Generalized Statistical Convergence in Sequence Spaces: A Modulus Function Perspective

Abstract

This study introduces a flexible framework for analyzing how sequences of numbers approach a limit, even when traditional convergence criteria fail. By incorporating modulus function mathematical tools that quantify growth rates, this research extends the concept of statistical convergence to handle sequences with irregular or sparse behavior. Key results establish connections between this generalized convergence theory and related properties, like boundedness, providing a unified approach to understanding sequence dynamics. The findings enhance our ability to model and analyze complex data patterns in mathematics and beyond.

1. Introduction

Imagine tracking the average temperature in a city over years: most days cluster near a seasonal norm, but rare extreme events (heatwaves, cold snaps) disrupt the pattern. Traditional convergence where every term in a sequence must get arbitrarily close to a limit fails to describe such scenarios, as the extreme values never settle down. Statistical convergence offers a solution: it ignores negligible exceptions (e.g., the 1% of days with extreme temperatures) to focus on the dominant trend. This paper enhances this idea using modulus functions to quantify how negligible those exceptions are, enabling a more nuanced analysis of sequence behavior.

The study by Ji-Huan et al. [1] demonstrated how homotopy perturbation methods can systematically optimize initial estimates in nonlinear systems. Similarly, modulus functions can be derived in a data-adaptive manner, tailored to the structural features of the data itself, further enhancing the descriptive power of the framework.

As a response to the limitations in classical convergence, statistical summability has found widespread application in functional analysis, approximation theory, and summability theory. Its relevance increases in situations involving data with stochastic elements or in spaces where standard norms do not fully capture the convergence behavior. One of its key advantages lies in tolerating deviations from the limit over a set of indices with zero density, which broadens the class of summable sequences. To refine this idea, researchers have introduced stronger versions, such as strong statistical summability, where the absolute difference between sequence terms and the limit is considered within a density framework. Moreover, the adaptation of $p-$norms to this context has led to the development of strong $p-$statistical summability, which combines statistical density and norm-based measurements to offer a more rigorous convergence criterion.

Statistical convergence, a broader form of classical convergence, was initially introduced in 1935 by Zygmund [2] in his debut edition published in Warsaw. The formal definition of statistical convergence was subsequently provided by Steinhaus [3] and Fast [4]. This concept shares important associations with the summability method, as studied by Schoenberg [5], Salat [6], Fridy ([7,8]), Connor [9], and Rath and Tripathy [10]. Recently, several mathematicians have delved into the exploration of statistical convergence.

Throughout this paper, The following notations will be used frequently:

$\mathbb{N}$	The set of all positive integers;
$c$	The set of all convergent sequences;
$l_{\infty }$	The set of all bounded sequences;
$f$	A modulus function; $f\left(u\right)$ denotes its value at a real number $u$;
$d\left(K\right)$	The natural density of a subset $K\subset \mathbb{N}$;
$d_f$(K)	The $f-$density of a subset $K\subset \mathbb{N}$;
$d_f^{\sigma }\left(K\right)$	The $f_{\sigma }$−density of a subset $K\subset \mathbb{N}$;
$S$	The set of all statistically convergent sequences;
$S\left(b\right)$	The set of all statistically bounded sequences;
$S_f\left(b\right)$	The set of all$f-$statistically bounded sequences;
$V_{\sigma }$	The set of all $\sigma -$convergent sequences;
[$V_{\sigma }$]	The set of strongly $\sigma -$convergent sequences;
$S^{\sigma }$	The set of all $\sigma -$statistically convergent;
$S_f^{\sigma }$	The set of all $f_{\sigma }$−statistically convergent sequences;
$S_{f_{\sigma }}$(b)	The set of all $f_{\sigma }$−statistically bounded sequences;
$L{\left[V_{f_{\sigma }}\right]}_q$	The space of sequences that are strongly $S_f^{\sigma }$−convergent to a limit $L$.

Let $\mathbb{N}$ be the set of positive integers. The natural density of a set $K\subset \mathbb{N}$ is defined by

$$d\left(K\right)=\underset{n\to \infty }{\mathit{lim}}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:k\in K\right\}\right|,$$

where $\left|\left\{k\le n:k\in K\right\}\right|$ indicates the number of elements of $K$ not exceeding $n$. One easily may see that $d\left(\mathbb{N}\right)=1$ and $d\left(K\right)=0$ if $K\subset \mathbb{N}$ is a finite set and $d\left(K^c\right)=d\left(\mathbb{N}\right)-d\left(K\right)=1-d\left(K\right)$, where $K^c=K-\mathbb{N}$.

A sequence $u=\left(u_k\right)$ is referred to as statistically convergent to $a$ if, for every $ϵ>0$,

$$\underset{n\to \infty }{\mathit{lim}}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:\left|u_k-a\right|\ge ϵ\right\}\right|=0.$$

If a sequence is statistically convergent to $a$, we denote this by $S-lim{u}_k=a$.

It is well known that every classically convergent sequence is also statistically convergent; however, the converse does not necessarily hold.

