theory of uniformly distributed sequences, initiated in [3]. There are two classical references for the subject: [4] and [5].Kakutani took the interval I=[0,1]I=\left[0,1], a number α∈]0,1[\alpha \in ]0,1{[}and divided the interval in proportion α:1−α\alpha :1-\alpha . Then, he divided the longest interval of this partition in the same proportion and iterated the procedure dividing always the longest interval of the nth partition, so as to obtain a sequence of partitions of ]0,1[]0,1{[}. If at a certain step there were two or more intervals of maximal length, they were divided simultaneously.Kakutani proved that this sequence of partitions (denote it by {αnI}\left\{{\alpha }^{n}I\right\}) is uniformly distributed, which means that if αkI={0<t1k<t2k<⋯<tNkn<1}{\alpha }^{k}I=\left\{0\lt {t}_{1}^{k}\lt {t}_{2}^{k}\hspace{0.33em}\lt \cdots \lt {t}_{{N}_{k}}^{n}\lt 1\right\}is the kth partition, then limk→∞1Nk∑i=1Nkf(tik)=∫01f(t)dt,\mathop{\mathrm{lim}}\limits_{k\to \infty }\frac{1}{{N}_{k}}\mathop{\sum }\limits_{i=1}^{{N}_{k}}f\left({t}_{i}^{k})=\underset{0}{\overset{1}{\int }}f\left(t){\rm{d}}t,for every continuous function ff.In other words, the discrete measure concentrated in the points tik{t}_{i}^{k}converges weakly to the Lebesgue measure on [0,1]\left[0,1].The construction has been generalized in [1]. Let ρ\rho be any non trivial finite partition of II.In the first step, the longest interval(s) of ρ\rho is subdivided positively homothetically to ρ\rho . The partition obtained in this manner is denoted by ρ2I{\rho }^{2}I. In the second step, the same procedure is repeated on the longest interval(s), operating with ρ\rho on ρ2I{\rho }^{2}I. Iteration of this procedure leads to a sequence of partitions {ρnI}\left\{{\rho }^{n}I\right\}.If ρ={[0,α[,[α,1[}\rho =\left\{{[}0,\alpha {[},{[}\alpha ,1{[}\right\}, one gets Kakutani’s sequence.The following theorem includes the results of the study by Kakutani ([1], Theorem 2.7]).Theorem 1The sequence {ρnI}\left\{{\rho }^{n}I\right\}is uniformly distributed.There are interesting connections between the theory of u.d. sequences of partitions and u.d. sequences of points. This connection is far reaching in the construction of a significant subclass of ρ\rho -refinements, the so-called LSLS-sequences. The subject was initiated by the present author in [6], and it is connected with the van der Corput sequences of points.LSLS-sequences are constructed starting from the partition ρLS{\rho }_{LS}made of L+SL+Sintervals (LLand SSare positive integers) of length β\beta and β2{\beta }^{2}, respectively, where β\beta is the positive solution of the equation Lβ+Sβ2=1L\beta +S{\beta }^{2}=1.The present paper is concerned with a result in the domain of uniformly distributed sequences of partitions, related to a proposition by von Neumann for uniformly sequences of points [7].Theorem 2If {xn}\left\{{x}_{n}\right\}is a dense sequence of points in [0,1]\left[0,1], then there exists a rearrangement of these points, {xnk}\left\{{x}_{{n}_{k}}\right\}, which is uniformly distributed.One of the consequences of von Neumann’s result is that there are many u.d. sequences of points.We will now introduce the definitions we need.DefinitionsGiven a partition π={[ti−1,ti],1≤i≤N}\pi =\left\{\left[{t}_{i-1},{t}_{i}],1\le i\le N\right\}, we denote by li=ti−ti−1{l}_{i}={t}_{i}-{t}_{i-1}the length of its ith interval.The diameter of π\pi , denoted by LL, is equal to