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In this paper, we consider the linear autoregressive model with varying coefficients θ n tending to the unit root. Cramér's moderate deviations of the least-squares estimator of the parameter θ n is discussed. [ABSTRACT FROM AUTHOR]

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]

In this paper, the weak laws of large numbers for weighted sums and random sums of H -valued dependent random vectors are established, which include and generalize some known results. As an application, the concept of CWOD random vectors is introduced and the corresponding weak laws of large numbers for weighted sums and random sums cases are obtained. Furthermore, to complete the work of Anh and Hien (2021), the weak law of large numbers for random sums of PCND random vectors is also presented. [ABSTRACT FROM AUTHOR]

In this paper we study the ratio of various order statistics based on samples from an exponential distribution and establish a central limit theorem and the almost sure central limit theorem for these statistics. [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 the present paper, we study the simple linear errors in variables (EV) model: , with i.i.d. errors . The consistency and asymptotic normality for the LS estimators and of the unknown parameters are obtained, which weaken some known conditions and improve some known results. Finally, the large deviation principle for and are given under the assumptions that possess normal distributions. [ABSTRACT FROM AUTHOR]

Abstract: In the paper, the law of the iterated logarithm for additive functionals of Markov chains is obtained under some weak conditions, which are weaker than the conditions of invariance principle of additive functionals of Markov chains in Maxwell and Woodroofe [2000. Central limit theorems for additive functionals of Markov chains. Ann. Probab. 28(2), 713–724]. [Copyright &y& Elsevier]

In the paper, the large deviation inequalities of the LS estimator for the nonlinear regression model with martingale differences errors are established. The assumptions for the errors are (conditional) exponential integrability which weaken the bounded condition in Hu (1993). As an application, we give the large deviation inequalities of LS estimator for the simple Michaelis–Menten model. [ABSTRACT FROM AUTHOR]

This paper investigates the convergence in L 2 of renormalized weighted quadratic variation associated to tempered fractional Brownian motion with Hurst index 0 < H < 1/4 and λ > 0. We first give four lemmas about tempered fractional Brownian motion by means of Malliavin calculus. Then the convergence for tempered fractional Brownian motion is derived in L 2 through these lemmas. Our main result extends findings of Nourdin (2008), concerning the weighted power variations of fractional Brownian motion. [ABSTRACT FROM AUTHOR]

In this paper, we consider the multi-dimensional Kesten algorithm studied in Xu and Dai (2012). The convergence in r th mean for the algorithm is established under the weaker assumptions than Xu and Dai (2012). Moreover, we obtain also the almost sure convergence of the algorithm under the conditions of Xu and Dai (2012). [ABSTRACT FROM AUTHOR]