Consider the sequence $u=\left(u_k\right)$ defined by

$$u_k=\left\{\begin{array}{c} 1 ; k=m^2, m=(\mathrm{1,2},3,..) \\ 0 ; k\ne m^2. \end{array}\right.$$

For any $ϵ>0$, since

$$\left|\left\{k\le n:\left|u_k\right|\ge ϵ\right\}\right|=\left|\left\{k\le n:u_k\ne 0\right\}\right|\le \sqrt{n},$$

we get

$$\underset{n}{\mathit{lim}}{\displaystyle \frac{1}{n}}\left|\left\{k\le n:u_k\ne 0\right\}\right|\le \underset{n}{\mathit{lim}}{\displaystyle \frac{\sqrt{n}}{n}}=0.$$

This shows that the statistical limit of the sequence is $0$. So, statistical convergence does not imply classical convergence.

Nakano [11] was the pioneering contributor to the conception of a modulus function.

The function $f:[0,\infty )\to [0,\infty )$ is referred to as the modulus function when it fulfills the following conditions: $f\left(u\right)=0$ if and only if $u=0$, $f(u+z)\le f\left(u\right)+f\left(z\right)$ for $u,z\ge 0$, $f$ is continuous from the right at $0$, and f is increasing.

$f$ is continuous everywhere over the modulus function $[0,\infty )$. Additionally, a modulus function can exhibit either bounded or unbounded behavior.

The modulus function has been used by many mathematicians in summability theory. In a later study, Aizpuru et al. [12] introduced the concept of $f-$density for a subset $K\subset \mathbb{N}$, where $f$is an unbounded modulus. The $f-$density, defined as

$$d_f\left(K\right)=\underset{n\to \infty }{\mathit{lim}}{\displaystyle \frac{f\left(\left\{k\le n:k\in K\right\}\right)}{f\left(n\right)}},$$

exists when the limit is well defined. They also introduced the notion of $f-$statistical convergence using an unbounded modulus $f,$ such as

$$d_f\left(\left\{k\in \mathbb{N}:\left|u_k-a\right|\ge ϵ\right\}\right)=0$$

which can be expressed as

$$\underset{n\to \infty }{\mathit{lim}}{\displaystyle \frac{1}{f\left(n\right)}}f\left(\left|k\in \mathbb{N}:\left|u_k-a\right|\ge ϵ\right|\right)=0.$$

It is important to observe that while every $f-$statistically convergent sequence also converges statistically, not all statistically convergent sequences are necessarily $f-$statistically convergent for all unbounded moduli $f$.

Lemma 1. 

The truth of the limit $\underset{t\to \infty }{\mathit{lim}}{\displaystyle \frac{f\left(t\right)}{t}}=\beta$stands pertaining to any modulus function $f$ (see [13]).

Theorem 1. 

Let us examine two unbounded modulus functions $f$ and $g$. So, for a subset $K\subseteq \mathbb{N}$:

(i) 

If the limit

$$\underset{t\to \infty }{\mathit{lim}}{\displaystyle \frac{f\left(t\right)}{g\left(t\right)}}>0,$$

(1)

then $d_g\left(K\right)=0$ implies $d_f\left(K\right)=0$, provided the limit exists.

(ii) 

If

$$0<\underset{t\to \infty }{\mathit{lim}}{\displaystyle \frac{f\left(t\right)}{g\left(t\right)}}=\alpha <\infty ,$$

(2)

then $d_g\left(K\right)=0$ if and only if $d_f\left(K\right)=0$, given that the limit exists (see [14]).

Corollary 1. 

If we have an unbounded modulus function $f$ and a subset $K\subseteq \mathbb{N}$, and the requirement

$$\underset{t\to \infty }{\mathit{lim}}{\displaystyle \frac{f\left(t\right)}{t}}>0$$

(3)

holds, then we can assert that $d_f\left(K\right)=0 \iff d\left(K\right)=0$ (see [14]).

The concept of statistical boundedness of sequences was first introduced in the well-known paper by Fridy and Orhan [15]. In contrast to statistical convergence, statistical boundedness has not received as much attention in the literature. Nevertheless, Bhardwaj et al. [16] extended this concept by developing generalizations based on f-statistical convergence.

Definition 1. 

The number sequence $u=\left(u_k\right)$ is considered statistically bounded if there exists a number $L>0$ for which $d\left(\left\{k\in \mathbb{N}:\left|u_k\right|>L\right\}\right)=0$. The collection of all statistically bounded sequences is denoted as $S\left(b\right)$ (see [15]).

Definition 2. 

The number sequence $u=\left(u_k\right)$ is referred to as $f-$statistically bounded if there exists a number $L>0$ for which $d_f\left(\left\{k\in \mathbb{N}:\left|u_k\right|>L\right\}\right)=0$. The set of all $f-$statistically bounded sequences is denoted as $S_f\left(b\right)$ (see [16]).

In this study, we introduce the following concepts:

Let the set of positive numbers $\mathbb{N}$ be mapped into itself by the expression $\sigma$. If a continuous linear functional $\phi$ is non-negative, normal, and $\phi \left(x\right)=\phi \left(\right(x_{\sigma \left(n\right)}\left)\right)$, it is said to have an invariant mean and is defined on the space $l_{\infty }$ of all limited sequences (see [17]).

