Složená Poissonova rozložení
Small and big probability worlds
Small and large time stability of the time taken for a Lévy process to cross curved boundaries
This paper is concerned with the small time behaviour of a Lévy process . In particular, we investigate thestabilitiesof the times, and , at which , started with , first leaves the space-time regions (one-sided exit), or (two-sided exit), , as . Thus essentially we determine whether or not these passage times behave like deterministic functions in the sense of different modes of convergence; specifically convergence in probability, almost surely and in . In many instances these are...
Small ball probabilities for stable convolutions
We investigate the small deviations under various norms for stable processes defined by the convolution of a smooth function with a real SαS Lévy process. We show that the small ball exponent is uniquely determined by the norm and by the behaviour of f at zero, which extends the results of Lifshits and Simon, Ann. Inst. H. Poincaré Probab. Statist.41 (2005) 725–752 where this was proved for f being a power function (Riemann-Liouville processes). In the Gaussian case, the same generality as...
Small ball probability estimates in terms of width
A certain inequality conjectured by Vershynin is studied. It is proved that for any symmetric convex body K ⊆ ℝⁿ with inradius w and γₙ(K) ≤ 1/2 we have for any s ∈ [0,1], where γₙ is the standard Gaussian probability measure. Some natural corollaries are deduced. Another conjecture of Vershynin is proved to be false.
Small counts in the infinite occupancy scheme.
Small deviation probabilities for a class of distributions with a polynomial decreasing at zero.
Small deviations for fractional stable processes
Small deviations of Gaussian random fields in -spaces.
Small deviations of iterated processes in the space of trajectories
We derive logarithmic asymptotics of probabilities of small deviations for iterated processes in the space of trajectories. We find conditions under which these asymptotics coincide with those of processes generating iterated processes. When these conditions fail the asymptotics are quite different.
Small diffusion and fast dying out asymptotics for superprocesses as non-Hamiltonian quasiclassics for evolution equations.
Small perturbation of diffusions in inhomogeneous media
Small perturbations with large effects on value-at-risk
We show that in the delta-normal model there exist perturbations of the Gaussian multivariate distribution of the returns of a portfolio such that the initial marginal distributions of the returns are statistically undistinguishable from the perturbed ones and such that the perturbed V@R is close to the worst possible V@R which, under some reasonable assumptions, is the sum of the V@Rs of each of the portfolio assets.
Small positive values for supercritical branching processes in random environment
Branching Processes in Random Environment (BPREs) are the generalization of Galton–Watson processes where in each generation the reproduction law is picked randomly in an i.i.d. manner. In the supercritical case, the process survives with positive probability and then almost surely grows geometrically. This paper focuses on rare events when the process takes positive but small values for large times. We describe the asymptotic behavior of , as . More precisely, we characterize the exponential...
Small random perturbations of infinite dimensional dynamical systems and nucleation theory
Small scale limit theorems for the intersection local times of Brownian motion.
Small tails for the supremum of a gaussian process
Small time asymptotics of Ornstein-Uhlenbeck densities in Hilbert spaces.
Small time expansions for transition probabilities of some Lev́y processes.
Smallest singular value of sparse random matrices
We extend probability estimates on the smallest singular value of random matrices with independent entries to a class of sparse random matrices. We show that one can relax a previously used condition of uniform boundedness of the variances from below. This allows us to consider matrices with null entries or, more generally, with entries having small variances. Our results do not assume identical distribution of the entries of a random matrix and help to clarify the role of the variances of the entries....