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- Albin, Patrik, 1960
(författare)
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A continuous non-Brownian motion martingale with Brownian motion marginal distributions
- 2008
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Ingår i: Statistics and Probability Letters. - 0167-7152. ; 78:6, s. 682-686
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Tidskriftsartikel (refereegranskat)abstract
- This note exhibits a continuous martingale $M$ which is not Brownian motion, but has the same univariate marginal distributions as Brownian motion. It is given by $M(t)=X_1(t)X_2(t)Y$, where $X_1$ and $X_2$ are independent copies of the diffusion $dX(t)=dB(t)(2X(t))^{-1},\ X(0)=0$, and $Y$ is an independent random variable with known density on $(0,\sqrt{2})$. The existence of such a martingale was an open problem until now.
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- Albin, Patrik, 1960, et al.
(författare)
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A New Proof of an Old Result by Pickands
- 2010
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Ingår i: Electronic Communications in Probability. - 1083-589X. ; 15, s. 339-345
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Tidskriftsartikel (refereegranskat)abstract
- Let {xi(t)}(t is an element of[0,h]) be a stationary Gaussian process with covariance function r such that r(t) = 1 - C vertical bar t vertical bar(alpha) + o(vertical bar t vertical bar(alpha)) as t -> 0. We give a new and direct proof of a result originally obtained by Pickands, on the asymptotic behaviour as u -> infinity of the probability P{sup(t is an element of vertical bar 0,h vertical bar) xi(t) > u} that the process xi exceeds the level u. As a by-product, we obtain a new expression for Pickands constant H alpha
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- Albin, Patrik, 1960, et al.
(författare)
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Extremes and limit theorems for difference of chi-type processes
- 2016
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Ingår i: ESAIM - Probability and Statistics. - : EDP Sciences. - 1262-3318 .- 1292-8100. ; 20, s. 349-366
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Tidskriftsartikel (refereegranskat)abstract
- © 2016 EDP Sciences, SMAI. Let { m,k (k) (t),t≥ 0},κ > 0 be random processes defined as the differences of two independent stationary chi-type processes with m and k degrees of freedom. In this paper we derive the asymptotics of P supt [0,T[ m,k (k) (t) > u →∞, u→∞ under some assumptions on the covariance structures of the underlying Gaussian processes. Further, we establish a Berman sojourn limit theorem and a Gumbel limit result.
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