A sequence $u=\left(u_k\right)$ is regarded as $\sigma -$convergent to the number $L$ when all of its $\sigma -$means coincide with $L$, which implies that $\phi \left(x\right)=a$ for all $\phi$. Similarly, a bounded sequence $u=\left(u_k\right)$ $\sigma -$converges to the number $L$ if the limit of $t_{pi}$ converges uniformly to $L$ as $i$ tends to infinity. Here,

$$t_{pi}={\displaystyle \frac{u_i+u_{\sigma \left(i\right)}+u_{{\sigma }^2\left(i\right)}+\cdots +u_{{\sigma }^p\left(i\right)}}{p+1}}$$

(see [18]).

Let us denote by $V_{\sigma }$ the collection of all sequences that are $\sigma -$convergent. In this context, we express the convergence as $V_{\sigma }$, where $L$ is referred to as the $\sigma -$limit of the sequence $u$.

It is important to highlight that a $\sigma -$mean is a generalization of the limit functional defined on the space $c$, satisfying $\phi \left(u\right)=limu$ for every $u\in c$, if and only if $\sigma$ has no finite orbits. In this framework, the inclusion $c\subset V_{\sigma }\subset l_{\infty }$ holds (see [19]). Moreover, when $\sigma$ is interpreted as a shift (or translation), the corresponding $\sigma -$mean is known as a Banach limit (refer to [20]). In such cases, $\sigma -$convergence coincides with the notion of almost convergence introduced by Lorentz (see [21]).

Definition 3. 

A bounded sequence $u=\left(u_k\right)$ is said to be strongly $\sigma -$converged to the number $L$ if

$$\underset{n\to \infty }{\mathit{lim}}{\displaystyle \frac{1}{p}}\sum _{j=1}^p\left|u_{{\sigma }^j\left(i\right)}-L\right|=0, uniformly in i.$$

The collection of all strongly $\sigma -$convergent sequences is denoted as $\left[V_{\sigma }\right]$, and it is expressed as $u_k\to L\left(V_{\sigma }\right)$ (see [22]). Taking $\sigma \left(n\right)=n+1$, we obtain $\left[V_{\sigma }\right]=\left[\widehat{c}\right]$ so that strong σ-convergence generalizes the concept of strong almost convergence. Note that $c\subset \left[V_{\sigma }\right]\subset V_{\sigma }\subset l_{\infty }.$

Definition 4. 

Let $K_{\sigma }(i+1,i+p)$ represent the cardinality of the set $\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):k\in K\right\}$, and define $N_p=\underset{i}{\mathit{min}}{K}_{\sigma }(i+1,i+p)$, $N^p=\underset{i}{\mathit{max}}{K}_{\sigma }(i+1,i+p)$. It can be observed that the limits ${\underset{\_}{\delta }}_{\sigma }\left(K\right)=\underset{p\to \infty }{\mathit{lim}}{\displaystyle \frac{N_p}{p}}$ and ${\overline{\delta }}^{\sigma }\left(K\right)=\underset{p\to \infty }{\mathit{lim}}{\displaystyle \frac{N^p}{p}}$ exist. These limits are referred to as the lower and upper $\sigma -$density of the set $K$, respectively. If ${\underset{\_}{\delta }}_{\sigma }\left(K\right)={\overline{\delta }}^{\sigma }\left(K\right)$, then this shared value ${\delta }_{\sigma }\left(K\right)$ is called the $\sigma -$density of the set $K$. Importantly, for $K\subseteq N$, it follows that ${\underset{\_}{\delta }}_{\sigma }\left(K\right)\le \underset{\_}{\delta }\left(K\right)\le \overline{\delta }\left(K\right)\le {\overline{\delta }}^{\sigma }\left(K\right)$. In the case where $\sigma \left(n\right)=n+1$, the $\sigma -$density is reduced to uniform density (see [23]).

Definition 5. 

A sequence $u=\left(u_k\right)$ is considered to be $\sigma -$statistically convergent to $L$ if, for every $ϵ>0$,

$$d^{\sigma }\left(\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-a\right|\ge ϵ\right\}\right)=0, uniformly in i$$

indicating that

$$\underset{p\to \infty }{\mathit{lim}}{\displaystyle \frac{\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-a\right|\ge ϵ\right\}\right|\right)}{p}}=0, uniformly in i.$$

We denote this as $\sigma -st lim u_k=a$ in such cases (see [24]). We define $S^{\sigma }=\{u=u_k for some a,\sigma -st lim u_k=a\}.$

Many studies have been carried out on sequence spaces, statistics, statistical convergence, etc. ([15,25,26,27,28,29,30,31,32,33,34,35]).

Definition 6. 

Let $X$ be a sequence space. Then, $X$ is called

(i) 

Solid (or normal), if (${\alpha }_k{u}_k$) ∈ X whenever $\left(u_k\right)\in X$ for all sequences (${\alpha }_k$) of scalar with $\left|{\alpha }_k\right|\le 1$, for all$k\in \mathbb{N}$;

(ii) 

Symmetric if (xk) ∈ X$\left(u_k\right)\in X$ implies $\left(u_{\pi \left(k\right)}\right)\in X$, where$\pi \left(k\right)$ is a permutation of $\mathbb{N}$;

(iii) 

Monotone, provided $X$ contains the canonical preimages of all its stepspace [36].

Lemma 2. 

(i) If a sequence space $E$ is solid, then $E$ is monotone.

(ii) 

$X$ is monotone if and only if $m_{o}X\subset X$.

$$m_o=sp\left\{A\right\}, where A is all sequences of zeros and ones.$$

It is clear that $m_o$ is monotone but not normal, and $c$ is not monotone and not normal [36].

