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In 2018, Das et al. [Characterization of rough weighted statistical statistical limit set, Math. Slovaca 68(4) (2018), 881–896] (or, Ghosal et al. [Effects on rough 𝓘-lacunary statistical convergence to induce the weighted sequence, Filomat 32(10) (2018), 3557–3568]) established the result: The diameter of rough weighted statistical limit set (or, rough weighted 𝓘-lacunary limit set) of a sequence x = {xn}n∈ℕ is 2 r lim inf n ∈ A t n $\begin{array}{} \frac{2r}{{\liminf\limits_{n\in A}} t_n} \end{array}$ if the weighted sequence {tn}n∈ℕ is statistically bounded (or, self weighted 𝓘-lacunary statistically bounded), where A = {k ∈ ℕ : tk < M} and M is a positive real number such that natural density (or, self weighted 𝓘-lacunary density) of A is 1 respectively. Generally this set has no smaller bound other than 2 r lim inf n ∈ A t n $\begin{array}{} \frac{2r}{{\liminf\limits_{n\in A}} t_n} \end{array}$. We concentrate on investigation that whether in a θ-metric space above mentioned result is satisfied for rough weighted 𝓘-limit set or not? Answer is no. In this paper we establish infinite as well as unbounded θ-metric space (which has not been done so far) by utilizing some non-trivial examples. In addition we introduce and investigate some problems concerning the sets of rough weighted 𝓘-limit points and weighted 𝓘-cluster points in θ-metric space and formalize how these sets could deviate from the existing basic results. [ABSTRACT FROM AUTHOR]

In this paper we introduce rough weighted statistical limit set and weighted statistical cluster points set which are natural generalizations of rough statistical limit set and statistical cluster points set of double sequences respectively. Some new examples are constructed to ensure the deviation of basic results. Both the sets don't follow the usual extension properties which will be discussed here. [ABSTRACT FROM AUTHOR]

Our focus is to generalize the definition of the weighted statistical convergence in a wider range of the weighted sequence {tn}n∈ℕ. We extend the concept of weighted statistical convergence and rough statistical convergence to renovate a new concept namely, rough weighted statistical convergence. On a continuation we also define rough weighted statistical limit set. In the year (2008) Aytar established the following results: The diameter of rough statistical limit set of a real sequence is ≤ 2r (where r is the degree of roughness) and in general it has no smaller bound. If the rough statistical limit set is non-empty then the sequence is statistically bounded. If x∗ and c belong to rough statistical limit set and statistical cluster point set respectively, then |x∗ − c| ≤ r. We investigate whether the above mentioned three results are satisfied for rough weighted statistical limit set or not? Answer is no. So our main objective is to interpret above mentioned different behaviors of the new convergence and characterize the rough weighted statistical limit set. Also we show that this set satisfies some topological properties like boundedness, compactness, path connectedness etc. [ABSTRACT FROM AUTHOR]