We consider the one-sided exit problem for (fractionally) integrated random walks and L 'evy processes. We prove that the rate of decrease of the non-exit probability -- the so-called survival exponent -- is universal in this class of processes. In particular, the survival exponent can be inferred from the (fractionally) integrated Brownian motion. This, in particular, extends Sinai's result on the survival exponent for the integrated simple random walk to general random walks with some finite exponential moment. Further, we prove existence and monotonicity of the survival exponent of fractionally integrated processes. We show that this exponent is related to a constant appearing in the study of random polynomials.
We further mention a recent work of Simon @cite_31 , where the problem is studied for certain integrated stable L vy processes (in particular, with heavy tails). Even though we also study integrated L vy processes in this paper, the results and techniques are completely disjoint.
@cite_31
2963290535 168790500 2092543127 950823788
In this paper, we consider the problem of linear regression with heavy-tailed distributions. Different from previous studies that use the squared loss to measure the performance, we choose the absolute loss, which is more robust in the presence of large prediction errors. To address the challenge that both the input and output could be heavy-tailed, we propose a truncated minimization problem, and demonstrate that it enjoys an O( √ d n ) excess risk, where d is the dimensionality and n is the number of samples. Compared with traditional work on l1-regression, the main advantage of our result is that we achieve a high-probability risk bound without exponential moment conditions on the input and output. Furthermore, if the input is bounded, we show that the classical empirical risk minimization is competent for l1-regression even when the output is heavy-tailed. Abstract We study the behavior at infinity of the tail of the stationary solution of a multidimensional linear auto-regressive process with random coefficients. We exhibit an extended class of multiplicative coefficients satisfying a condition of irreducibility and proximality that yield to a heavy tail behavior. To cite this article: B. de , C. R. Acad. Sci. Paris, Ser. I 339 (2004). We describe the wavelet–vaguelette decomposition (WVD) of a linear inverse problem. It is a substitute for the singular value decomposition (SVD) of an inverse problem, and it exists for a class of special inverse problems of homogeneous type—such as numerical differentiation, inversion of Abel-type transforms, certain convolution transforms, and the Radon transform. We propose to solve ill-posed linear inverse problems by nonlinearly "shrinking" the WVD coefficients of the noisy, indirect data. Our approach offers significant advantages over traditional SVD inversion in recovering spatially inhomogeneous objects. We suppose that observations are contaminated by white noise and that the object is an unknown element of a Besov space. We prove that nonlinear WVD shrinkage can be tuned to attain the minimax rate of convergence, for L2 loss, over the entire scale of Besov spaces. The important case of Besov spaces Bσp,q, p < 2, which model spatial inhomogeneity, is included. In comparison, linear procedures— SVD included—cannot attain optimal rates of convergence over such classes in the case p < 2. For example, our methods achieve faster rates of convergence for objects known to lie in the bump algebra or in bounded variation than any linear procedure. Using a finite element method to solve the Helmholtz equation leads to a sparse system of equations which in three dimensions is too large to solve directly. It is also non-Hermitian and highly indefinite and consequently difficult to solve iteratively. The approach taken in this paper is to precondition this linear system with a new preconditioner and then solve it iteratively using a Krylov subspace method. Numerical analysis shows the preconditioner to be effective on a simple 1D test problem, and results are presented showing considerable convergence acceleration for a number of different Krylov methods for more complex problems in 2D, as well as for the more general problem of harmonic disturbances to a non-stagnant steady flow.