On distribution of continuous sequences

Milan Pasteka

arXiv:1706.00167·math.NT·June 29, 2017

On distribution of continuous sequences

Milan Pasteka

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TL;DR

This paper investigates the properties of polyadicly continuous sequences within probability theory, aiming to understand their distributional characteristics and potential applications.

Contribution

It introduces a new perspective on polyadicly continuous sequences by analyzing their distributional properties through probabilistic methods.

Findings

01

Characterization of distributional behavior of polyadicly continuous sequences

02

Identification of conditions for convergence in distribution

03

Potential applications in probabilistic modeling

Abstract

We study certain polyadicly continuous sequences from point of view the probability theory.

Equations114

d (a, b) = n = 1 \sum \infty \frac{ψ _{n} ( a - b )}{2 ^{n}},

d (a, b) = n = 1 \sum \infty \frac{ψ _{n} ( a - b )}{2 ^{n}},

\forall a, b \in N; a \equiv b (mod m) \Rightarrow ∣ v (a) - v (b) ∣ < ε .

\forall a, b \in N; a \equiv b (mod m) \Rightarrow ∣ v (a) - v (b) ∣ < ε .

\tilde{v} (α) = j \to \infty lim v (n_{j})

\tilde{v} (α) = j \to \infty lim v (n_{j})

s + m Ω = {s + m α; α \in Ω}

s + m Ω = {s + m α; α \in Ω}

Ω = s = 0 ⋃ m - 1 s + m Ω.

Ω = s = 0 ⋃ m - 1 s + m Ω.

P (s + m Ω) = \frac{1}{m}

P (s + m Ω) = \frac{1}{m}

n \to \infty lim \frac{1}{n} c a r d (S \cap [1, n]) := d (S)

n \to \infty lim \frac{1}{n} c a r d (S \cap [1, n]) := d (S)

d (r + (m)) = \frac{1}{m} .

d (r + (m)) = \frac{1}{m} .

\mu^{\ast}(S)=\inf\Big{\{}\sum_{\ell=1}^{k}\frac{1}{m_{\ell}};S\subset\bigcup_{\ell=1}^{k}r_{\ell}+(m_{\ell})\Big{\}}

\mu^{\ast}(S)=\inf\Big{\{}\sum_{\ell=1}^{k}\frac{1}{m_{\ell}};S\subset\bigcup_{\ell=1}^{k}r_{\ell}+(m_{\ell})\Big{\}}

μ^{*} (S) = P (c l (S))

μ^{*} (S) = P (c l (S))

D_{μ} = {S \subset N; μ^{*} (S) + μ^{*} (N ∖ S)}

D_{μ} = {S \subset N; μ^{*} (S) + μ^{*} (N ∖ S)}

d ({n \in N; v (n) < x}) = x .

d ({n \in N; v (n) < x}) = x .

F (x) = μ ({n \in N; v (n) < x})

F (x) = μ ({n \in N; v (n) < x})

F (x) = 0, x < 0, F (x) = x, x \in [0, 1], F (x) = 1, x > 1.

F (x) = 0, x < 0, F (x) = x, x \in [0, 1], F (x) = 1, x > 1.

n = a_{0} + a_{1} Q_{1} + ... + a_{k} Q_{k}

n = a_{0} + a_{1} Q_{1} + ... + a_{k} Q_{k}

γ (n) = \frac{a _{0}}{Q _{1}} + \dots + \frac{a _{k}}{Q _{k + 1}} .

γ (n) = \frac{a _{0}}{Q _{1}} + \dots + \frac{a _{k}}{Q _{k + 1}} .

μ ({n \in N; v (n) < x}) = F (x) .

μ ({n \in N; v (n) < x}) = F (x) .

P (\tilde{v} < x) = F (x) = P (\tilde{v} \leq x)

P (\tilde{v} < x) = F (x) = P (\tilde{v} \leq x)

{n \in N; v (n) < x} \subset {α \in Ω; \tilde{v} (α) \leq x}

{n \in N; v (n) < x} \subset {α \in Ω; \tilde{v} (α) \leq x}

c l ({n \in N; v (n) < x}) \subset {α \in Ω; \tilde{v} (α) \leq x} .

c l ({n \in N; v (n) < x}) \subset {α \in Ω; \tilde{v} (α) \leq x} .

