A Central Limit Theorem for $m$-Dependent Random Variables with Unbounded $m$

Berk, Kenneth N.

doi:10.1214/aop/1176996992

Cited by 91 publications

(100 citation statements)

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“…. , n} are m-dependent, the convergence (5.48) follows by the main theorem of [11], and the proof of (5.48) is complete.…”

Section: Proof Of Theorem 12(ii) To Prove Theorem 12(ii) We Startmentioning

confidence: 83%

Power variation for a class of stationary increments Lévy driven moving averages

Basse-O'Connor¹,

Podolskij²

2017

Ann. Probab.

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In this paper, we present some new limit theorems for power variation of kth order increments of stationary increments Lévy driven moving averages. In the infill asymptotic setting, where the sampling frequency converges to zero while the time span remains fixed, the asymptotic theory gives novel results, which (partially) have no counterpart in the theory of discrete moving averages. More specifically, we show that the first-order limit theory and the mode of convergence strongly depend on the interplay between the given order of the increments k ≥ 1, the considered power p > 0, the BlumenthalGetoor index β ∈ [0, 2) of the driving pure jump Lévy process L and the behaviour of the kernel function g at 0 determined by the power α. First-order asymptotic theory essentially comprises three cases: stable convergence towards a certain infinitely divisible distribution, an ergodic type limit theorem and convergence in probability towards an integrated random process. We also prove a second-order limit theorem connected to the ergodic type result. When the driving Lévy process L is a symmetric β-stable process, we obtain two different limits: a central limit theorem and convergence in distribution towards a (k − α)β-stable totally right skewed random variable.

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“…. , n} are m-dependent, the convergence (5.48) follows by the main theorem of [11], and the proof of (5.48) is complete.…”

Section: Proof Of Theorem 12(ii) To Prove Theorem 12(ii) We Startmentioning

confidence: 83%

Power variation for a class of stationary increments Lévy driven moving averages

Basse-O'Connor¹,

Podolskij²

2017

Ann. Probab.

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“…Remark 2.2 When p = o(n α ), α < 1/2, the asymptotic properties of the MPCLE can be shown by the use of the central limit theorem for m-dependent random variables (Berk, 1973).…”

Section: Pairwise Comparison Pseudo Likelihoodmentioning

confidence: 99%

Revisiting the Bradley-Terry model and its application to information retrieval

Jeon¹,

Kim²

2013

Journal of the Korean Data and Information Science Society

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The Bradley-Terry model is widely used for analysis of pairwise preference data. We explain that the popularity of Bradley-Terry model is gained due to not only easy computation but also some nice asymptotic properties when the model is misspecified. For information retrieval required to analyze big ranking data, we propose to use a pseudo likelihood based on the Bradley-Terry model even when the true model is different from the Bradley-Terry model. We justify using the Bradley-Terry model by proving that the estimated ranking based on the proposed pseudo likelihood is consistent when the true model belongs to the class of Thurstone models, which is much bigger than the Bradley-Terry model.

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“…the central limit theorem for m-dependent random variables, with m fixed [34] or unbounded [35], tells that N i=1 X i converges to a Gaussian random variable with mean G and variance G + K.…”

Section: Appendix a Proof Of Theoremmentioning

confidence: 99%

A Protocol-Independent Approach for Analyzing the Optimal Operation Point of CSMA/CA Protocols

2009

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Abstract-This paper presents a protocol-independent approach to reveal a new insight into the performance of carrier sense multiple access with collision avoidance (CSMA/CA) protocols: the family of CSMA/CA protocols, independent of implementation details, share the same optimal operation point where the maximum protocol capacity is achieved. The protocolindependent analysis is inspired by the concept of virtual time slot. At the timescale of virtual-slot, all the CSMA/CA protocols show the same behavior pattern and, therefore, a generic virtualslot based S-G (VS S-G) analysis is developed to compute the optimal operation point. The accuracy of the VS S-G analysis is benchmarked against the precise protocol-specific analysis, in particular, for the 802.11 distributed coordination function (DCF) and the 802.15.4 contention access period (CAP). Furthermore, this paper discusses how to integrate the network-layer queueing analysis with the VS S-G analysis at the medium access control (MAC) layer to form a generic cross-layer framework for calllevel network capacity analysis.

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A Central Limit Theorem for $m$-Dependent Random Variables with Unbounded $m$

Cited by 91 publications

References 0 publications

Power variation for a class of stationary increments Lévy driven moving averages

Power variation for a class of stationary increments Lévy driven moving averages

Revisiting the Bradley-Terry model and its application to information retrieval

A Protocol-Independent Approach for Analyzing the Optimal Operation Point of CSMA/CA Protocols

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