max1≤i≤Nli{\max }_{1\le i\le N}{l}_{i}.Given a sequence of partitions {πk}\left\{{\pi }_{k}\right\}, we say that it is dense if, denoted by Lk{L}_{k}the diameter of πk{\pi }_{k}, limk→∞Lk=0{\mathrm{lim}}_{k\to \infty }{L}_{k}=0.If π={[ti−1,ti],1≤i≤N}\pi =\left\{\left[{t}_{i-1},{t}_{i}],1\le i\le N\right\}is a partition, its random permutation is a partition π′={[sh−1,sh],1≤h≤N}\pi ^{\prime} =\left\{\left[{s}_{h-1},{s}_{h}],1\le h\le N\right\}defined by the points sh=∑j=0hlij,{s}_{h}={\sum }_{j=0}^{h}{l}_{{i}_{j}},for 0≤h≤N0\le h\le N, where s0=0{s}_{0}=0and the indices {ij}\left\{{i}_{j}\right\}, are successively taken at random, with probability 1N\frac{1}{N}, from the set {is:1≤s≤N}\left\{{i}_{s}:1\le s\le N\right\}.We will denote by π!\pi \hspace{0.1em}\text{&#x0021;}\hspace{0.1em}the set of all the N!N\&#x0021;permutations of π\pi .2Main resultsIn a previous paper [8], we proved the following result.Proposition 3If {πn}\left\{{\pi }_{n}\right\}is a dense sequence of partitions, then there exists a sequence of partitions {σn}\left\{{\sigma }_{n}\right\}, with σn∈πn!{\sigma }_{n}\in {\pi }_{n}\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}, which is uniformly distributed.In the same paper, we made the following conjecture.ConjectureIf {πn}\left\{{\pi }_{n}\right\}is a dense sequence of partitions and we select at random a partition σk∈πn!{\sigma }_{k}\in {\pi }_{n}\hspace{0.1em}\text{&#x0021;}\hspace{0.1em}, then {σk}\left\{{\sigma }_{k}\right\}is uniformly distributed with probability 1.We need some preliminary calculations.Let q∈]0,1[q\in ]0,1{[}and denote by Nk(q){N}_{k}\left(q)the integer such that Nk(q)Nk≤q<Nk(q)+1Nk.\frac{{N}_{k}\left(q)}{{N}_{k}}\le q\lt \frac{{N}_{k}\left(q)+1}{{N}_{k}}.Consider a sequence {qm}\left\{{q}_{m}\right\}of points, which is dense in [0,1]\left[0,1]. For later convenience, we will denote by Nk(m){N}_{k}\left(m)the integer Nk(qm){N}_{k}\left({q}_{m}).Select at random from the Nk{N}_{k}intervals of πk{\pi }_{k}, with probability 1Nk\frac{1}{{N}_{k}}, Nk(m){N}_{k}\left(m)intervals. Denote by ξik{\xi }_{i}^{k}the length of the interval selected in the ith draw (1≤i≤Nk(m)1\le i\le {N}_{k}\left(m)) and consider the random variable ηkm=∑i=1Nk(m)ξik.{\eta }_{k}^{m}=\mathop{\sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}.Obviously, E(ηkm)=∑i=1Nk(m)E(ξik)=∑i=1Nk(m)1Nk=Nk(m)Nk,E\left({\eta }_{k}^{m})=\mathop{\sum }\limits_{i=1}^{{N}_{k}\left(m)}E\left({\xi }_{i}^{k})=\mathop{\sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}}=\frac{{N}_{k}\left(m)}{{N}_{k}},hence, ∣E(ηkm)−qm∣≤1Nk.| E\left({\eta }_{k}^{m})-{q}_{m}| \le \frac{1}{{N}_{k}}.It is easy to see that the second moment of ηkm{\eta }_{k}^{m}is uniformly bounded for any sequence of partitions.This, together with the independence of the ηkm{\eta }_{k}^{m}’s (for k∈Nk\in {\mathbb{N}}), would allow us to apply the strong law of large numbers and to conclude that, when kktends to infinity, the sequence ηkm{\eta }_{k}^{m}tends to qm{q}_{m}in the Cesàro mean (and nothing more, at least following this line of thought).But this is not what we were looking for.If we want to identify a class for which the conjecture is true, we have to make some assumptions on the sequence {πk}\left\{{\pi }_{k}\right\}.A simple sufficient assumption is expressed as follows: ∑i=1∞Lk2<∞.