Statistical Order Convergence

• Statistical order convergence is a framework that generalizes classical order convergence by relaxing inequality conditions outside sets of negligible measure, enabling robust analysis in Riesz spaces.
• It unifies density-based convergence with operator methods, extending to nets, fuzzy numbers, and partial metric spaces to address cases where uniform tail decay fails.
• Applications include stability analysis of order-bounded operators, convergence characterizations in Banach lattices, and innovative approaches to summability in functional analysis.

Statistical order convergence generalizes classical notions of order convergence in vector lattices (Riesz spaces) and related structures by requiring that the inequalities governing convergence hold outside sets of "small" measure, typically quantified by asymptotic density or finitely additive measures. This paradigm allows for limiting behavior where classical uniform tail decay fails, but exceptional indices are sparse. Recent developments include the extension to net-indexed families via directed-set measures, analysis in operator spaces, connections to lattice operations, and adaptation to partial metric spaces and fuzzy numbers.

1. Statistical Order Convergence in Riesz Spaces

Let $E$ be a Riesz space and $(A,\le)$ an infinite directed set. The classical order convergence requires, for a net $(x_a)_{a\in A}\subset E$ convergent to $x\in E$, the existence of a tail net $(y_a)_{a\in A}\subset E^+$ with $\inf_a y_a=0$ such that $|x_a-x|\le y_a$ for all $a$. Statistical order convergence alters this by introducing a directed-set measure $(\mathcal{M},p)$: a finitely additive measure on an interval-field $\mathcal{M}\subset\mathcal{P}(A)$, with $p(A)=1$ and $p([a,b])=0$ for each finite interval. The statistical order convergence to $x$ (denoted $x_a\xrightarrow{\mathrm{st}_p\text{-}\mathrm{o}}x$) requires

• a full-measure tail set $E\in\mathcal{M}$, $p(E)=1$,
• a net $(r_a)_{a\in E}\subset E^+$ with $\inf_{a\in E} r_a=0$, $r_a$ $p$-statistically monotone decreasing to $0$, such that $|x_a-x|\le r_a$ for all $a\in E$ (Aydın et al., 2021).

This structure admits convergence even for nets whose exceptional indices form sets of measure zero but infinite cardinality.

2. Core Properties and Relationships

Statistical order convergence is strictly weaker than classical order convergence except in monotone or order-bounded cases. If $(x_a)$ is order convergent, then $x_a\xrightarrow{\mathrm{o}}x$ implies $x_a\xrightarrow{\mathrm{st}_p\text{-}\mathrm{o}}x$. The converse generally fails unless $(x_a)$ is monotone or the space is Dedekind complete and $(x_a)$ is order bounded (Aydın et al., 2021):

• Monotonicity: If $(x_a)$ is monotone and $p$-statistically order convergent, then classical order convergence follows ($x_a\xrightarrow{\mathrm{o}}x$).
• Order-bounded monotone nets: In Dedekind complete spaces, any order-bounded monotone net is $p$-statistically order convergent.
• Counterexample: The sequence $(e_1,0,e_2,0,e_3,0,\dots)$ in $c_0$ is statistically order convergent but not order convergent since the set of zero terms has density $1/2$ and the net is not order-bounded.
• Lattice operations: Statistical order convergence is preserved under lattice operations; for nets $x_a\to x$ and $w_a\to w$, $$x_a\vee w_a\xrightarrow{\mathrm{st}_p\text{-}\mathrm{o}} x\vee w$$, and similarly for $\wedge$, modulus, positive, and negative parts.

3. Statistical Order Convergence for Operators

In the context of order bounded linear operators $R_n$ between Riesz spaces, statistical order convergence is defined using natural density on $\mathbb{N}$ (Aydın et al., 27 Dec 2025):

• A sequence $(R_n)$ converges statistically in order to $R$ if there exists a control sequence $(S_n)$ statistically order decreasing to $0$ and a set $J\subseteq\mathbb N$ of density one such that $|R_j-R|\le S_j$ for all $j\in J$.
• Pointwise variant: For each $u$ in the positive cone, the pointwise modulus $|R_j(u)-R(u)|$ is controlled by $S_j(u)\downarrow_{st}0$ outside a density-zero exceptional set.
• Uniqueness and stability: The statistical order limit is unique, and lattice operations are stable under statistical order convergence.

A key result provides a density-subsequence characterization: statistical order convergence is equivalent to the existence of a subsequence of density one that is classically order convergent to the same limit.

4. Extensions: Deferred and Unbounded Statistical Order

Deferred statistical order convergence, studied in (Küçükaslan et al., 2022), replaces asymptotic density with deferred density calculated over moving intervals $[p_n+1,q_n]$, permitting further flexibility and capturing lacunary/block behaviors. The deferred variant sits strictly between statistical and classical order convergence, recovering the usual case for bounded interval lengths.

Unbounded statistical order convergence (Wang et al., 2019) generalizes in Banach lattices and Riesz spaces via truncations:

• A net $(x_\alpha)$ is statistically unbounded order convergent to $x$ if for all $u\in E^+$, $|x_\alpha-x|\wedge u$ is statistically order convergent to $0$.
• This unifies statistical order convergence, unbounded order convergence (uo), and statistical unbounded norm/absolute weak convergence.

KB-space and order continuity in Banach lattices can be characterized in terms of statistical (unbounded) Cauchy sequences.

5. Order Convergence in Generalized and Weighted Settings

Statistical order convergence notions extend to metric spaces, partial metric spaces (Bayram et al., 2023), and sequences of fuzzy numbers (Altinok et al., 2016):

• In partial metric spaces $(X,p)$, statistical convergence of order $\alpha$ is defined relative to the self-distance $p(L,L)$, with inclusion theorems relating Cesàro, classical, and $\lambda$-statistical types.
• For fuzzy numbers, $f$-statistical convergence of order $\beta$ uses an unbounded modulus function $f$ in the density normalization:

$$d_f^{\beta}(A) = \lim_{n\to\infty} \frac{f\left(|\{k\le n : k \in A\}|\right)}{f(n^\beta)},$$

and convergence to $X_0$ requires $$d_f^\beta\left(\{k: d(X_k,X_0)\ge\varepsilon\}\right)=0$$ for all $\varepsilon>0$ (Altinok et al., 2016).

• Inclusion and ordering theorems hold for various parameters ($\beta$, modulus $f$), with distinct hierarchies for weighted/statistical Cesàro summability and $f$-statistical convergence.
• Examples exhibit strictness of inclusions and non-uniqueness in certain parameter regimes (e.g., when $\beta>1$ for $f$-statistical convergence).

6. Statistical Order Convergence: Applications and Implications

The statistical order convergence framework:

• Unifies and extends classical order convergence, summability, and density-based methods in vector lattices and spaces of functions.
• Provides robust tools for the analysis of limit behavior where uniform tail decay is not attainable, relevant in functional analysis, approximation theory, and statistical mechanics.
• Facilitates the study of operator theory in Banach lattices, including stability under composition and band/ideal closure.
• Admits generalization to partial metric spaces, fuzzy numbers, and spaces equipped with modulus functions, with applications in signal processing, approximation and ergodic theory, and probabilistic metric settings.
• Opens directions for the study of statistical nets, filters, duality, and relationships to locally solid topologies.

The flexibility of statistical order convergence allows the handling of convergence phenomena in which nontrivial configurations of exceptional indices dominate classical approaches, making it an essential tool for the contemporary theory of summability, function spaces, and lattice structures.