A Maximal Inequality and Dependent Strong Laws

McLeish, D. L.

doi:10.1214/aop/1176996269

Cited by 324 publications

(188 citation statements)

References 13 publications

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“…In particular, (M i ) i∈N is a square-integrable martingale difference sequence and i E[M 2 i ]/i 2 < ∞. By Corollary 1.9. in McLeish (1975),…”

Section: Estimationmentioning

confidence: 99%

“…Except for the computation of the estimators, this procedure does not require any additional effort, since the simulation results can be recycled for this purpose after s * has been estimated. The proofs of the consistency of the estimators relies on a law of large numbers for martingale difference sequences or, more generally, mixingales, see McLeish (1975) and Chow (1967).…”

Section: Estimator Of the Asymptotic Variancementioning

confidence: 99%

See 1 more Smart Citation

Stochastic Root Finding and Efficient Estimation of Convex Risk Measures

Dunkel

Weber

2010

Operations Research

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Reliable risk measurement is a key problem for financial institutions and regulatory authorities. The current industry standard Value-at-Risk has several deficiencies. Improved risk measures have been suggested and analyzed in the recent literature, but their computational implementation has largely been neglected so far. We propose and investigate stochastic approximation algorithms for the convex risk measure Utility-based Shortfall Risk. Our approach combines stochastic root finding schemes with importance sampling. We prove that the resulting Shortfall Risk estimators are consistent and asymptotically normal, and provide formulas for confidence intervals. The performance of the proposed algorithms is tested numerically. We finally apply our techniques to the Normal Copula Model, which is also known as the industry model CreditMetrics. This provides guidance for future implementations in practice.

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“…In particular, (M i ) i∈N is a square-integrable martingale difference sequence and i E[M 2 i ]/i 2 < ∞. By Corollary 1.9. in McLeish (1975),…”

Section: Estimationmentioning

confidence: 99%

Section: Estimator Of the Asymptotic Variancementioning

confidence: 99%

Stochastic Root Finding and Efficient Estimation of Convex Risk Measures

Dunkel

Weber

2010

Operations Research

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“…The NED condition was introduced by Billingsley (1968) (under the rubric "functions of mixing processes"). McLeish (1975aMcLeish ( ,b, 1977 provides an LLN, a central limit, and an invariance principle for NED sequences that extend lim-�cni <oo. D n°"' � kn i=l…”

Section: --Oomentioning

confidence: 99%

Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables

Andrews¹

1988

Econ Theory

221

189

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This paper provides L 1 and weak laws of large numbers for uniformly integrable L 1 -mixingales. The L 1 -mixingale condition is a condition of asymptotic weak temporal dependence that is weaker than most conditions considered in the literature. Processes covered by the laws of large numbers include martingale difference, 4>0. p(·), and a(·) mixing, autoregressive moving average, infinite order moving average, near epoch dependent, L 1 -near epoch dependent, and mixingale sequences and triangular arrays. The random variables need not possess more than one moment finite and the L 1 -mixingale numbers need not decay to zero at any particular rate. The proof of the results is remarkably simple and completely self-contained.

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“…The NED property dates in various forms to Ibragimov (1962) and McLeish (1975), with a myriad subsequent extensions in Bierens (1987), Gallant and White (1988), and recently Nze et al (2002) and Wu and Min (2005). See Nze and Doukhan (2004) for compendium details on NED and similar concepts, and their usage in econometrics.…”

Section: Tail Memory: Events and Exceedancesmentioning

confidence: 99%

“…L 0 -APP is useful for heavy tailed processes that may not satisfy certain moment conditions, as opposed to moment-based Near Epoch Dependence, mixingale (McLeish 1975), and so-called -Weak Dependence (Nze et al 2002) and L p -Weak Dependence (Wu and Min 2005). Further, L p -NED implies L 0 -APP (Davidson 1994: p. 274), and as with NED and E-NED if z t is adapted to X t then fX t g is trivially L 0 -APP on itself.…”

Section: Tail Memory: Events and Exceedancesmentioning

confidence: 99%

Tail and Nontail Memory With Applications to Extreme Value and Robust Statistics

Hill¹

2011

Econom. Theory

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New notions of tail and non-tail dependence are used to characterize separately extremal and non-extremal information, including tail log-exceedances and events, and tail-trimmed levels. We prove Near Epoch Dependence (McLeish 1975, Gallant andWhite 1988) and L 0 -Approximability (Pötscher and Prucha 1991) are equivalent for tail events and tail-trimmed levels, ensuring a Gaussian central limit theory for important extreme value and robust statistics under general conditions. We apply the theory to characterize the extremal and nonextremal memory properties of possibly very heavy tailed GARCH processes and distributed lags. This in turn is used to verify Gaussian limits for tail index, tail dependence and tail trimmed sums of these data, allowing for Gaussian asymptotics for a new Tail-Trimmed Least Squares estimator for heavy tailed processes.

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A Maximal Inequality and Dependent Strong Laws

Cited by 324 publications

References 13 publications

Stochastic Root Finding and Efficient Estimation of Convex Risk Measures

Stochastic Root Finding and Efficient Estimation of Convex Risk Measures

Laws of Large Numbers for Dependent Non-Identically Distributed Random Variables

Tail and Nontail Memory With Applications to Extreme Value and Robust Statistics

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