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Let $(X _i)_{i\geq1}$ be a stationary sequence. Denote $m=\lfloor n^\alpha \rfloor, 0< \alpha < 1,$ and $ k=\lfloor n/m \rfloor,$ where $\lfloor a \rfloor$ stands for the integer part of $a.$ Set $S_{j}^\circ = \sum_{i=1}^m X_{m(j-1)+i}, 1\leq j \leq k,$ and $ (V_k^\circ)^2 = \sum_{j=1}^k (S_{j}^\circ)^2.$ We prove a Cram\'er type moderate deviation expansion for $\mathbb{P}( \sum_{j=1}^k S_{j}^\circ /V_k^\circ \geq x)$ as $n\to \infty.$ Applications to mixing type sequences, contracting Markov chains, expanding maps and confidence intervals are discussed., Comment: 30 pages

Let ( X i , Y i ) i ∈ Z be a stationary sequence of R 2 -valued random variables. To test if X 1 and Y 1 are correlated in the sense of Kendall, we propose a robust correction of the usual Kendall ...

A limiting curve of a stationary process in discrete time was defined by E. Janvresse, T. de la Rue, and Y. Velenik as the uniform limit of the functions$$ t\mapsto \left(S\left(t{l}_n\right)- tS\left({l}_n\right)\right)/{R}_n\in C\left(\left[0,1\right]\right), $$ where S stands for the piecewise linear extension of the partial sum, Rn:= sup |S(tln) − tS(ln))|, and (ln) = (ln(ω)) is a suitable sequence of integers. We determine the limiting curves for the stationary sequence (f ∘ Tn(ω)) where T is the dyadic odometer on {0, 1}ℕ and $$ f\left(\left({\omega}_i\right)\right)=\sum \limits_{i\ge 0}{\omega}_i{q}^{i+1} $$ for 1/2 < |q| < 1. Namely, we prove that for a.e. ω there exists a sequence (ln(ω)) such that the limiting curve exists and is equal to (−1) times the Tagaki–Landsberg function with parameter 1/2q. The result can be obtained as a corollary of a generalization of the Trollope–Delange formula to the q-weighted case.

The risk of catastrophes is related to the possibility of occurring extreme values. Several statistical methodologies have been developed in order to evaluate the propensity of a process for the occurrence of high values and the permanence of these in time. The extremal index. (Leadbetter in Z Wahrscheinlichkeitstheor Verw Geb 65:291306, 1983) allows to infer the tendency for clustering of high values, but does not allow to evaluate the greater or less amount of oscillations in a cluster. The estimation of. entails the validation of local dependence conditions regulating the distance between high levels oscillations of the process, which is difficult to implement in practice. In this work, we propose a smoothness coefficient to evaluate the degree of smoothness/oscillation in the trajectory of a process, with an intuitive reading and simple estimation. Application in some examples will be provided. We will see that, in a stationary sequence, it coincides with the tail dependence coefficient. (Sibuya in Ann Inst Stat Math 11:195-210, 1960; Joe in Multivariate models and dependence concepts. Monographs on statistics and applied probability, vol 73. Chapman and Hall, London, 1997), providing a new interpretation of the latter. This relationship will inspire a new estimator for., and its performance will be evaluated based on a simulation study. We illustrate with an application to financial series., The authors thank the reviewers and the associated editor for the important and valuable comments that contributed to the improvement of this work. The first author was partially supported by the research unit Centre of Mathematics and Applications of University of Beira Interior UIDB/00212/2020 - FCT (Fundação para a Ciencia e a Tecnologia). The second author was financed by Portuguese Funds through FCT-Fundao para a Cincia e a Tecnologia within the Projects UIDB/00013/2020 and UIDP/00013/2020 of Centre of Mathematics of the University of Minho, UIDB/00006/2020 of Centre of Statistics and its Applications of University of Lisbon and PTDC/MAT-STA/28243/2017.

In this paper, we consider a strictly stationary sequence of m-dependent random variables through a compatible sequence of independent and identically distributed random variables by the moving averages processes. Using the Zolotarev distance, we estimate some rates of convergence in the weak limit theorems for normalized geometric random sums of the strictly stationary sequence of m-dependent random variables. The obtained results are extensions and generalizations of several known results on geometric random sums of independent and identically distributed random variables.

When collecting samples, sometimes failures of observations occur and consequently missing data. This can have an impact on the analysis and subsequent inference, especially if the study focuses on the extreme values where the data is more scarce. In this work, we analyze the effect of different types of failures on the dependence within the tail of a stationary series. We will also present some examples., The first author was supported by Portuguese Funds through FCT — Fundação para a Ciência e a Tecnologia within the Projects UID/MAT/00013/2013 of Centre of Mathematics of the University of Minho, UID/Multi/04621/2019 of Center for Computational and Stochastic Mathematics, UID/MAT/00006/2019 of Centre of Statistics and Its Applications, and PTDC/MAT-STA/28243/2017. The second author was partially supported by the research unit UID/MAT/00212/2019 — FCT (Funda¸c˜ao para a Ciˆencia e a Tecnologia).

The consecutive hit probability of antiaircraft artillery corresponds to the multivariate normal probability distribution. The computational complexity depends on the length of the firing error sequence (i.e., the integral dimension may exceed 100). The traditional numerical integration and the Monte Carlo method are too slow for this calculation. This paper established the state equation of the firing error sequence, which was the bridge between the multivariate normal probability distribution and stochastic process theory. The recursive calculation model was given after the rigorous derivation process. The accuracy and computational complexity of the model were quantified by theoretical analysis and expressed intuitively by examples. The model shows the upper bound for the absolute error, and the computational efficiency is significantly improved.

This paper probabilistically explores a class of stationary count time series models built by superpositioning (or otherwise combining) independent copies of a binary stationary sequence of zeroes and ones. Superpositioning methods have proven useful in devising stationary count time series having prespecified marginal distributions. Here, basic properties of this model class are established and the idea is further developed. Specifically, stationary series with binomial, Poisson, negative binomial, discrete uniform, and multinomial marginal distributions are constructed; other marginal distributions are possible. Our primary goal is to derive the autocovariance function of the resulting series.

In this paper, we obtain necessary as well as sufficient conditions for exponential rate of decrease of the variance of the best linear unbiased estimator (BLUE) for the unknown mean of a stationary sequence possessing a spectral density. In particular, we show that a necessary condition for variance of BLUE to decrease to zero exponentially is that the spectral density vanishes on a set of positive Lebesgue measure in any vicinity of zero.