Basic Properties of Strong Mixing Conditions. A Survey and Some Open Questions

Bradley, Richard C.

doi:10.1214/154957805100000104

Cited by 755 publications

(614 citation statements)

References 139 publications

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“…Our measure is reminiscent of quantities used when defining mixing conditions, see, e.g., Ref. [5] . For instance, with t s defining the -field (Q u : s ≤ u ≤ t ), and…”

Section: Resultsmentioning

confidence: 99%

On the Dependence Structure of Gaussian Queues

Es-Saghouani¹,

Mandjes

2009

Stochastic Models

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To cite this Article Es-Saghouani, Abdelghafour andMandjes, Michel (2009) Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden.The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.

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“…Our measure is reminiscent of quantities used when defining mixing conditions, see, e.g., Ref. [5] . For instance, with t s defining the -field (Q u : s ≤ u ≤ t ), and…”

Section: Resultsmentioning

confidence: 99%

On the Dependence Structure of Gaussian Queues

Es-Saghouani¹,

Mandjes

2009

Stochastic Models

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“…Given this bound for each case, we set b n = n to satisfy the rate given in (11). Then the second condition in (34) is satis…ed for any a large enough (for example, if we let b n = n T 1=(1+ ) for 2 (0; 1) then "a 64" is enough).…”

Section: Resultsmentioning

confidence: 99%

“…This is a standard de…nition of triangular arrays (see Bradley, 2005). The mixing coe¢ cients represent the degree of time-series dependence of fX n;i g. We note that the coe¢ cients may depend on n, since n F n m+k and n F k 1 are allowed to depend on n. The dependence on n is relevant in this study, since we consider the cases where the degree of time-series dependence between consecutive observations changes with n (as in Example 2 of a near-unit-root process).…”

Section: Basic Setupmentioning

confidence: 99%

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

Kanaya

2016

Econom. Theory

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The convergence rates of the sums of -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 …rst to investigate the convergence rates of the sums of -mixing triangular arrays whose mixing coe¢ cients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is …xed for any sample size n; and the other, called the in…ll assumption, is that it shrinks to zero as n tends to in…nity. Our convergence theorems indicate that an explicit trade-o¤ exists between the rate of convergence and the degree of dependence. While the results under the in…ll assumption can be seen as a direct extension of those under the …xed-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 di¤usion process with weak mean reversion, and a near-unit-root process.

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“…We got the clue from the works of [1]- [3]. Cooke raised the idea of the use of the indicator random variables (r.v.…”

Section: Introductionmentioning

confidence: 99%

“…Let 1  and 2  be two σ -sub-algebras of  . The survey of [3] proposes the measure ( ) ( ) ( ) ( ) …”

Section: Introductionmentioning

confidence: 99%