Deposits of clastic carbonate-dominated (calciclastic) sedimentary slope systems in the rock record have been identified mostly as linearly-consistent carbonate apron deposits, even though most ancient clastic carbonate slope deposits fit the submarine fan systems better. Calciclastic submarine fans are consequently rarely described and are poorly understood. Subsequently, very little is known especially in mud-dominated calciclastic submarine fan systems. Presented in this study are a sedimentological core and petrographic characterisation of samples from eleven boreholes from the Lower Carboniferous of Bowland Basin (Northwest England) that reveals a >250 m thick calciturbidite complex deposited in a calciclastic submarine fan setting. Seven facies are recognised from core and thin section characterisation and are grouped into three carbonate turbidite sequences. They include: 1) Calciturbidites, comprising mostly of highto low-density, wavy-laminated bioclast-rich facies; 2) low-density densite mudstones which are characterised by planar laminated and unlaminated muddominated facies; and 3) Calcidebrites which are muddy or hyper-concentrated debrisflow deposits occurring as poorly-sorted, chaotic, mud-supported floatstones. These

Phd by thesis

The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Stable non-Gaussian random processes , by G. Samorodnitsky and M. S. Taqqu. Pp. 632. £49.50. 1994. ISBN 0-412-05171-0 (Chapman and Hall).

Preface to the revised edition Remarks on notation 1. Basic examples 2. Characterization and existence 3. Stable processes and their extensions 4. The Levy-Ito decomposition of sample functions 5. Distributional properties of Levy processes 6. Subordination and density transformation 7. Recurrence and transience 8. Potential theory for Levy processes 9. Wiener-Hopf factorizations 10. More distributional properties Supplement Solutions to exercises References and author index Subject index.

Lévy processes and infinitely divisible distributions

Stochastic differential equations and diffusion processes

Let ${\bf L}$ be the unit exponential random variable and ${\bf Z}_\alpha$ the standard positive $\alpha$-stable random variable. We prove that $\{(1-\alpha)\alpha^{\gamma_\alpha} {\bf Z}_\alpha^{-\gamma_\alpha}, 0< \alpha <1\}$ is decreasing for the optimal stochastic order and that $\{(1-\alpha){\bf Z}_\alpha^{ \gamma_\alpha}, 0< \alpha < 1\}$ is increasing for the convex order, with $\gamma_\alpha = \alpha/(1-\alpha).$ We also show that $\{\Gamma(1+\alpha) {\bf Z}_\alpha^{-\alpha}, 1/2\le \alpha \le 1\}$ is decreasing for the convex order, that ${\bf Z}_\alpha^{ \alpha}\,\prec_{st}\, \Gamma(1-\alpha) {\bf L}$ and that $\Gamma(1+\alpha){\bf Z}_\alpha^{-\alpha} \,\prec_{cx}\,{\bf L}.$ This allows to compare ${\bf Z}_\alpha$ with the two extremal Frechet distributions corresponding to the behaviour of its density at zero and at infinity. We also discuss the applications of these bounds to the strange behaviour of the median of ${\bf Z}_\alpha$ and ${\bf Z}_\alpha^{-\alpha}$ and to some uniform estimates on the classical Mittag-Leffler function. Along the way, we obtain a canonical factorization of ${\bf Z}_\alpha$ for $\alpha$ rational in terms of Beta random variables. The latter extends to the one-sided branches of real strictly stable densities.

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Comparing Fréchet and positive stable laws

This article deals with the asymptotic behavior as $t \rightarrow +\infty $ of the survival function $\mathbb{P}[T > t]$, where T is the first passage time above a non negative level of a random process starting from zero. In many cases of physical significance, the behavior is of the type $\mathbb{P}[T > t] = t^{-\theta +o(1)}$ for a known or unknown positive parameter $\theta$ which is called the persistence exponent. The problem is well understood for random walks or Levy processes but becomes more difficult for integrals of such processes, which are more related to physics. We survey recent results and open problems in this field.

/pdf/persistence-probabilities-and-exponents-3bf0bsge4p.pdf

Persistence Probabilities and Exponents

https://orbilu.uni.lu/bitstream/10993/19298/1/nourdin-simon-spl.pdf

On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion

Let { R t , 0 ⩽ t ⩽ 1 } be a symmetric α-stable Riemann–Liouville process with Hurst parameter H > 0 . Consider a translation invariant, β-self-similar, and p-pseudo-additive functional semi-norm ‖ ⋅ ‖ . We show that if H > β + 1 / p and γ = ( H − β − 1 / p ) −1 , then lim ɛ ↓ 0 ɛ γ log P [ ‖ R ‖ ⩽ ɛ ] = − K ∈ [ − ∞ , 0 ) , with K finite in the Gaussian case α = 2 . If α 2 , we prove that K is finite when R is continuous and H > β + 1 / p + 1 / α . We also show that under the above assumptions, lim ɛ ↓ 0 ɛ γ log P [ ‖ X ‖ ⩽ ɛ ] = − K ∈ ( − ∞ , 0 ) , where X is the linear α-stable fractional motion with Hurst parameter H ∈ ( 0 , 1 ) (if α = 2 , then X is the classical fractional Brownian motion). These general results cover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and non-Gaussian frameworks.

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Small deviations for fractional stable processes

Let R be a symmetric a-stable Riemann-Liouville process with Hurst parameter H > 0. Consider ||.|| a translation invariant, b-self-similar, and p-pseudo-additive functional semi-norm. We show that if H > (b + 1/p) and c = (H - b - 1/p), then x log P [ log ||R|| - k 0, with k finite in the Gaussian case a = 2. If a (b + 1/p + 1/a). We also show that under the above assumptions, x log P [ log ||X|| - k 0, where k is finite and X is the linear a-stable fractional motion with Hurst parameter 0 < H < 1 (if a = 2, then X is the classical fractional Brownian motion). These general results recover many cases previously studied in the literature, and also prove the existence of new small deviation constants, both in Gaussian and Non-Gaussian frameworks.

/pdf/small-deviations-for-fractional-stable-processes-4m73zp85k6.pdf

Thomas Simon

Papers

Comparing Fréchet and positive stable laws

Persistence Probabilities and Exponents

On the absolute continuity of one-dimensional SDEs driven by a fractional Brownian motion

Small deviations for fractional stable processes

Small deviations for fractional stable processes