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Any of these conditions can be waived by permission of the Corresponding Author. ecp@iam.uni-bonn.de (Anton Bovier (Chief Editor)) ejpecp@chafai.net (Djalil Chafaï) Thu, 09 Jan 2014 00:05:12 -0800 OJS http://blogs.law.harvard.edu/tech/rss 60 A Note on the $S_2(\delta)$ Distribution and the Riemann Xi Function http://ecp.ejpecp.org/article/view/3608 The theory of $S_2(\delta)$ family of probability distributions is used to give a derivation of the functional equation of the Riemann xi function. The $\delta$ deformation of the xi function is formulated in terms of the $S_2(\delta)$ distribution and shown to satisfy Riemann's functional equation.  Criteria for simplicity of roots of the xi function and for its simple roots to satisfy the Riemann hypothesis are formulated in terms of a differentiability property of the $S_2(\delta)$ family. For application, the values of the Riemann zeta function at the integers and of the Riemann xi function in the complex plane are represented as integrals involving the Laplace transform of $S_2.$ Dmitry Ostrovsky http://ecp.ejpecp.org/article/view/3608 Thu, 11 Dec 2014 01:59:56 -0800 On the range of subordinators http://ecp.ejpecp.org/article/view/3629 In this note we look into detail into the box-counting dimension of subordinators. Given that X is a non-decreasing Levy process which is not Compound Poisson process we show that in the limit, a.s., the minimum number of boxes of size $a$ that cover the range of $(X_s)_{s\leq t}$ is a.s. of order $t/U(a)$, where U is the potential function of X. This is a more rened result than the lower and upper index of the box-counting dimension computed by Jean Bertoin in his 1999 book, which deals with the asymptotic of the number of boxes at logarithmic scale. Mladen Svetoslavov Savov http://ecp.ejpecp.org/article/view/3629 Thu, 11 Dec 2014 00:50:24 -0800 Lower bounds on the smallest eigenvalue of a sample covariance matrix. http://ecp.ejpecp.org/article/view/3807 We provide tight lower bounds on the smallest eigenvalue of a sample covariance matrix of a centred isotropic random vector under weak or no assumptions on its components. Pavel Yaskov http://ecp.ejpecp.org/article/view/3807 Sat, 06 Dec 2014 07:11:11 -0800 Large gaps asymptotics for the 1-dimensional random Schr¨odinger operator http://ecp.ejpecp.org/article/view/2724 We show that in the Schr\"{o}dinger point process, Sch$_\tau$, $\tau&gt;0,$ the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by <br />\[ <br />\exp\left(-\frac{\lambda^2}{4\tau}+\left(\frac{2}{\tau}-\frac{1}{4}\right)\lambda +o(\lambda)\right), <br />\] <br />as $\lambda\to\infty.$ It is a slightly more precise version than the one given in a previous work. Stephanie S.M. Jacquot http://ecp.ejpecp.org/article/view/2724 Wed, 26 Nov 2014 10:08:25 -0800 A note on the strong formulation of stochastic control problems with model uncertainty http://ecp.ejpecp.org/article/view/3436 We consider a  Markovian stochastic control problem with  model uncertainty. The controller (intelligent player) observes only the state, and, therefore, uses feedback (closed-loop) strategies.  The adverse player (nature) who does not have a direct interest in the payoff, chooses open-loop controls that parametrize Knightian uncertainty. This creates a two-step optimization  problem (like half of a game) over feedback strategies and open-loop controls. The main result is to show that, under some assumptions, this provides the  same value as the  (half of) the zero-sum symmetric game where the adverse player  also plays feedback strategies and actively tries to minimize the payoff. The value function is independent of the filtration accessible to the adverse player. Aside from the modeling issue, the present note is a technical companion to a previous work. Mihai Sirbu http://ecp.ejpecp.org/article/view/3436 Wed, 26 Nov 2014 10:00:38 -0800