2. Main Results

In this investigation, we aim to introduce the concepts of $f_{\sigma }-$density and $f_{\sigma }-$statistical convergence. Additionally, we will delve into the interrelations linking $f_{\sigma }-$statistical convergence and $g_{\sigma }-$statistical convergence.

Definition 7. 

The $f_{\sigma }-$density of a set $K\subseteq \mathbb{N}$ is denoted as

$$d_f^{\sigma }\left(K\right)=\underset{p\to \infty }{\mathit{lim}}{\displaystyle \frac{f\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):k\in K\right\}\right|\right)}{f\left(p\right)}}=0,$$

provided a limit exists, where $f$ represents an unbounded modulus function.

Definition 8. 

A sequence $u=\left(u_k\right)$ is defined as $f_{\sigma }-$statistically convergent to $a$ if, for any $ϵ>0$,

$$d_f^{\sigma }\left(\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-a\right|\ge ϵ\right\}\right)=0, uniformly in i$$

(or $K_f^{\sigma }\left(ϵ\right):=f\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-a\right|\ge ϵ\right\}\right|\right)$ meaning that

$$\underset{p\to \infty }{\mathit{lim}}{\displaystyle \frac{f\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-a\right|\ge ϵ\right\}\right|\right)}{f\left(p\right)}}=0, uniformly in i,$$

and it is represented as $f_{\sigma }-st lim u_k=a$. Hereafter, we assume that f is an unbounded modulus unless otherwise stated. We will use $S_f^{\sigma }$ to represent the set of sequences.

Theorem 2. 

(i) Under condition (1) of Theorem 1 (i), if a sequence $\left(u_k\right)$ is $g_{\sigma }-$statistically convergent, then it is also $f_{\sigma }-$statistically convergent (with the same limit), which means

$$S_g^{\sigma }\subset S_f^{\sigma }.$$

(ii) If condition (2) of Theorem 1 (ii) is satisfied, then a sequence $\left(u_k\right)$ is $f_{\sigma }-$statistically convergent⇔ it is $g_{\sigma }-$statistically convergent, which means

$$S_g^{\sigma }=S_f^{\sigma }.$$

The functions $f$ and $g$ are both unbounded modulus functions.

Proof. 

(i) Assume that $\left(u_k\right)$ is statistically $g_{\sigma }-$convergent to $a$, indicated by $S_g^{\sigma }-lim{u}_k=a$. Define $K=\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-a\right|\ge ϵ\right\}$. Then,

$$\underset{p\to \infty }{\mathit{lim}}{\displaystyle \frac{g\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-a\right|\ge ϵ\right\}\right|\right)}{g\left(p\right)}}=0, \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y} \mathrm{i}\mathrm{n} i$$

which implies

$$\underset{p\to \infty }{\mathit{lim}}{\displaystyle \frac{f\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-a\right|\ge ϵ\right\}\right|\right)}{f\left(p\right)}}=0, \mathrm{u}\mathrm{n}\mathrm{i}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{m}\mathrm{l}\mathrm{y} \mathrm{i}\mathrm{n} i$$

if condition (1) holds, as stated in Theorem 1 (i). This implies that $\left(u_k\right)$ exhibits $f_{\sigma }-$statistical convergence to $a$. □

The proof of (ii) can be derived using the condition (2) of Theorem 1 (ii).

Under condition (1) of Theorem 1 (i), the overall picture regarding inclusions among the already existing spaces $S_f^{\sigma }$, $S^{\sigma }$, $S$, $S_f,$ and the newly introduced space $S_g^{\sigma }$ is as shown below:

$$\begin{gathered} \begin{array}{ccc}{ S}^{\sigma }& \subset & S\end{array} \begin{array}{cc}\supset & c\end{array} \\ \begin{array}{ccc}\bigcap & & \bigcap \end{array} \\ \begin{array}{ccc}\begin{array}{ccc}{S}_g^{\sigma }& \subset & S_f^{\sigma }\end{array}& \subset & S_f\end{array} \end{gathered}$$

Definition 9. 

A sequence $u=\left(u_k\right)$ in $X$ is categorized as exhibiting $f_{\sigma }-$statistical Cauchy behavior if, for any $ϵ>0$, there exists a positive integer $n=n\left(ϵ\right)$ such that

$$d_f^{\sigma }\left(\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-u_n\right|\ge ϵ\right\}\right)=0,uniformly in i.$$

Here, $f$ represents an unbounded modulus function.

Theorem 3. 

(i) In the event that condition (1) is met, a $g_{\sigma }-$statistically Cauchy sequence also holds the status of being an $f_{\sigma }-$statistically Cauchy sequence.

(ii) Conversely, when condition (2) is fulfilled, a sequence $\left(u_k\right)$ qualifies as a $g_{\sigma }-$statistically Cauchy sequence if and only if it also aligns with the criteria for being an $f_{\sigma }-$statistically Cauchy sequence. In this context, the functions $f$ and $g$ symbolize unbounded modulus functions.

Definition 10. 

A number sequence $u=\left(u_k\right)$ is considered $f_{\sigma }-$statistically bounded if there exists an $M>0,$ such that

$$d_f^{\sigma }\left(\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k\right|>M\right\}\right)=0.$$

The space of all $f_{\sigma }-$statistically bounded sequences is symbolized by $S_{f_{\sigma }}\left(b\right)$. Here, $f$ represents an unbounded modulus function.