\mathop{\sum }\limits_{i=1}^{\infty }{L}_{k}^{2}\lt \infty .Theorem 4If the series of squares of diameters of {πk}\left\{{\pi }_{k}\right\}is convergent, then its random permutations σk{\sigma }_{k}are uniformly distributed with probability 1.ProofWe have Var(ηkm)=E∑i=1Nk(m)ξik−∑i=1Nk(m)1Nk2=E∑i=1Nk(m)ξik−1Nk2=E∑i=1Nk(m)(ξik)2−2∑i=1Nk(m)ξik1Nk+∑i=1Nk(m)1Nk2=E∑i=1Nk(m)(ξik)2−21NkE∑i=1Nk(m)ξik+E∑i=1Nk(m)1Nk2≤E∑i=1Nk(ξik)2−2NkE∑i=1Nkξik+∑i=1Nk1Nk=E∑i=1Nk(ξik)2−∑i=1Nk1Nk≤∑i=1NkLk2−1<∑i=1NkLk2.\begin{array}{rcl}\hspace{0.1em}\text{Var}\hspace{0.1em}\left({\eta }_{k}^{m})& =& E\left({\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}-\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}}\right)}^{2}\right)\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\left({\xi }_{i}^{k}-\frac{1}{{N}_{k}}\right)}^{2}\right)\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\left({\xi }_{i}^{k})}^{2}-2\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}\frac{1}{{N}_{k}}+\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}^{2}}\right)\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\left({\xi }_{i}^{k})}^{2}\right)-2\frac{1}{{N}_{k}}E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}{\xi }_{i}^{k}\right)+E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}\left(m)}\frac{1}{{N}_{k}^{2}}\right)\\ & \le & E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{\left({\xi }_{i}^{k})}^{2}\right)-\frac{2}{{N}_{k}}E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{\xi }_{i}^{k}\right)+\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}\frac{1}{{N}_{k}}\\ & =& E\left(\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{\left({\xi }_{i}^{k})}^{2}\right)-\mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}\frac{1}{{N}_{k}}\le \mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{L}_{k}^{2}-1\lt \mathop{\displaystyle \sum }\limits_{i=1}^{{N}_{k}}{L}_{k}^{2}.\end{array}Apply now the Čebišëv inequality. By our assumption, we have, for every ε>0\varepsilon \gt 0(and every mm), ∑k=1∞P(∣ηkm−E(ηkm)∣>ε)≤∑n=1∞Var(ηkm)ε2<∞.\mathop{\sum }\limits_{k=1}^{\infty }P\left(| {\eta }_{k}^{m}-E\left({\eta }_{k}^{m})| \gt \varepsilon )\le \mathop{\sum }\limits_{n=1}^{\infty }\frac{\hspace{0.1em}\text{Var}\hspace{0.1em}\left({\eta }_{k}^{m})}{{\varepsilon }^{2}}\lt \infty .Recalling that E(ηkm)E\left({\eta }_{k}^{m})tends to qm{q}_{m}and applying the Borel-Cantelli lemma [9, Theorem 4.2.1], we obtain that limk→∞ηkm=qm\mathop{\mathrm{lim}}\limits_{k\to \infty }{\eta }_{k}^{m}={q}_{m}almost surely for every m∈Nm\in {\mathbb{N}}.The set {qm}\left\{{q}_{m}\right\}is countable; therefore, the aforementioned limit holds almost surely for all the values of mmsimultaneously.Observe now that limk→∞Nk(q)Nk{\mathrm{lim}}_{k\to \infty }\frac{{N}_{k}\left(q)}{{N}_{k}}is an increasing function of qq. Therefore, it follows that, almost surely, limk→∞Nk(q)Nk=q\mathop{\mathrm{lim}}\limits_{k\to \infty }\frac{{N}_{k}\left(q)}{{N}_{k}}=qfor every q∈[0,1]q\in \left[0,1].In other words, the empirical distribution function Fk{F}_{k}of σk{\sigma }_{k}tends almost surely to the distribution function of the random variable UUuniformly distributed on [0,1]\left[0,1].On the other hand, convergence in distribution is known to be equivalent to weak convergence, so the desired conclusion follows.□

Journal

Open Mathematicsde Gruyter

Published: Jan 1, 2022

Keywords: number theory; discrepancy; uniformly distributed sequences of partitions; probability theory; 11-xx; 40-xx; 60-xx

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