A maximal 𝕃_{𝕡}-inequality for stationary sequences and its applications

Peligrad, Magda; Utev, Sergey; Wu, Wei Biao

doi:10.1090/s0002-9939-06-08488-7

Cited by 47 publications

(57 citation statements)

References 18 publications

(23 reference statements)

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“…The increment sequences (X n − X n−1 ) and (Y n − Y n−1 ) n∈Z also form stationary and ergodic sequences by Lemma 4.4. Therefore, we can invoke Theorem 1 in [27] and conclude that E max k=1,...,n…”

Section: Lemma 42 Thus Givesmentioning

confidence: 92%

Einstein Relation for Random Walk in a One-Dimensional Percolation Model

2019

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We consider random walks on the infinite cluster of a conditional bond percolation model on the infinite ladder graph. In a companion paper, we have shown that if the random walk is pulled to the right by a positive bias λ > 0, then its asymptotic linear speed v is continuous in the variable λ > 0 and differentiable for all sufficiently small λ > 0. In the paper at hand, we complement this result by proving that v is differentiable at λ = 0. Further, we show the Einstein relation for the model, i.e., that the derivative of the speed at λ = 0 equals the diffusivity of the unbiased walk. p ,λ . If the walk starts at v = 0, we sometimes omit the superscript 0. Further, if λ = 0, we sometimes omit λ as a subscript, and write p ω for p ω,0 , and P for P 0 .

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Section: Lemma 42 Thus Givesmentioning

confidence: 92%

Einstein Relation for Random Walk in a One-Dimensional Percolation Model

2019

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“…We consider here the rather general condition of "martingale difference sequences" (MDS, see Conditions (A) below) since, on the one hand, there are many models satisfying this notion (see the next section) and since, on the other hand, studying this framework of dependency makes it possible to study the most general case of strictly stationary random variables as is done by Peligrad, Utev and Wu (2007). Note also, that MDS are uncorrelated dependent random variables, so that the above GCV-type criterion, G X , is well adapted.…”

Section: Model Selection Criteria and Useful Toolsmentioning

confidence: 99%

On the excess of average squared error for data-driven bandwidths in nonparametric trend estimation

Benhenni¹,

Girard²,

Louhichi³

2021

APAM

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We consider the problem of the optimal selection of the smoothing parameter h in kernel estimation of a trend in nonparametric regression models with strictly stationary errors. We suppose that the errors are stochastic volatility sequences. Three types of volatility sequences are studied: the log-normal volatility, the Gamma volatility and the log-linear volatility with Bernoulli innovations. We take the weighted average squared error (ASE) as the global measure of performance of the trend estimation using h and we study two classical criteria for selecting h from the data, namely the adjusted generalized cross validation and Mallows-type criteria. We establish the asymptotic distribution of the gap between the ASE evaluated at one of these selectors and the smallest possible ASE. A Monte-Carlo simulation for a log-normal stochastic volatility model illustrates that this asymptotic approximation can be accurate even for small sample sizes.

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“…Our main technical toolbox for devising an UCB-type learning scheme is a general type of high probability maximal Hoeffding-type of concentration inequality which controls the deviations of the random sum for the stationary real-valued (mixingale-type) of stochastic process (X t ) t∈N . The result is due to Peligrad et al (2007) and we provide it below for completness.…”

Section: Concentration Toolboxmentioning

confidence: 99%

Restless dependent bandits with fading memory

Zadorozhnyi,

Blanchard,

Carpentier

2019

Preprint

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We study the stochastic multi-armed bandit problem in the case when the arm samples are dependent over time and generated from so-called weak C-mixing processes. We establish a C−Mix Improved UCB agorithm and provide both problem-dependent and independent regret analysis in two different scenarios. In the first, so-called fast-mixing scenario, we show that pseudo-regret enjoys the same upper bound (up to a factor) as for i.i.d. observations; whereas in the second, slow mixing scenario, we discover a surprising effect, that the regret upper bound is similar to the independent case, with an incremental additive term which does not depend on the number of arms. The analysis of slow mixing scenario is supported with a minmax lower bound, which (up to a log(T ) factor) matches the obtained upper bound.

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A maximal 𝕃_{𝕡}-inequality for stationary sequences and its applications

Cited by 47 publications

References 18 publications

Einstein Relation for Random Walk in a One-Dimensional Percolation Model

Einstein Relation for Random Walk in a One-Dimensional Percolation Model

On the excess of average squared error for data-driven bandwidths in nonparametric trend estimation

Restless dependent bandits with fading memory

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