An insensitivity property of Lundberg's estimate for delayed claims

This short note shows that the Lundberg's exponential upperbound in the ruin problem of non-life insurance with compound Poisson claims is also valid for the Poisson shot noise delayed claims model, and that the optimal exponent depends only on the distribution of the total claim per accident, not on the time it takes to honor the claim. This result holds under Cramer's condition

An insensitivity property for light-tail shot noise traffic overflow asymptotics

In this short note, we derive the large deviations estimate of the tail of the buffer occupancy distribution in a communications link with a very general integrated Poisson shot noise model for the total input. The result is obtained by a straightforward application of the results of Duffield and O'Connell. The interesting outcome of these computations is that, in the light-tail case, it is largely model independent, and this makes the statistical analysis of traffic in view of link dimensioning, with a procedure implementable by a simple network-based software

Projections, Pseudo-Stopping Times and the Immersion Property

Given two filtrations $\mathbb F \subset \mathbb G$, we study under which conditions the $\mathbb F$-optional projection and the $\mathbb F$-dual optional projection coincide for the class of $\mathbb G$-optional processes with integrable variation. It turns out that this property is equivalent to the immersion property for $\mathbb F$ and $\mathbb G$, that is every $\mathbb F$-local martingale is a $\mathbb G$-local martingale, which, equivalently, may be characterised using the class of $\mathbb F$-pseudo-stopping times. We also show that every $\mathbb G$-stopping time can be decomposed into the minimum of two barrier hitting times

Stationary IPA Estimates for Non-Smooth G/G/1/∞\infty Functionals via Palm Inversion and Level-Crossing Analysis

We give stationary estimates for the derivative of the expectation of a non-smooth function of bounded variation f of the workload in a G/G/1/$\infty$ queue, with respect to a parameter influencing the distribu- tion of the input process. For this, we use an idea of Konstantopoulos and Zazanis based on the Palm inversion formula, however avoiding a limiting argument by performing the level-crossing analysis thereof globally, via Fubini's theorem. This method of proof allows to treat the case where the workload distribution has a mass at discontinuities of f and where the formula has to be modified. The case where the parameter is the speed of service or/and the time scale factor of the input process is also treated using the same approach

A Markov model for inferring flows in directed contact networks

Directed contact networks (DCNs) are a particularly flexible and convenient class of temporal networks, useful for modeling and analyzing the transfer of discrete quantities in communications, transportation, epidemiology, etc. Transfers modeled by contacts typically underlie flows that associate multiple contacts based on their spatiotemporal relationships. To infer these flows, we introduce a simple inhomogeneous Markov model associated to a DCN and show how it can be effectively used for data reduction and anomaly detection through an example of kernel-level information transfers within a computer.Comment: 12 page

A Theorem on the origin of Phase Transitions

For physical systems described by smooth, finite-range and confining microscopic interaction potentials V with continuously varying coordinates, we announce and outline the proof of a theorem that establishes that unless the equipotential hypersurfaces of configuration space \Sigma_v ={(q_1,...,q_N)\in R^N | V(q_1,...,q_N) = v}, v \in R, change topology at some v_c in a given interval [v_0, v_1] of values v of V, the Helmoltz free energy must be at least twice differentiable in the corresponding interval of inverse temperature (\beta(v_0), \beta(v_1)) also in the N -> \infty$limit. Thus the occurrence of a phase transition at some \beta_c =\beta(v_c) is necessarily the consequence of the loss of diffeomorphicity among the {\Sigma_v}_{v < v_c}$ and the {\Sigma_v}_{v > v_c}, which is the consequence of the existence of critical points of V on \Sigma_{v=v_c}, that is points where \nabla V=0.Comment: 10 pages, Statistical Mechanics, Phase Transitions, General Theory. Phys. Rev. Lett., in pres

A stochastic-Lagrangian particle system for the Navier-Stokes equations

This paper is based on a formulation of the Navier-Stokes equations developed by P. Constantin and the first author (\texttt{arxiv:math.PR/0511067}, to appear), where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. In this paper, we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with $\frac{1}{N}$ times the sum over these $N$ copies. (We remark that our formulation requires one to keep track of $N$ stochastic flows of diffeomorphisms, and not just the motion of $N$ particles.) We prove that in two dimensions, this system of interacting diffeomorphisms has (time) global solutions with initial data in the space \holderspace{1}{\alpha} which consists of differentiable functions whose first derivative is $\alpha$ H\"older continuous (see Section \ref{sGexist} for the precise definition). Further, we show that as $N \to \infty$ the system converges to the solution of Navier-Stokes equations on any finite interval $[0,T]$. However for fixed $N$, we prove that this system retains roughly $O(\frac{1}{N})$ times its original energy as $t \to \infty$. Hence the limit $N \to \infty$ and $T\to \infty$ do not commute. For general flows, we only provide a lower bound to this effect. In the special case of shear flows, we compute the behaviour as $t \to \infty$ explicitly.Comment: v3: Typo fixes, and a few stylistic changes. 17 pages, 2 figure

Metastability in Markov processes

We present a formalism to describe slowly decaying systems in the context of finite Markov chains obeying detailed balance. We show that phase space can be partitioned into approximately decoupled regions, in which one may introduce restricted Markov chains which are close to the original process but do not leave these regions. Within this context, we identify the conditions under which the decaying system can be considered to be in a metastable state. Furthermore, we show that such metastable states can be described in thermodynamic terms and define their free energy. This is accomplished showing that the probability distribution describing the metastable state is indeed proportional to the equilibrium distribution, as is commonly assumed. We test the formalism numerically in the case of the two-dimensional kinetic Ising model, using the Wang--Landau algorithm to show this proportionality explicitly, and confirm that the proportionality constant is as derived in the theory. Finally, we extend the formalism to situations in which a system can have several metastable states.Comment: 30 pages, 5 figures; version with one higher quality figure available at http://www.fis.unam.mx/~dsanders

Optimally coherent sets in geophysical flows: A new approach to delimiting the stratospheric polar vortex

The "edge" of the Antarctic polar vortex is known to behave as a barrier to the meridional (poleward) transport of ozone during the austral winter. This chemical isolation of the polar vortex from the middle and low latitudes produces an ozone minimum in the vortex region, intensifying the ozone hole relative to that which would be produced by photochemical processes alone. Observational determination of the vortex edge remains an active field of research. In this letter, we obtain objective estimates of the structure of the polar vortex by introducing a new technique based on transfer operators that aims to find regions with minimal external transport. Applying this new technique to European Centre for Medium-Range Weather Forecasts (ECMWF) ERA-40 three-dimensional velocity data we produce an improved three-dimensional estimate of the vortex location in the upper stratosphere where the vortex is most pronounced. This novel computational approach has wide potential application in detecting and analysing mixing structures in a variety of atmospheric, oceanographic, and general fluid dynamical settings

Analytical probabilistic approach to the ground state of lattice quantum systems: exact results in terms of a cumulant expansion

We present a large deviation analysis of a recently proposed probabilistic approach to the study of the ground-state properties of lattice quantum systems. The ground-state energy, as well as the correlation functions in the ground state, are exactly determined as a series expansion in the cumulants of the multiplicities of the potential and hopping energies assumed by the system during its long-time evolution. Once these cumulants are known, even at a finite order, our approach provides the ground state analytically as a function of the Hamiltonian parameters. A scenario of possible applications of this analyticity property is discussed.Comment: 26 pages, 5 figure