Theorem 4. 

Any sequence that is $f_{\sigma }-$statistically convergent is necessarily $f_{\sigma }-$statistically bounded. Nevertheless, the reverse implication does not always hold.

Proof. 

The result shows that

$$\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k\right|>\left|L\right|+ϵ\right\}\subset \left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-a\right|>ϵ\right\}.$$

Regarding the converse aspect, selecting $f\left(u\right)=u$, the identity map, and define the sequence $u=\left(u_k\right)$ by $u_k={\left(-1\right)}^k=\left(-1, 1,-1, 1,-\mathrm{1,1}, \dots \right)$. For all $k\in \mathbb{N},$ we have $\left(u_k\right)\in s_{f_{\sigma }}\left(b\right)$, but $\left(u_k\right)\notin s_{f_{\sigma }}$, the space of $f_{\sigma }-$statistically convergent sequences of scalars. □

Example 1. 

Consider the function $f\left(u\right)=log(u+1)$ and the sequence $\left(u_k\right)=(1, 0, 0, 4, 0, 0, 0, 0, 9,\dots )$. Let $A=\{\mathrm{1,4},9,\dots \}$, the set of squares of natural numbers. For any $M>0$,

$$\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k\right|>M\right\}=A-a$$

is a finite subset of $\mathbb{N}$. Since $d_f^{\sigma }\left(A\right)=1/2\ne 0$ and $d\left(A\right)=0$, $\left(u_k\right)\notin s_{f_{\sigma }}\left(b\right)$ and $\left(u_k\right)\in s\left(b\right)$. Consequently, $s_{f_{\sigma }}\left(b\right)⊊s\left(b\right)$.

Example 2. 

Let $X=\mathbb{C}$, the space of complex numbers, and $f\left(u\right)=u^t$ with $0<t\le 1$. Consider the sequence $\left(u_k\right)=(\mathrm{1,0},\mathrm{0,4},\mathrm{0,0},\mathrm{0,0},9,\dots )$. Now

$$d_f^{\sigma }\left(\left\{\sigma \left(i\right)\le m\le {\sigma }^p\left(i\right):\left|u_m-0\right|>ϵ\right\}\right)=d_f^{\sigma }\left(A\right)$$

for every $ϵ>0$ where $A=\{\mathrm{1,4},9,...\}$. Then, $\left|A\right(k\left)\right|=\left|\right\{n\le k:n\in A\left\}\right|\le \sqrt{k}$ for every $k\in \mathbb{N}$, and therefore,

$${\displaystyle \frac{f\left(\left|A\left(k\right)\right|\right)}{f\left(k\right)}}\le {\displaystyle \frac{{\left(k^{\frac{1}{2}}\right)}^t}{k^t}}\to 0 \mathrm{a}\mathrm{s} \mathrm{k}\to \mathrm{\infty };$$

that is, $d_f^{\sigma }\left(A\right)=0$. Hence, $\left(u_k\right)$ is $f_{\sigma }-$statistically convergent; otherwise, $(\mathrm{1,4},9,...)$ is a subsequence of $\left(u_k\right)$, which is not $f_{\sigma }-$statistically convergent.

Theorem 5. 

Every bounded sequence is $f_{\sigma }-$statistically bounded, but the converse need not be true.

Proof. 

The result shows that the empty set has zero $f_{\sigma }-$density for every unbounded modulus $f$. Regarding the opposite aspect, the sequence $u=\left(u_k\right)=(\mathrm{1,0},\mathrm{0,4},\mathrm{0,0},\mathrm{0,0},9,...)$ of Example 2, the purpose. □

Theorem 6. 

(i) When condition (1) is fulfilled, a sequence that is $g_{\sigma }-$statistically bounded is simultaneously $f_{\sigma }-$statistically bounded. In other words, the set $S_{g_{\sigma }}\left(b\right)$ is contained within $S_{f_{\sigma }}\left(b\right)$.

(ii) When condition (2) is met, a sequence attains $g_{\sigma }-$statistical boundedness if and only if it achieves $f_{\sigma }-$statistical boundedness. In this case, the sets $S_{g_{\sigma }}\left(b\right)$ and $S_{f_{\sigma }}\left(b\right)$ are equivalent.

Proof. 

Consider the sequence $\left(u_k\right)$ being $g_{\sigma }-$statistically bounded. This implies the existence of a real number $M>0,$ such that

$$d_g^{\sigma }\left(\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k\right|>M\right\}\right)=0.$$

By taking

$$K=\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k\right|>M\right\},$$

the verifications for (i) and (ii) can be inferred from Theorem 1 (i) and (ii) in the cited reference [14], correspondingly. □

Corollary 2. 

The following is true for each $f$ unbounded modulus function:

(i) $S_{f_{\sigma }}\left(b\right)\subseteq S_{\sigma }\left(b\right)$;

(ii) If condition (2) is satisfied, then $S_{f_{\sigma }}\left(b\right)=S_{\sigma }\left(b\right)$.

Proof. 

(i) is derived due to the reality that “given a set $K\subseteq \mathbb{N}$, $d_f^{\sigma }\left(\mathrm{K}\right)=0$ implies $d\left(K\right)=0$ for any unbounded modulus f”, while (ii) is based on Corollary 1. □

Theorem 7. 