μ^{*} ({n \in N; v (n) < x}) \leq F (x)

μ^{*} ({n \in N; v (n) < x}) \leq F (x)

N ∖ {n \in N; v (n) < x} = {n \in N; v (n) \geq x},

N ∖ {n \in N; v (n) < x} = {n \in N; v (n) \geq x},

c l (N ∖ {n \in N; v (n) < x}) \subset {α \in Ω; \tilde{v} (α) \geq x} .

c l (N ∖ {n \in N; v (n) < x}) \subset {α \in Ω; \tilde{v} (α) \geq x} .

μ^{*} (N ∖ {n \in N; v (n) < x}) \leq 1 - F (x)

μ^{*} (N ∖ {n \in N; v (n) < x}) \leq 1 - F (x)

μ^{*} ({n \in N; v (n) < x}) = F (x)

μ^{*} ({n \in N; v (n) < x}) = F (x)

μ ({n \in N; v (n) < x}) = F (x) .

μ ({n \in N; v (n) < x}) = F (x) .

{n \in N; v (n) < x} \subset {α \in Ω; \tilde{v} (α) \leq x})

{n \in N; v (n) < x} \subset {α \in Ω; \tilde{v} (α) \leq x})

{α \in Ω; \tilde{v} (α) < x}) \subset c l ({n \in N; v (n) \leq x})

{α \in Ω; \tilde{v} (α) < x}) \subset c l ({n \in N; v (n) \leq x})

μ (j = 1 ⋂ r {n \in N; v_{j} (n) < x_{j}}) = j = 1 \prod r μ ({n \in N; v_{j} (n) < x_{j}}) .

μ (j = 1 ⋂ r {n \in N; v_{j} (n) < x_{j}}) = j = 1 \prod r μ ({n \in N; v_{j} (n) < x_{j}}) .

μ ({n \in N; v_{j} (n) < x} = F_{j} (x)

μ ({n \in N; v_{j} (n) < x} = F_{j} (x)

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TopicsMathematical Approximation and Integration · Approximation Theory and Sequence Spaces · advanced mathematical theories

Full text

ON DISTRIBUTION OF CONTINUOUS SEQUENCES

Milan Paštéka

Let ${\mathbb{N}}$ be the set of positive integers and $\Omega$ compact metric ring of polyadic integers.(see [N], [N1]) Let us remark that $\Omega$ is the completion of ${\mathbb{N}}$ with respect to polyadic metric

[TABLE]

where $\psi(x)=0$ if $n|x$ and $\psi(x)=1$ otherwise. We shall use two synonymous : sequence of real numbers and arithmetic function. A sequence of real numbers $\{v(n)\}$ is called polyadicly continuous if and only for each $\varepsilon>0$ such $m\in{\mathbb{N}}$ exists that

[TABLE]

It is well known that polyadicly continuous sequence of real numbers is uniformly continuous with respect polyadic metric $\mathfrak{d}$ and so each polyadicly continuous sequence of real numbers $\{v(n)\}$ can be extended by the natural way to a real valued continuous function $\tilde{v}$ defined on $\Omega$ such that

[TABLE]

where $\{n_{j}\}$ is such sequence of positive integers that $n_{j}\to\alpha$ for $j\to\infty$ with respect the polyadic metric. The compact ring $\Omega$ is equipped with Haar probability measure $P$ and so the function $\tilde{v}$ can be considered as random variable in the probability space $(\Omega,P)$ . As usually if $h$ ia a random variable on $\Omega$ the we denote $E(h)=\int hdP$ the mean value of $h$ .

Let $m\in{\mathbb{N}}$ and $s=0,1,\dots m-1$ . Put

[TABLE]

The ring $\Omega$ can be represented as disjoint decomposition

[TABLE]

(see [N], [N1]). And so for the Haar probability measure $P$ we have

[TABLE]

for $m\in{\mathbb{N}},s=0,\dots m-1$ .

We say that a set $S\subset{\mathbb{N}}$ has asymptotic density if and only if the limit

[TABLE]

exists. The value $d(S)$ is then called asymptotic density of $S$ . see [NAR], [G], [G1]). The system of all subsets of ${\mathbb{N}}$ having asymptotic density we shall denote $\mathcal{D}$ . The function $d$ is a finitely additive probability measure on $\mathcal{D}$ . If $r+(m)$ , $m\in{\mathbb{N}},r\in{\mathbb{Z}}$ is the set of all positive integers congruent to $r$ modulo $m$ , then it is easy to check that $r+(m)\in\text{$ \mathcal{D} $}$ and

[TABLE]

By the symbol $\mu^{\ast}(S)$ we shall denote the *Buck’s measure density * of the set $S\subset{\mathbb{N}}$ constructed in [BUC] as follows

[TABLE]

for $S\subset{\mathbb{N}}$ .

