[Paper Review] Statistical convergence of order $α$ in probability

Pratulananda Das, Sanjoy Ghosal+1 more|SHILAP Revista de lepidopterología|May 18, 2016

Approximation Theory and Sequence SpacesMathematics29 references18 citations

TL;DR

This paper introduces and analyzes four new modes of convergence for sequences of random variables in probability: statistical convergence of order α in probability, strong p-Cesàro summability of order α in probability, lacunary statistical convergence (Sθ-convergence) of order α in probability, and Nθ-convergence of order α in probability. The key contribution is establishing conditions under which these convergence types imply almost sure equality of limits, particularly showing that if a sequence converges in both statistical and lacunary statistical sense of order α and β respectively, with liminf q_r > 1 and β ≤ α, then the limits are equal almost surely.

ABSTRACT

In this paper ideas of different types of convergence of a sequence of random variables in probability, namely, statistical convergence of order $α$ in probability, strong $p$-Ces$\grave{\mbox{a}}$ro summability of order $α$ in probability, lacunary statistical convergence or $S_θ$-convergence of order $α$ in probability, ${N_θ}$-convergence of order $α$ in probability have been introduced and their certain basic properties have been studied.

Motivation & Objective

• To extend the concept of statistical convergence to order α in the context of probability theory.
• To define and study four new summability methods: statistical convergence of order α in probability, strong p-Cesàro summability of order α in probability, lacunary statistical convergence (Sθ-convergence) of order α in probability, and Nθ-convergence of order α in probability.
• To establish relationships between these four convergence types and derive sufficient conditions for the almost sure equality of their limits.
• To generalize existing results in statistical convergence and summability theory in the probabilistic setting, particularly improving upon prior work by Bhunia et al., Colak, Fridy & Orhan, and others.

Proposed method

• Introduces the notion of α-natural density for subsets of natural numbers, extending classical statistical convergence to order α.
• Defines statistical convergence of order α in probability via the limit of the normalized count of indices where P(|X_n - X| ≥ ε) ≥ δ, scaled by n^α.
• Applies the concept of lacunary sequences (θ = {k_r}) to define Sθ-convergence of order α in probability, using h_r = k_r - k_{r-1} as the lacuna size.
• Uses the Cesàro mean-type summability with exponent p and order α to define strong p-Cesàro summability of order α in probability.
• Employs the Nθ-convergence method based on the ratio of the number of indices in lacunary intervals to the total index, scaled by h_r^α.
• Applies probabilistic inequalities and asymptotic analysis to compare convergence types and derive conditions for limit equivalence.

Experimental results

Research questions

• RQ1 Under what conditions does statistical convergence of order α in probability imply convergence in the lacunary statistical sense of order β, and vice versa?
• RQ2 What is the relationship between strong p-Cesàro summability of order α in probability and other convergence types?
• RQ3 How do the convergence modes of order α and β relate when the ratio q_r = k_r / k_{r-1} satisfies liminf q_r > 1?
• RQ4 Can a sequence converge in both statistical and lacunary statistical sense of different orders α and β, and when does this imply almost sure equality of the limits?
• RQ5 What are the necessary and sufficient conditions for the limit of a sequence to be unique across different convergence types of order α?

Key findings

• If a sequence of random variables converges in both statistical convergence of order α and lacunary statistical convergence of order β in probability, and liminf q_r > 1 with β ≤ α, then the limits are equal almost surely.
• The paper proves that the condition liminf q_r > 1 is necessary for the implication PS^α ⊂ PSθ^β to hold, and provides a counterexample when this condition fails.
• It is shown that the sequence defined with X_n = ±1 on lacunary intervals and X_n = 0 or 1 with small probability off these intervals is not Sθ-convergent of order β in probability, even if it is statistically convergent of order 1.
• The paper establishes that the limit of a sequence converging in probability via statistical convergence of order α is almost surely equal to the limit via lacunary statistical convergence of order β under the stated conditions.
• The authors demonstrate that the convergence modes are not equivalent and that the order α and β play a crucial role in determining the relationship between the convergence types.
• The proof technique relies on bounding the measure of sets where P(|X_n - X| ≥ ε) ≥ δ using α- and β-normalized densities, leading to a contradiction when the limit is not almost surely unique.