If condition (1) is satisfied, then a $g_{\sigma }-$statistically convergent sequence is also $f_{\sigma }-$statistically bounded, which means $S_{g_{\sigma }}\left(b\right)=S_{f_{\sigma }}\left(b\right)$.

Corollary 3. 

A sequence that achieves $f_{\sigma }-$statistical convergence also demonstrates statistical bounded denoting that the set $S_{f_{\sigma }}$ is a subset of $S_{\sigma }\left(b\right)$ for any unbounded modulus $f$.

Definition 11. 

A sequence $u=\left(u_k\right)$ is said to be statistically $f_{\sigma }-$convergent to $L$ if, for every $ϵ>0,$ the set $K_{ϵ}\left(f\right):=\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):f\left(\left|u_k-L\right|\right)\ge ϵ\right\}\right|\right)$ has natural density zero, i.e., $\delta \left(K_{ϵ}\left(f\right)\right)=0$. We can write $\delta \left(f_{\sigma }\right)-limx=L.$

This signifies that

$$\underset{n\to \infty }{\mathit{lim}}{\displaystyle \frac{f\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|t_{pi}-L\right|\ge ϵ\right\}\right|\right)}{f\left(n\right)}}=0, uniformly in i.$$

Remark 1. 

(i) The sequence $u$ exhibits $f_{\sigma }-$statistical convergence. This implies that $u$ is also statistically convergent and satisfies $f_{\sigma }\left(st\right)-\mathit{lim}u=st-lim u$.

(ii) The notion of statistical convergence implies $f_{\sigma }-$statistical convergence, which is established by (i).

(iii) While $\sigma -$convergence ensures statistical $f_{\sigma }-$convergence, it does not guarantee $f_{\sigma }-$statistical convergence.

Examples 3. 

Consider the set $P$ comprising all prime numbers, and let $f\left(u\right)=u$. Define the sequence $u=\left(u_k\right)$ by

$$u_k=\left\{\begin{array}{c}1 ; if k\in P, \\ 0 ; otherwise. \end{array}\right.$$

In this case, $u_k$ is not convergent; however, it demonstrates $f_{\sigma }-$statistical convergence due to the property that $d_f^{\sigma }\left(P\right)=0$. As a result, based on Remark 1 (ii), it becomes both statistically convergent and statistically $f_{\sigma }-$convergent.

Examples 4. 

The sequence $u=\left(u_k\right)$ and the function $f\left(u\right)=u$, defined as

$$u_k=\left\{\begin{array}{c} 1 ; if k is odd,\\ 0 ; if k is even,\end{array}\right.$$

$\sigma -$converges to $1/2$ (where $\sigma \left(n\right)=n+1$). Consequently, it is statistically $f_{\sigma }-$convergent to $1/2$. However, it does not exhibit both statistical convergence and $f_{\sigma }-$statistical convergence.

Definition 12. 

A sequence $u=\left(u_k\right)$ is considered to be strongly $f_{{\sigma }_q}-$convergent (where $0<q<\infty )$ to the limit $L$ if

$$\underset{p\to \infty }{\mathit{lim}}{\displaystyle \frac{1}{f\left(p\right)}}\sum _{j=1}^{p}f\left({\left|u_{{\sigma }^j\left(i\right)}-L\right|}^q\right)=0, uniformly in i,$$

and this is denoted as $u_k\to L{\left[V_{f_{\sigma }}\right]}_q$. In this context, $L$ is referred to as the ${\left[V_{f_{\sigma }}\right]}_q-limit$ of $u$. It is important to emphasize that when $q=1$, ${\left[V_{\sigma }\right]}_q\cap l_{\infty }=\left[V_{\sigma }\right]$.

The spaces we give in Definition 8, Definition 11, and Definition 12 are quite general. By making special choices of $\sigma$, f and q, we obtain some spaces that have been studied before. For example;

If we take $f\left(u\right)=u$ in Definition 8 and Definition 12, we obtain the concepts of $\sigma -$statistical convergence and strong ${\sigma }_q-$convergence, which were defined and studied by Mursaleen and Edely in [24], respectively.

Theorem 8. 

Assume that a sequence $u=\left(u_k\right)$ is strongly $f_{{\sigma }_q}-$convergent to the limit $L$, with $0<q<\infty .$ In this case, the sequence is also $f_{\sigma }-$statistically convergent to $L.$

Proof. 

When $0<i<\mathrm{\infty }$ and $u_k\to \mathrm{L}{\left[V_{\sigma }\right]}_q$, then as $p\to \mathrm{\infty }$,

$$0\leftarrow {\displaystyle \frac{1}{p}}\sum _{j=1}^p{\left|u_{{\sigma }^j\left(i\right)}-L\right|}^q\ge {\displaystyle \frac{{ϵ}^q}{p}}{K}_{\sigma }\left(i+1, i+p\right)\ge {\displaystyle \frac{{ϵ}^q}{p}}{N}^p.$$

In other words, $\underset{p\to \mathrm{\infty }}{\lim }{\displaystyle \frac{N^p}{p}}=0$ and so $d_f^{\sigma }\left(K_f^{\sigma }\left(ϵ\right)\right)=0$, where

$$K_f^{\sigma }\left(ϵ\right):=f\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-L\right|\ge ϵ\right\}\right|\right).$$

Thus, the sequence $u=\left(u_k\right)$ is $f_{\sigma }-$statistically convergent to $L$. □

Theorem 9. 