In [PAS4] is proved

[TABLE]

for each $S\subset{\mathbb{N}}$ , where $cl(S)$ denote the topological closure of $S$ in $\Omega$ . It is known that $\mu^{\ast}$ is a strong submeasure , the system of Buck’s measurables sets

[TABLE]

is an algebra of sets and the restriction $\mu=\mu^{\ast}|_{\text{$ \mathcal{D} $}_{\mu}}$ is a finitely additive probability measure on $\text{$ \mathcal{D} $}_{\mu}$ . Moreover $\text{$ \mathcal{D} $}_{\mu}\subset\text{$ \mathcal{D} $}$ and $\mu(S)=d(S)$ for each $S\in\text{$ \mathcal{D} $}_{\mu}$ .

The notion of uniform distribution of sequences was introduced and studied first time by H. Weyl in his work [Wey]. $\{n\in{\mathbb{N}};v(n)<x\}$ belongs to $\mathcal{D}$ and

[TABLE]

In the paper [PAS2], (see also [PAS3]) is this concept transfer for the case of Buck’s measure density and Buck’s measurability.

We say that a sequence $\{v(n)\}$ is Buck’s measurable if and only if for every $x$ - real number the set $\{n\in{\mathbb{N}};v(n)<x\}$ belong to $\text{$ \mathcal{D} $}_{\mu}$ .

A Buck’s measurable sequence $\{v(n)\}$ is called Buck’s uniformly distributed if and only if for the function

[TABLE]

we have

[TABLE]

Example 1.

Let $\{Q_{k}\}$ be an increasing sequence of integers such that $Q_{0}=1$ and $Q_{k}|Q_{k+1},k=1,2,3,\dots$ . Each positive integer $n$ can be uniquely represented in the form

[TABLE]

where $a_{j}<\frac{Q_{j+1}}{Q_{j}},j=1,\dots,k$ . To this $n$ can be associated an element of unit interval $\gamma(n)$ in the form

[TABLE]

The sequence $\{\gamma(n)\}$ is known as van der Corput sequence with base $\{Q_{k}\}$ and in [PAS2] is proved that it is Buck’s uniformly distributed and polyadicly continuous.

Theorem 1.

Let $\{v(n)\}$ a polyadicly continuous sequence of real numbers. Suppose that $F:(-\infty,\infty)\to[0,1]$ is a continuous function. If $F$ is the distribution function of random variable $\tilde{v}$ then for each $x$ - real number we have that $\{n\in{\mathbb{N}};v(n)<x\}\in\text{$ \mathcal{D} $}_{\mu}$ and**

[TABLE]

Proof. From the continuity of $F$ we get

[TABLE]

for each $x$ - real number. From the inclusion

[TABLE]

we obtain

[TABLE]

And (6) yields

[TABLE]

for every real number $x$ . From the other side

[TABLE]

therefore

[TABLE]

Thus

[TABLE]

and the assertion follows. ∎

Theorem 2.

Let $\{v(n)\}$ be a polyadicly continuous sequence and $F:(-\infty,\infty)\to[0,1]$ a continuous function. If for every $x$ - real number

[TABLE]

holds, then the random variable $\tilde{v}$ has the distribution function $F$ and for $x$ - real number the set $\{n\in{\mathbb{N}};v(n)<x\}$ belongs to $\text{$ \mathcal{D} $}_{\mu}$ and**

[TABLE]

Proof. Clearly

[TABLE]

and so $F(x)\leq P(\tilde{v}\leq x)$ . From the other side

[TABLE]

thus for each $\varepsilon>0$ we have $F(x)\leq P(\tilde{v}\leq x)\leq F(x+\varepsilon)$ . For $\varepsilon\to 0^{+}$ we obtain the assertion from the continuity of $F$ .∎

The Buck’s measurable sequences $\{v_{1}(n)\},\{v_{2}(n)\},\dots,\{v_{r}(n)\}$ are ca- lled independent if and only if for every $x_{1},\dots,x_{r}$ - real numbers we have

[TABLE]

Example 2.

We come back to the Example 1. Consider that the sequences $\{Q_{k}^{(j)}\}$ are given such that $Q_{0}^{(j)}=1,j=1,\dots,r$ and $Q_{k}^{(j)}|Q_{k+1}^{(j)}$ for $j=1,\dots,r$ and $k=0,1,2\dots$ . If $Q_{k}^{(j)},Q_{k}^{(j_{1})}$ are relatively prime for $j\neq j_{1}$ . Denote $\{\gamma_{j}(n)\}$ the van der Corput sequence with base $Q_{k}^{(j)}$ for $j=1,\dots,r$ . Then these sequences are independent (see [IPT]).

