Appendix Multidimensional infinitely divisible variables and processes. Part I: Stable case

LePage, Raoul

doi:10.1007/bfb0083389

Cited by 25 publications

(33 citation statements)

References 6 publications

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“…This series is convergent with probability one (see LePage, 1980). With the definitions given by (7) (10) we have the following limit behavior of W n .…”

Section: Conditional Invariance Principle For Errors Attracted To Stamentioning

confidence: 93%

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Resampling Permutations in Regression without Second Moments

LePage

Podgórski²

1996

Journal of Multivariate Analysis

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For given observations we consider the resampling distribution obtained by permuting residuals versus the distribution of errors conditional on their order statistics. We observe that with high probability both are approximately equal on contrast vectors after suitable normalization. No momentÂdistribution assumptions on the underlying errors other than exchangeability are required. These results remain valid if we consider reduced regression models derived from the original one by removing data for which the corresponding residuals take extreme values. We obtain general conditions under which confidence regions produced by such methods are accurate. In the absence of finite second moments randomness of the limiting bootstrap distributions has been considered as a failure of the traditional bootstrap and no practical meaning has been given to the phenomenon. For our method this type of randomness is still present in the limit but it has clear probabilistic interpretation as a conditional distribution which can be used, e.g., to obtain conditional confidence sets. We study this phenomenon in detail for the case of independent errors with distribution from the domain of attraction of stable laws.

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“…This series is convergent with probability one (see LePage, 1980). With the definitions given by (7) (10) we have the following limit behavior of W n .…”

Section: Conditional Invariance Principle For Errors Attracted To Stamentioning

confidence: 93%

“…random variables from the domain of attraction of a symmetric stable law with index of stability :. In fact for x>1 we have P(|= n, i | >x)=x &: (see also LePage, 1980). On a sequence v n =(v i, n ) n i=1 of contrast vectors we impose the form…”

Section: Conditional Invariance Principle For Errors Attracted To Stamentioning

confidence: 94%

Resampling Permutations in Regression without Second Moments

LePage

Podgórski²

1996

Journal of Multivariate Analysis

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“…5.3 of infinite Lévy measure ν(R 0 ) = +∞ with finite variance σ 2 < +∞, as well as the decomposition (i) Inverse Lévy measure representation (Ferguson and Klass 1972;LePage 1980):…”

Section: Finite Truncation Of Infinite Series Representationmentioning

confidence: 99%

“…We construct approximate discrete-time simulation schemes based on the so-called series representation of infinitely divisible laws and associated Lévy processes (Bondesson 1982;Ferguson and Klass 1972;LePage 1980;Rosiński 2001), as was done in (Todorov and Tauchen 2006) for non-negative Lévy-driven CARMA processes (Section 6), with a view towards application in financial economics. The aim of the present paper is twofold; to extend this approach to the classes of stable and general second-order CARMA processes, as well as to develop the verifiable error analysis, not only of academic interest but directly useful for actual numerical implementation.…”

Section: Introductionmentioning

confidence: 99%

Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

Kawai

2015

Methodol Comput Appl Probab

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We address the issue of sample path simulation of Lévy-driven continuous-time autoregressive moving average (CARMA) processes. Approximate discrete-time simulation schemes are constructed along with quantifiable error analysis for stable, second-order and non-negative CARMA processes, based upon the so-called series representation of infinitely divisible laws and associated Lévy processes. We prove that under suitable conditions, the simulation scheme can be improved in terms of second-order structure, finite dimensional laws as well as sample path properties. The simulation procedure is often quite simple and allows one to conduct super-sampling without running the algorithm once again. The computational complexity of the proposed scheme is not affected much by the sampling scheme, such as sampling frequency and irregular spacing. Numerical results are presented throughout to illustrate the effectiveness of the proposed simulation scheme.

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“…Let The following list of properties of (I n i (u; ·)) n i=1 is veriÿed in [10,9]. For ÿxed t ∈ [0; 1] a sequence (I n i (u; t)) n i=1 consists of exactly k = [nt] ones and can be interpreted as a combination of k-elements from n-elements.…”

Section: Invariance Principlementioning

confidence: 99%

Strong and conditional invariance principles for samples attracted to stable laws

LePage¹,

Podgórski²,

Ryznar³

1997

Probab Theory Relat Fields

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We prove almost sure convergence of a representation of normalized partial sum processes of a sequence of i.i.d. random variables from the domain of attraction of an -stable law, ¡2. We obtain an explicit form of the limit in terms of the LePage series representation of stable laws. One consequence of these results is a conditional invariance principle having applications to option pricing as well as to resampling by signs and permutations.

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Appendix Multidimensional infinitely divisible variables and processes. Part I: Stable case

Cited by 25 publications

References 6 publications

Resampling Permutations in Regression without Second Moments

Resampling Permutations in Regression without Second Moments

Sample Path Generation of Lévy-Driven Continuous-Time Autoregressive Moving Average Processes

Strong and conditional invariance principles for samples attracted to stable laws

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