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Open Access Research

On ℐ-asymptotically lacunary statistical equivalent sequences

Ekrem Savaş

Author Affiliations

Department of Mathematics, Istanbul Commerce University, Üsküdar, Istanbul, Turkey

Advances in Difference Equations 2013, 2013:111  doi:10.1186/1687-1847-2013-111

The electronic version of this article is the complete one and can be found online at: http://www.advancesindifferenceequations.com/content/2013/1/111


Received:1 February 2013
Accepted:1 April 2013
Published:18 April 2013

© 2013 Savaş; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

This paper presents the following definition, which is a natural combination of the definitions for asymptotically equivalent, ℐ-statistically limit and ℐ-lacunary statistical convergence. Let θ be a lacunary sequence; the two nonnegative sequences <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2">View MathML</a> are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3">View MathML</a>, and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M5">View MathML</a>

(denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6">View MathML</a>) and simply ℐ-asymptotically lacunary statistical equivalent if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7">View MathML</a>.

MSC: 40A99, 40A05.

Keywords:
asymptotical equivalent; ideal convergence; lacunary sequence; statistical convergence

1 Introduction

In 1993, Marouf [1] presented definitions for asymptotically equivalent sequences and asymptotic regular matrices. In 2003, Patterson [2] extended these concepts by presenting an asymptotically statistical equivalent analog of these definitions and natural regularity conditions for nonnegative summability matrices.

In [3], asymptotically lacunary statistical equivalent, which is a natural combination of the definitions for asymptotically equivalent, statistical convergence and lacunary sequences. Later on, the extension asymptotically lacunary statistical equivalent sequences is presented (see [4]).

Recently, Das, Savaş and Ghosal [5] introduced new notions, namely ℐ-statistical convergence and ℐ-lacunary statistical convergence by using ideal.

In this short paper, we shall use asymptotical equivalent and lacunary sequence to introduce the concepts ℐ-asymptotically statistical equivalent and ℐ-asymptotically lacunary statistical equivalent. In addition to these definitions, natural inclusion theorems shall also be presented.

First, we introduce some definitions.

2 Definitions and notations

Definition 2.1 (Marouf [1])

Two nonnegative sequences <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2">View MathML</a> are said to be asymptotically equivalent if

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M10">View MathML</a>

(denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M11">View MathML</a>).

Definition 2.2 (Fridy [6])

The sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> has statistic limitL, denoted by st-<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M13">View MathML</a> provided that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M15">View MathML</a>

The next definition is natural combination of Definitions 2.1 and 2.2.

Definition 2.3 (Patterson [2])

Two nonnegative sequences <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2">View MathML</a> are said to be asymptotically statistical equivalent of multiple L provided that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M19">View MathML</a>

(denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M20">View MathML</a>) and simply asymptotically statistical equivalent if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7">View MathML</a>.

By a lacunary <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M22">View MathML</a>; <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M23">View MathML</a> , where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M24">View MathML</a>, we shall mean an increasing sequence of nonnegative integers with <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M25">View MathML</a> as <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M26">View MathML</a>. The intervals determined by θ will be denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M27">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M28">View MathML</a>. The ratio <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M29">View MathML</a> will be denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M30">View MathML</a>.

Definition 2.4 ([3])

Let θ be a lacunary sequence; the two nonnegative sequences <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2">View MathML</a> are said to be asymptotically lacunary statistical equivalent of multiple L provided that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M34">View MathML</a>

(denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M35">View MathML</a>) and simply asymptotically lacunary statistical equivalent if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7">View MathML</a>.

More investigations in this direction and more applications of asymptotically statistical equivalent can be found in [7,8] where many important references can be found.

The following definitions and notions will be needed.

Definition 2.5 ([9])

A nonempty family <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M37">View MathML</a> of subsets a nonempty set Y is said to be an ideal in Y if the following conditions hold:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M38">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M39">View MathML</a>;

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M40">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M41">View MathML</a> imply <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M42">View MathML</a>.

Definition 2.6 ([10])

A nonempty family <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M43">View MathML</a> is said to be a filter of ℕ if the following conditions hold:

(i) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M44">View MathML</a>;

(ii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M45">View MathML</a> implies <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M46">View MathML</a>;

(iii) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M47">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M41">View MathML</a> imply <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M49">View MathML</a>.

If ℐ is proper ideal of ℕ (i.e., <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M50">View MathML</a>), then the family of sets <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M51">View MathML</a> is a filter of ℕ. It is called the filter associated with the ideal.

Definition 2.7 ([9,10])

A proper ideal ℐ is said to be admissible if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M52">View MathML</a> for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M53">View MathML</a>.

Throughout ℐ will stand for a proper admissible ideal of ℕ, and by sequence we always mean sequences of real numbers.

Definition 2.8 ([9])

Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M54">View MathML</a> be a proper admissible ideal in ℕ.

The sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M55">View MathML</a> of elements of ℝ is said to be ℐ-convergent to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M56">View MathML</a> if for each <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3">View MathML</a> the set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M58">View MathML</a>.

Following these results, we introduce two new notions ℐ-asymptotically lacunary statistical equivalent of multiple L and strong ℐ-asymptotically lacunary equivalent of multiple L.

The following definitions are given in [5].

Definition 2.9 A sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> is said to be ℐ-statistically convergent to L or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M60">View MathML</a>-convergent to L if, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M61">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M63">View MathML</a>

In this case, we write <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M64">View MathML</a>. The class of all ℐ-statistically convergent sequences will be denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M60">View MathML</a>.