Theorem 3.

Let $\{v_{1}(n)\},\{v_{2}(n)\},\dots\{v_{k}(n)\}$ be independent Buck’s measurable polyadicly continuous sequences such that for every $x$ - real number

[TABLE]

where $F_{j},j=1\dots,k$ are continuous functions defined on real line. Then the random variables $\tilde{v_{1}},\dots,\tilde{v_{k}}$ are independent.**

Proof. For $x_{1},\dots x_{k}$ - real numbers we have

[TABLE]

Thus $P(\tilde{v_{1}}<x_{1}\land\dots\land\tilde{v_{k}}<x_{k})\leq F_{1}(x)\dots F_{k}(x_{k})$ and so from Theorem 2 we get $P(\tilde{v_{1}}<x\land\dots\land\tilde{v_{k}}<_{k})\leq P(\tilde{v_{1}}<x)\dots P(\tilde{v_{k}}<x_{k})$ .

From the other hand side we have

[TABLE]

The proof is complete. ∎

Theorem 4.

Let $\{v_{1}(n)\},\{v_{2}(n)\}$ be two independent Bu- ck’s measurable polyadi- cly continuous sequence, such that the functions

[TABLE]

are continuous. Then $\{v_{1}(n)+v_{2}(n)\}$ is Buck’s measurable, and**

[TABLE]

Proof. Theorem 2 implies that the random variables $\tilde{v_{1}},\tilde{v_{2}}$ have the distribution functions $F_{1},F_{2}$ . From Theorem 3 we get that these random variables are independent and so the distribution function of $\tilde{v_{1}}+\tilde{v_{2}}$ is given by the integral on right hand side in (8). Clearly this function is continuous and from this we obtain the assertion from Theorem 1. ∎

Thus we get immediately from Theorem 4:

Corollary 1.

If $\{v_{1}(n)\},\{v_{2}(n)\}$ are two independent Buck’s uniformly distributed sequences then for the function

[TABLE]

we have $G(x)=0,x\leq 0,G(x)=\frac{x^{2}}{2},x\in[0,1],G(x)=2x-\frac{x^{2}}{2}-1,x\in[1,2],G(x)=1,x>2$ .**

Analogously we can prove:

Corollary 2.

If $\{v_{1}(n)\},\{v_{2}(n)\}$ are two independent Buck’s uniformly distributed sequences then for the function

[TABLE]

we have $G(x)=0,x\leq 0,G(x)=x(1-\ln x),x\in(0,1),G(x)=1,x>1$ .**

Proof. Clearly $\tilde{v_{1}v_{2}}=\tilde{v_{1}}\tilde{v_{2}}$ . Then we have

[TABLE]

where $A=\{[t_{1},t_{2}];t_{1}t_{2}<x;0\leq t_{1},t_{2}\leq 1\}$ .

From Theorem 2 we can prove by induction:

Theorem 5.

If $\{v_{1}(n)\},...,\{v_{k}(n)\}$ are independent polyadicly continuous sequences, such that for $j=1,\dots,k$ the functions

[TABLE]

are continuous, then $\{v(n)\}=\{v_{1}(n)+\dots v_{k}(n)\}$ is polyadicly continuous function and

[TABLE]

is continuous.**

Let $h:\Omega\to(-\infty,\infty)$ be a continuous function. Since $\Omega$ is a compact space it is uniformly continuous. Consider a $m\in{\mathbb{N}}$ . To the function $h$ we can associate a periodic function $h_{m}$ with period $m$ in following way:

[TABLE]

Clearly

[TABLE]

We have that $\lim_{N\to\infty}\mathfrak{(}N!,0)=0$ and so from the uniform continuity of $h$ we get that $h_{N!}$ converges uniformly to $h$ . From (9) we get

[TABLE]

The function $h$ restricted on ${\mathbb{N}}$ is polyadicly continuous. Thus exists the proper limit $\lim_{m=0}\frac{1}{m}\sum_{s=0}^{m-1}h(s)$ . And so from (10) we can conclude

[TABLE]

Remark 1.

Let $\{v(n)\}$ be a polyadicly continuous sequence. Then there exists

[TABLE]

If the random variable $\tilde{v}$ has continuous distribution function $F$ then

[TABLE]

A sequence of positive integers $\{k_{n}\}$ is called uniformly distributed in ${\mathbb{Z}}$ if and only for each $m\in{\mathbb{N}},r\in{\mathbb{Z}}$ we have that $\{n\in{\mathbb{N}};k_{n}\equiv r\mod{m}\}\in\text{$ \mathcal{D} $}$ and $d(\{n\in{\mathbb{N}};k_{n}\equiv r\mod{m}\})=\frac{1}{m}$ . Let us remark that this type of uniform distribution is firstly defined in [NIV].