If a sequence $u=\left(u_k\right)$ is $f_{\sigma }-$statistically convergent to $L$ and bounded, it is also statistically $f_{\sigma }-$convergent to $L$. However, the reverse is not necessarily true.

Proof. 

When a sequence is $f_{\sigma }-$statistically convergent to $L$ and $u=\left(u_k\right)$ is bounded, it can be deduced that $K_f^{\sigma }\left(ϵ\right):=f\left(\left|\left\{\sigma \left(i\right)\le k\le {\sigma }^p\left(i\right):\left|u_k-L\right|\ge ϵ\right\}\right|\right).$ Subsequently,

$$\begin{gathered} \left|t_{pi}-L\right|=\left|{\displaystyle \frac{1}{p}}\sum _{j=1}^p{u}_{{\sigma }^j\left(i\right)}-L\right|=\left|{\displaystyle \frac{1}{p}}\sum _{k=\sigma \left(i\right)}^{{\sigma }^p\left(i\right)}\left(u_k-L\right)\right| \\ \le {\displaystyle \frac{1}{p}}\left(\underset{k}{\mathit{sup}}\left|u_k-L\right|\right)\left(K_f^{\sigma }\left(i+1, i+p\right)\right) \\ ={\displaystyle \frac{1}{p}}\left(\underset{k}{\mathit{sup}}\left|u_k-L\right|\right)N^p\to \infty \mathrm{a}\mathrm{s} \mathrm{p}\to \mathrm{\infty }. \end{gathered}$$

This leads to the deduction that $t_{pi}\to L$ as $\mathrm{p}\to \mathrm{\infty }$ uniformly in $i$. Consequently, $u$ displays $f_{\sigma }-$convergence to $L$ and concurrently, it manifests statistical $f_{\sigma }-$convergence to $L$.

Now, considering the opposite scenario, let us assume $\sigma \left(n\right)=n+1$, and let the sequence $u=\left(u_k\right)$ and the function $f\left(u\right)=u$ be defined as

$$u_k=\left\{\begin{array}{c} 1 ; if k is odd,\\ -1 ; if k is even.\end{array}\right.$$

Hence, this sequence is not $f_{\sigma }-$statistically convergent. However, $u$ is $f_{\sigma }-$convergent to $0$ and, hence, statistically $f_{\sigma }-$convergent to $0$. □

Theorem 10. 

Suppose $f_{\sigma } is$ statistically convergent to $L$ and $u=\left(u_k\right)$ is bounded. Then, $u_k\to L{\left[V_{\sigma }\right]}_q$.

Proof. 

Suppose that $f_{\sigma } \mathrm{is}$ statistically convergent to $L$ and $u=\left(u_k\right)$ is bounded. Then, for $ϵ>0$, we have $d_f^{\sigma }\left(K_{ϵ}\right)=0$. Since $u\in l_{\infty }$, there is $M>0,$ such that $\left|u_k-L\right|\le M (\mathrm{k}=\mathrm{1,2},\dots )$. For every $i\in \mathbb{N}$, we get

$$\begin{gathered} {\displaystyle \frac{1}{p}}\sum _{j=1}^p{\left|u_{{\sigma }^j\left(i\right)}-L\right|}^q={\displaystyle \frac{1}{p}}\sum _{\begin{array}{c}k=\sigma \left(i\right)\\ k\notin K_{ϵ}\end{array}}^{{\sigma }^p\left(i\right)}{\left|u_k-L\right|}^q+{\displaystyle \frac{1}{p}}\sum _{\begin{array}{c}k=\sigma \left(i\right)\\ k\in K_{ϵ}\end{array}}^{{\sigma }^p\left(i\right)}{\left|u_k-L\right|}^q \\ =G_1(i,p)+G_2(i,p), \end{gathered}$$

where $G_1\left(i,p\right)={\displaystyle \frac{1}{p}}\sum _{\begin{array}{c}k=\sigma \left(i\right)\\ k\notin K_{ϵ}\end{array}}^{{\sigma }^p\left(i\right)}{\left|u_k-L\right|}^q$ and $G_2\left(i,p\right)={\displaystyle \frac{1}{p}}\sum _{\begin{array}{c}k=\sigma \left(i\right)\\ k\in K_{ϵ}\end{array}}^{{\sigma }^p\left(i\right)}{\left|u_k-L\right|}^q$.

Now, if $k\notin K_{ϵ},$ then $G_1\left(i,p\right)<{ϵ}^q$. For $k\in K_{ϵ}$, the expression

$$G_2\left(i,p\right)\le \left(sup\left|u_k-L\right|\right)\left(\underset{m\ge 0}{\mathit{max}}{K}_{\sigma }\left(i+1, i+p\right)/p\right)\le \mathrm{M}{\displaystyle \frac{N^p}{p}}\to 0,$$

holds true as $p$ approaches infinity, due to the fact that $d_f^{\sigma }\left(K_f^{\sigma }\left(ϵ\right)\right)=0$. Consequently, $u_k\to L{\left[V_{\sigma }\right]}_q$. □

Theorem 11. 

