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Let 0 < p < 2 and θ > 0. Let { X , X n ; n ≥ 1 } be a sequence of independent and identically distributed B -valued random variables and set S n = ∑ i = 1 n X i , n ≥ 1. In this note, a general logarithmic asymptotic behavior for { S n ; n ≥ 1 } is established. We show that if S n / n 1 / p → P 0 , then, for all s > 0 , lim sup n → ∞ log P ‖ S n ‖ > s n 1 / p (log n) θ = − p − θ ζ ¯ (p , θ) , lim inf n → ∞ log P ‖ S n ‖ > s n 1 / p (log n) θ = − p − θ ζ ̲ (p , θ) , where ζ ¯ (p , θ) = − lim sup t → ∞ log e p t P (log ‖ X ‖ > t) t θ and ζ ̲ (p , θ) = − lim inf t → ∞ log e p t P (log ‖ X ‖ > t) t θ . The main tools used to prove this result are the symmetrization technique, an auxiliary lemma for the maximum of i.i.d. random variables, a moment inequality, and an exponential inequality. [ABSTRACT FROM AUTHOR]

This paper presents a general result that allows for establishing a link between the Kolmogorov–Marcinkiewicz– Zygmund strong law of large numbers and Feller’s strong law of large numbers in a Banach space setting. Let { X , X n ; n ≥ 1 } be a sequence of independent and identically distributed Banach space valued random variables and set S n = ∑ i = 1 n X i , n ≥ 1 . Let { a n ; n ≥ 1 } and { b n ; n ≥ 1 } be increasing sequences of positive real numbers such that lim n → ∞ a n = ∞ and b n ∕ a n ; n ≥ 1 is a nondecreasing sequence. We show that S n − n E X I { ‖ X ‖ ≤ b n } b n → 0 almost surely for every Banach space valued random variable X with ∑ n = 1 ∞ P ( ‖ X ‖ > b n ) < ∞ if S n ∕ a n → 0 almost surely for every symmetric Banach space valued random variable X with ∑ n = 1 ∞ P ( ‖ X ‖ > a n ) < ∞ . To establish this result, we invoke two tools (obtained recently by Li, Liang, and Rosalsky): a symmetrization procedure for the strong law of large numbers and a probability inequality for sums of independent Banach space valued random variables. [ABSTRACT FROM AUTHOR]

In this note, general symmetrization procedures for both the weak and strong laws of large numbers concerning the row sums from arrays of rowwise independent Banach space valued random variables are established. Necessary and sufficient conditions are provided for the weak and strong laws of large numbers to hold in terms of corresponding laws of large numbers for a symmetrized version of the array. Several corollaries of the main result are provided. [ABSTRACT FROM AUTHOR]

Let { X , X n ; n ≥ 1 } be a sequence of i.i.d. real-valued random variables, and let S n = ∑ i = 1 n X i , n ≥ 1 . Write log x = log e ( e ∨ x ) , x ≥ 0 . In this note we establish a limit theorem which is related to the classical Hartman–Wintner–Strassen law of the iterated logarithm and the classical Chover law of the iterated logarithm. That is, for 1 ≤ p < ∞ , we show that lim sup n → ∞ | S n n | ( log log log n ) − 1 = e p / 2 almost surely if and only if E X = 0 and inf { b > 0 : lim x → ∞ ( log log x ) 1 − b E ( X 2 I ( | X | ≤ x ) ) = 0 } = p . [ABSTRACT FROM AUTHOR]

In this paper, a general large deviation result for partial sums of independent and identically distributed random variables with super-heavy tailed distribution is established. Our main result extends in particular the results of Stoica (2008) and Nakata (2022). The symmetrization technique, one of Lévy's inequalities, and two preliminary results on slowly and regularly varying functions are paramount in the proof of our main result. [ABSTRACT FROM AUTHOR]

Abstract: In a recent paper by Chen (in press) the limit law of the iterated logarithm for the partial sums of i.i.d. real-valued random variables has been established. In this note we look at the corresponding problem in Banach space setting. Let be a real separable Banach space with topological dual . Let be a sequence of i.i.d. -valued random variables and set and . We show that provided that where and is the unit ball of . [Copyright &y& Elsevier]

Abstract: Let be a sequence of i.i.d. random variables, and set . For , if , and , we prove that thus extending a result by . For and , we establish conditions for the convergence of the related series and derive its precise asymptotic as . [Copyright &y& Elsevier]

Abstract: We prove a Kolmogorov–Feller weak law of large numbers for exchangeable sequences, under a second order hypothesis on the truncated mixands. [Copyright &y& Elsevier]

Abstract: For a sequence of i.i.d. mean 0 random variables with weighted partial sums where is a Lipschitz function of order 1 with , necessary and sufficient conditions are provided for to enjoy iterated logarithm type behavior of the form almost surely where is a positive, nondecreasing function which is slowly varying at infinity. Some corollaries are presented for particular choices of . [Copyright &y& Elsevier]

Abstract: Let k and m be two fixed positive integers with and . Let which is the th smallest value of k real numbers . We call this f the th order function of k real numbers . Let be a nonconstant random variable with cumulative distribution . Let be the unique solution in (0, 1) to the equation where , which is the (cumulative) Beta distribution function with parameters m and . Write and . Define the hierarchical sequence of random variables by with being independent random variables identically distributed as . In this note it is shown that and a.s. and a.s. It follows that a.s. if and only if . This result generalizes and improves Propositions 4.3 and 4.4 of Li and Rogers [1999. Asymptotic behavior for iterated functions of random variables. Ann. Appl. Probab. 9, 1175–1201]. [Copyright &y& Elsevier]

Abstract: Let be i.i.d. random variables with partial sums , put , and assume there exist functions g and h, such that whenever and . We prove the converse result, namely that and imply and . [Copyright &y& Elsevier]

Abstract: We consider an investment model based on MacQueen–Heyde martingales and find the asymptotics of the expected gains for regular or informed investors. [Copyright &y& Elsevier]

Medical devices utilize auxetic tubes with a negative Poisson's ratio in esophageal and vascular stents to guard against embolism. Designing stents with auxetic structures provides several advantages compared to traditional stents. Therefore, the current investigation focuses on analyzing the small/large amplitude vibration of the pre/post-buckled geometrically imperfect re-entrant auxetic tube. In order to achieve this, von-Kármán nonlinear assumption along with the first order shear deformation Timoshenko beam theory are employed to obtain the governing equations of the auxetic tube with geometrical imperfection subjected to axial mechanical load. These motion equations are established by applying the Hamilton's principle. Based on Malek-Gibson relations, the effective elastic characteristics of the auxetic honeycomb tube are extracted. Then, the mechanical properties of polylactic acid (PLA) biopolymer are derived experimentally through tensile tests and the digital image correlation (DIC) method. In addition, to investigate the factors affecting the difference in Young's modulus as well as ultimate tensile strength values in dogbone specimens, scanning electron microscopy (SEM) images are taken. The nonlinear equations, using the Ritz method with Chebyshev polynomials, are discretized. The direct displacement control strategy and the Newton–Raphson iterative method are applied to assess the small/large amplitude vibration of the re-entrant auxetic tube. After verifying the created mathematical model, a detailed analysis is undertaken to evaluate the impact of cell angle, vertical to inclined cell wall length ratio, cell thickness to inclined cell wall length ratio, the proportion between the inner and the outer radius of the tube, imperfection parameter, and edge supports on the vibrational behavior of the tubular metastructure. [ABSTRACT FROM AUTHOR]