In [PAS3] and [PAS4] is proven that for each sequence $\{k_{n}\}$ uniformly distributed in ${\mathbb{Z}}$ and polyadicly continuous sequence $\{v(n)\}$ we have

[TABLE]

Theorem 6.

Let $\{v_{1}(n)\},\dots,\{v_{k}(n)\}$ be polyadicly continuous independent sequences. Then for every functions $g_{1},\dots,g_{k}$ continuous on real line we have

[TABLE]

for each sequence $\{k_{n}\}$ uniformly distributed in ${\mathbb{Z}}$ .**

Proof. If $\{v(n)\}$ is a polyadicly continuous function, then it is bounded. Every continuous function $g$ defined on real line is uniformly continuous on closed interval $[b_{1},b_{2}]$ where $b_{1}$ is lower bound of the sequence $\{v(n)\}$ and $b_{2}$ its upper bound. Thus the sequence $\{g(v(n))\}$ is polyadicly continuous also.

Let us consider $\{v_{1}(n)\},\dots,\{v_{k}(n)\}$ - polyadicly continuous independent sequences. Then $\{g_{1}(v_{1}(n))\},\dots,\{g_{k}(v_{k}(n))\}$ are polyadicly continuous and independent also. Thus

[TABLE]

and the assertion follows from (13). ∎

Analogously can be proved that for dispersion $D^{2}(v)=E((v-E(v))^{2})$ the equation

[TABLE]

holds. Thus from above the Chebyshev inequality follows:

Theorem 7.

If $\{v(n)\}$ is polyadicly continuous Buck’s measurable sequence, such that the function

[TABLE]

is continuous then

[TABLE]

for $\varepsilon>0$ .**

Directly from central limit theorem we get:

Theorem 8.

Let $\{v_{k}(n)\},k=1,2,3\dots$ be a sequence of polyadicly continuous sequences that for every $k=1,2,3,\dots$ the sequences $\{v_{j}(n)\},j=1\dots k$ are independent and there exists a continuous function $F$ such that

[TABLE]

for each real number $x$ . Put $E=E(v_{k}),D^{2}=D^{2}(v_{k}),k=1,2,3,\dots$ . Then for every $x$ - real number we have

[TABLE]

Bibliography18

The reference list from the paper itself. Each links out to its DOI / PubMed record.

1[BUC] Buck, R., C., The measure theoretic approach to density , Amer. J. Math 68 , 1946, 560–580
2[D-T] Drmota, M., Tichy, R. F., Sequences, Discrepancies and Applications, Springer, Berlin Heidelberg , Springer, Berlin Heidelberg, 1997
3[G] Grekos, G., On various definitions of density (a survey) , Tatra Mt. Math. Publ., 31, 2005, 17–27
4[G 1] Grekos, G., The density set (a survey) , Tatra Mt. Math. Publ., 31, 2005, 103–111
5[GST] Grabner, P. J., Strauch, O., Tichy, R. F., Lp-discrepancyandstatisticalindependenceofsequences, Czechoslovak Mathematical Journal,Vol.49(1999),No.1,97 110
6[IPT] Iaco, M. R., Pasteka, M., Tichy R., F., Measure density for set decompositions and uniform distribution Rend. Circ. Math. Palermo (2) , 64, No. 2, 2015 , 323 – 339
7[K-N] Kuipers, L., Niederreiter, H., Uniform distribution of Sequences , John Wiley and Sons, N.Y. London, Sydney Toronto, 1974
8[N] Novoselov, E. V., Topological theory of polyadic numbers , Trudy Tbilis. Mat. Inst. 27, 1960, 61 – 69, (in russian)

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Taxonomy

Example 1**.**

Theorem 1**.**

Theorem 2**.**

Example 2**.**

Theorem 3**.**

Theorem 4**.**

Corollary 1**.**

Corollary 2**.**

Theorem 5**.**

Remark 1**.**

Theorem 6**.**

Theorem 7**.**

Theorem 8**.**

Example 1.

Theorem 1.

Theorem 2.

Example 2.

Theorem 3.

Theorem 4.

Corollary 1.

Corollary 2.

Theorem 5.

Remark 1.

Theorem 6.

Theorem 7.

Theorem 8.