A note on the convergence to Gaussian laws of sums of stationary φ-mixing triangular arrays

1985 ◽  
pp. 387-399 ◽  
Author(s):  
Jorge D. Samur
Keyword(s):  
Triangular Arrays

scholarly journals CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES

2016 ◽  
Vol 33 (5) ◽  
pp. 1121-1153
Author(s):  
Shin Kanaya
Keyword(s):  
Continuous Time ◽  
Convergence Rates ◽  
Moving Average ◽  
Function Estimation ◽  
Triangular Arrays ◽  
Large Numbers ◽  
Strongly Mixing ◽  
Time Distance ◽  
Continuous Time Processes ◽  
Near Unit Root

The convergence rates of the sums of α-mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences/triangular arrays, but their convergence rates were not fully investigated. This study is the first to investigate the convergence rates of the sums of α-mixing triangular arrays whose mixing coefficients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is fixed for any sample size n; and the other, called the infill assumption, is that it shrinks to zero as n tends to infinity. Our convergence theorems indicate that an explicit trade-off exists between the rate of convergence and the degree of dependence. While the results under the infill assumption can be seen as a direct extension of those under the fixed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time diffusion process with weak mean reversion, and a near-unit-root process.

Investigation of In-Flow Fluidelastic Instability of Square Tube Arrays Subjected to Air Cross Flow

2015 ◽  
Author(s):  
Tomomichi Nakamura ◽  
Shinichiro Hagiwara ◽  
Joji Yamada ◽  
Kenji Usuki
Keyword(s):  
Nuclear Power ◽  
Flow Instability ◽  
Flow Direction ◽  
Cross Flow ◽  
Diameter Ratio ◽  
Nuclear Industry ◽  
Flexible Cylinder ◽  
Triangular Arrays ◽  
Tube Arrays ◽  
Fluidelastic Instability

In-flow instability of tube arrays is a recent major issue in heat exchanger design since the event at a nuclear power plant in California [1]. In our previous tests [2], the effect of the pitch-to-diameter ratio on fluidelastic instability in triangular arrays is reported. This is one of the present major issues in the nuclear industry. However, tube arrays in some heat exchangers are arranged as a square array configuration. Then, it is important to study the in-flow instability on the case of square arrays. The in-flow fluidelastic instability of square arrays is investigated in this report. It was easy to observe the in-flow instability of triangular arrays, but not for square arrays. The pitch-to-diameter ratio, P/D, is changed from 1.2 to 1.5. In-flow fluidelastic instability was not observed in the in-flow direction. Contrarily, the transverse instability is observed in all cases including the case of a single flexible cylinder. The test results are finally reported including the comparison with the triangular arrays.

WEAK CONVERGENCE OF NONLINEAR TRANSFORMATIONS OF INTEGRATED PROCESSES: THE MULTIVARIATE CASE

2009 ◽  
Vol 25 (5) ◽  
pp. 1180-1207 ◽  
Author(s):  
Norbert Christopeit
Keyword(s):  
Weak Convergence ◽  
Recent Work ◽  
Invariance Principle ◽  
Nonlinear Transformations ◽  
New Class ◽  
Triangular Arrays ◽  
Integrable Functions ◽  
Gaussian Case ◽  
Non Gaussian ◽  
Integrated Processes

We consider weak convergence of sample averages of nonlinearly transformed stochastic triangular arrays satisfying a functional invariance principle. A fundamental paradigm for such processes is constituted by integrated processes. The results obtained are extensions of recent work in the literature to the multivariate and non-Gaussian case. As admissible nonlinear transformation, a new class of functionals (so-called locally p-integrable functions) is introduced that adapts the concept of locally integrable functions in Pötscher (2004, Econometric Theory 20, 1–22) to the multidimensional setting.

scholarly journals Maxima of Dirichlet and triangular arrays of gamma variables

2008 ◽  
Vol 78 (16) ◽  
pp. 2811-2820 ◽  
Author(s):  
Arup Bose ◽  
Amites Dasgupta ◽  
Krishanu Maulik
Keyword(s):  
Triangular Arrays

Limit theorems for numbers satisfying a class of triangular arrays

2021 ◽  
Vol 56 (2) ◽  
pp. 195-223
Author(s):  
Igoris Belovas ◽  
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Convergence Rate ◽  
Asymptotic Normality ◽  
Limit Theorems ◽  
Limiting Distribution ◽  
Linear Recurrence ◽  
Triangular Arrays ◽  
Exponential Generating Function ◽  
Analytical Expressions

The paper extends the investigations of limit theorems for numbers satisfying a class of triangular arrays, defined by a bivariate linear recurrence with bivariate linear coefficients. We obtain the partial differential equation and special analytical expressions for the numbers using a semi-exponential generating function. We apply the results to prove the asymptotic normality of special classes of the numbers and specify the convergence rate to the limiting distribution. We demonstrate that the limiting distribution is not always Gaussian.

Moment Inequalities and Gaussian Approximation for Martingales

2019 ◽  
pp. 27-61
Author(s):  
Florence Merlevède ◽  
Magda Peligrad ◽  
Sergey Utev
Keyword(s):  
Functional Form ◽  
Gaussian Approximation ◽  
Sufficient Conditions ◽  
Maximal Inequality ◽  
Partial Sums ◽  
Moment Inequalities ◽  
Exponential Inequalities ◽  
Moderate Deviations Principle ◽  
Triangular Arrays ◽  
Burkholder Inequality

The aim of this chapter is to present useful tools for analyzing the asymptotic behavior of partial sums associated with dependent sequences, by approximating them with martingales. We start by collecting maximal and moment inequalities for martingales such as the Doob maximal inequality, the Burkholder inequality, and the Rosenthal inequality. Exponential inequalities for martingales are also provided. We then present several sufficient conditions for the central limit behavior and its functional form for triangular arrays of martingales. The last part of the chapter is devoted to the moderate deviations principle and its functional form for triangular arrays of martingale difference sequences.

scholarly journals Limit theorems for triangular arrays under a relaxed asymptotic negligibility condition

1987 ◽  
Vol 42 (1) ◽  
pp. 117-128 ◽  
Author(s):  
Miloslav Jirina
Keyword(s):  
Asymptotic Behavior ◽  
Classical Theory ◽  
Limit Theorems ◽  
Random Variables ◽  
Triangular Array ◽  
Independent Random Variables ◽  
Triangular Arrays

AbstractLet {Xnk} be a triangular array of independent random variables satisfying the so-called tail-negligibility condition, i.e. such that Prob{|Xnk| > a} → 0 as both k, n → ∞. It is also assumed that for each fixed k, Xnk converges in distribution as n → ∞. Theorems on the asymptotic behavior of the row sums of the array, analogous to those of the classical theory under the uniform negligibility condition, are presented.

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