A sequence $u=\left(u_k\right)$ is statistically $f_{\sigma }-$convergent to $L$ if and only if there exists a set $K=\{k_1<k_2<...<k_n<...\}\subseteq \mathbb{N},$ such that its natural density $d\left(K\right)=1$, and the $f_{\sigma }-lim{u}_{k_n}=L$.

Proof. 

Suppose there exists a set $K=\{k_1<k_2<...<k_n<...\}\subseteq \mathbb{N}$ such that its natural density $d\left(K\right)$ is $1$ and $\sigma -\mathrm{l}\mathrm{i}\mathrm{m}{u}_{k_n}=L$. In this situation, let $\mathbb{N}$ be a positive integer, such that for $n>\mathbb{N}$,

$$\phi \left(\left|u_k-L\right|\right)<ϵ.$$

(4)

Define $\left(K_f^{\sigma }\left(ϵ\right)\right)\left(\phi \right):=\left\{n\in \mathbb{N}:\phi \left(\left|u_k-L\right|\right)\ge ϵ\right\}$ and let $K^{\prime }=\left\{k_{N+1},k_{N+2},\dots \right\}$. Then, $d\left(K^{\prime }\right)=1$ and $K\left(\phi \right)\subseteq \mathbb{N}-K^{\prime },$ which provides that $d\left(\left(K_f^{\sigma }\left(ϵ\right)\right)\left(\phi \right)\right)=0$. Therefore, $u=\left(u_k\right)$ is statistically $f_{\sigma }-$convergent to $L$.

On the flip side, suppose $u=\left(u_k\right)$ is statistically $f_{\sigma }-$convergent to $L$. For each positive integer $r (r=\mathrm{1,2},3,...),$ define

$$K_r\left(\phi \right):=\left\{j\in \mathbb{N}:\phi \left(\left|u_{k_j}-L\right|\right)\ge 1/r\right\} \mathrm{and} M_r\left(\phi \right):=\left\{j\in \mathbb{N}:\phi \left(\left|u_{k_j}-L\right|\right)<1/r\right\}.$$

Consequently, we have $d\left(K_r\left(\phi \right)\right)=0$, as well as

$$M_1\left(\phi \right)\supset M_2\left(\phi \right)\supset \dots \supset M_i\left(\phi \right)\supset M_{i+1}\left(\phi \right)\supset$$

(5)

and

$$d\left(M_r\left(\phi \right)\right)=1 r=1, 2, 3, \dots$$

(6)

It is essential to emphasize that for ${j\in M}_r\left(\phi \right)$, the sequence ($u_{k_j}$) is $\sigma -$convergent to $L$. Suppose, for the sake of contradiction, that ($u_{k_j}$) is not $\sigma -$convergent to $L.$ This implies the existence of $\epsilon >0,$ such that $\phi \left(\left|u_{k_j}-L\right|\right)\ge \epsilon$ for an infinite number of terms. Now, define

$$M\left(\phi \right):=\left\{j\in \mathbb{N}:\phi \left(\left|u_{k_j}-L\right|\right)<\epsilon \right\},$$

and select $\epsilon >1/r$ (where $r=\mathrm{1,2},3,\dots$), resulting in

$$d\left(M_{ϵ}\left(\phi \right)\right)=0.$$

Using the relationship stated in Equation (5), we can establish that $M_r\left(\phi \right)\subset M\left(\phi \right)$. Consequently, it follows that $d\left(M_r\left(\phi \right)\right)=0$, which contradicts Equation (6). As a result, the assumption that ($u_{k_j}$) is not $\sigma -$convergent to $L$ leads to a contradiction, confirming that ($u_{k_j}$) is indeed $\sigma -$convergent to $L$. □

We give the following theorem without proof.

Theorem 12. 

(i) $S_{f_{\sigma }}\left(b\right)$ is solid and, therefore, monotone.

(ii)$S_{f_{\sigma }}\left(b\right)$ is a sequence algebra.

(iii) $S_{f_{\sigma }}\left(b\right)$ is not symmetric, generally.

3. Conclusions

The framework developed here has direct implications for analyzing noisy or high-dimensional data in machine learning. For instance, in time-series forecasting (e.g., stock prices, sensor networks), sequences often contain sparse outliers or non-stationary patterns. By using modulus functions to weight deviations from a trend, researchers can design more robust algorithms that distinguish between meaningful signals and negligible noise. Additionally, the theory may aid in statistical physics by modeling particle trajectories that exhibit rare but significant deviations from average behavior.

The conclusion’s suggestion to combine modulus functions with fuzzy set theory or uncertainty spaces holds particular promise for addressing semantic uncertainties in natural language processing or social science data. Such integration could enable the framework to handle not only numerical sparsity but also qualitative ambiguities, thereby expanding its interdisciplinary applicability.

In addition, extending the framework of f-convergence to high-dimensional sequences or stochastic processes (such as martingales) would significantly enhance its utility for modeling complex systems. This extension could address challenges in multi-variate time-series analysis or financial modeling, where data exhibits both dimensional complexity and probabilistic dynamics. Furthermore, the conclusion’s suggestion to combine modulus functions with fuzzy set theory or uncertainty spaces holds particular promise for addressing semantic uncertainties in natural language processing and social science data. Such an integration could enable the framework to handle not only numerical sparsity but also qualitative ambiguities, thereby expanding its interdisciplinary applicability.