Dependent Central Limit Theorems and Invariance Principles

McLeish, D. L.

doi:10.1214/aop/1176996608

Cited by 462 publications

(247 citation statements)

References 7 publications

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“…Since the σ-fields are nested, that is, F nt ⊆ F n,t+1 for all t ≤ n, using Lemma 1 given in Appendix C, , (a, b, c, d)Σ(a, b, c, d) McLeish, 1974, Theorem 2.3 and subsequent discussion], and Theorem 1 follows.…”

Section: Proofs Of the Main Resultsmentioning

confidence: 97%

Covariate Adjusted Correlation Analysis via Varying Coefficient Models

Şentürk

Müller

2005

Scand J Stat

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ABSTRACT. We propose covariate adjusted correlation (Cadcor) analysis to target the correlation between two hidden variables that are observed after being multiplied by an unknown function of a common observable confounding variable. The distorting effects of this confounding may alter the correlation relation between the hidden variables. Covariate adjusted correlation analysis enables consistent estimation of this correlation, by targeting the definition of correlation through the slopes of the regressions of the hidden variables on each other and by establishing a connection to varying-coefficient regression.The asymptotic distribution of the resulting adjusted correlation estimate is established.These distribution results, when combined with proposed consistent estimates of the asymptotic variance, lead to the construction of approximate confidence intervals and inference for adjusted correlations. We illustrate our approach through an application to the Boston house price data. Finite sample properties of the proposed procedures are investigated through a simulation study.

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Section: Proofs Of the Main Resultsmentioning

confidence: 97%

Covariate Adjusted Correlation Analysis via Varying Coefficient Models

Şentürk

Müller

2005

Scand J Stat

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“…By this fact, along with a CLT for martingale differences (Theorem 2.3 of [11] or Theorem 2.10 of [14]), it suffices to prove the conditions…”

Section: Proof Of Theorem 41: the Poisson Casementioning

confidence: 99%

“…Thus Slutsky's theorem yields the first (Poisson) part of (11). To obtain the second (binomial) part of (11), we use the coupling of Lemma 7.4.…”

Section: Lemma 82mentioning

confidence: 99%

On the total length of the random minimal directed spanning tree

Penrose

Wade

2006

Adv. Appl. Probab.

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In Bhatt and Roy's minimal directed spanning tree (MDST) construction for a random partially ordered set of points in the unit square, all edges must respect the "coordinatewise" partial order and there must be a directed path from each vertex to a minimal element. We study the asymptotic behaviour of the total length of this graph with power weighted edges. The limiting distribution is given by the sum of a normal component away from the boundary and a contribution introduced by the boundary effects, which can be characterized by a fixed point equation, and is reminiscent of limits arising in the probabilistic analysis of certain algorithms. As the exponent of the power weighting increases, the distribution undergoes a phase transition from the normal contribution being dominant to the boundary effects dominating. In the critical case where the weight is simple Euclidean length, both effects contribute significantly to the limit law. We also give a law of large numbers for the total weight of the graph.

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“…In Section 5.1, we will use an abstract theorem due to Durrett and Resnick [9], based on the invariance principle for martingale difference arrays with bounded variables (Freedman,[14] and [15]), together with a random change of time (see, for example, Helland [19] and Billingsley [7]). The underlying central limit theorem for martingale difference arrays can be found in Dvoretzky [10], [11] (see also [25], [19] and references therein). The alternative proof, in Section 6, is based on the convergence of the moments to the moments of a Brownian motion, under some asymptotic factorization conditions, and it uses combinatorial techniques.…”

Section: Introductionmentioning

confidence: 99%