Definition 2.10 Let θ be a lacunary sequence. A sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> is said to be ℐ-lacunary statistically convergent to L or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M67">View MathML</a>-convergent to L if, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M61">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M70">View MathML</a>

In this case, we write <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M71">View MathML</a>. The class of all ℐ-lacunary statistically convergent sequences will be denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M72">View MathML</a>.

Definition 2.11 Let θ be a lacunary sequence. A sequence <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> is said to be strong ℐ-lacunary convergent to L or <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M74">View MathML</a>-convergent to L if, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M75">View MathML</a>

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M76">View MathML</a>

In this case, we write <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M77">View MathML</a>. The class of all strong ℐ-lacunary statistically convergent sequences will be denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M74">View MathML</a>.

3 New definitions

The next definitions are combination of Definitions 2.1, 2.9, 2.10 and 2.11.

Definition 3.1 Two nonnegative sequences <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2">View MathML</a> are said to be ℐ-asymptotically statistical equivalent of multiple L provided that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M83">View MathML</a>

(denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M84">View MathML</a>) and simply ℐ-asymptotically statistical equivalent if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7">View MathML</a>.

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M86">View MathML</a>, ℐ-asymptotically statistical equivalent of multiple L coincides with asymptotically statistical equivalent of multiple L, which is defined in [3].

Definition 3.2 Let θ be a lacunary sequence; the two nonnegative sequences <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2">View MathML</a> are said to be ℐ-asymptotically lacunary statistical equivalent of multiple L provided that for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M91">View MathML</a>

(denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6">View MathML</a>) and simply ℐ-asymptotically lacunary statistical equivalent if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7">View MathML</a>.

For <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M86">View MathML</a>, ℐ-asymptotically lacunary statistical equivalent of multiple L coincides with asymptotically lacunary statistical equivalent of multiple L, which is defined in [3].

Definition 3.3 Let θ be a lacunary sequence; two number sequences <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2">View MathML</a> are strong ℐ-asymptotically lacunary equivalent of multiple L provided that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M97">View MathML</a>

(denoted by <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98">View MathML</a>) and strong simply ℐ-asymptotically lacunary equivalent if <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M7">View MathML</a>.

4 Main result

In this section, we state and prove the results of this article.

Theorem 4.1Let<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M100">View MathML</a>be a lacunary sequence then

(1)

(a) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98">View MathML</a>then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6">View MathML</a>,

(b) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98">View MathML</a>is a proper subset of<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6">View MathML</a>;

(2) If<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M105">View MathML</a>and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6">View MathML</a>then<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98">View MathML</a>;

(3) <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M108">View MathML</a>,

where<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M109">View MathML</a>denote the set of bounded sequences.

Proof Part (1a): If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98">View MathML</a> then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M112">View MathML</a>

and so

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M113">View MathML</a>

Then, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M114">View MathML</a>,

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M115">View MathML</a>

Hence, we have <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6">View MathML</a>.

Part (1b): <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M117">View MathML</a>, let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> be defined as follows: <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M119">View MathML</a> to be <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M120">View MathML</a> at the first <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M121">View MathML</a> integers in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M122">View MathML</a> and zero otherwise. <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M123">View MathML</a> for all k. These two satisfy the following <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6">View MathML</a>, but the following fails <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98">View MathML</a>.

Part (2): Suppose <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M1">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M2">View MathML</a> are in <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M109">View MathML</a> and <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M129">View MathML</a>. Then we can assume that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M130">View MathML</a>

Given <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M3">View MathML</a>, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M132">View MathML</a>

Consequently, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M133">View MathML</a>

Therefore, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M98">View MathML</a>.

Part (3): Follows from (1) and (2). □

Theorem 4.2Letis an ideal and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M100">View MathML</a>is a lacunary sequence with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M136">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M137">View MathML</a>

Proof Suppose first that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M136">View MathML</a>, then there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M4">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M140">View MathML</a> for sufficiently large r, which implies

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M141">View MathML</a>

If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M6">View MathML</a>, then for every <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M143">View MathML</a> and for sufficiently large r, we have

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M144">View MathML</a>

Then, for any <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M145">View MathML</a>, we get

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M146">View MathML</a>

This completes the proof. □

For the next result we assume that the lacunary sequence θ satisfies the condition that for any set <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M147">View MathML</a>, <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M148">View MathML</a>.

Theorem 4.3Letis an ideal and<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M22">View MathML</a>is a lacunary sequence with<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M150">View MathML</a>, then

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M151">View MathML</a>

Proof If <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M152">View MathML</a>, then without any loss of generality, we can assume that there exists a <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M153">View MathML</a> such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M154">View MathML</a> for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M155">View MathML</a>. Suppose that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M35">View MathML</a> and for <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M157">View MathML</a> define the sets

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M158">View MathML</a>

and

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M159">View MathML</a>

It is obvious from our assumption that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M147">View MathML</a>, the filter associated with the ideal ℐ. Further observe that

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M161">View MathML</a>

for all <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M162">View MathML</a>. Let <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M163">View MathML</a> be such that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M164">View MathML</a> for some <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M165">View MathML</a>. Now

<a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M166">View MathML</a>

Choosing <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M167">View MathML</a> and in view of the fact that <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M168">View MathML</a> where <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M147">View MathML</a>, it follows from our assumption on θ that the set T also belongs to <a onClick="popup('http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.advancesindifferenceequations.com/content/2013/1/111/mathml/M170">View MathML</a> and this completes the proof of the theorem. □

Competing interests

The author declares that they have no competing interests.

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