We consider a centered random walk with finite variance and investigate the asymptotic behaviour of the probability that the area under this walk remains positive up to a large time n. Assuming that the moment of order 2 + δ is finite, we show that the exact asymptotics for this probability are n−1/4. To show these asymptotics we develop a discrete potential theory for integrated random walks.

We will introduce a class of (m-times) integrated Brownian motions. The exact asymptotic expansions for the L2 small ball probabilities will be discussed for members of this class, of which the usual m-times integrated Brownian motion is an example. Another example will be what we call an Euler-integrated Brownian motion. We will also find very sharp estimates for the asymptotics of the eigenva...

A trace formula for Toeplitz operators was proved by Boutet de Monvel and Guillemin in the setting of general Toeplitz structures. Here we give a local version of this result for a class of Toeplitz operators related to continuous groups of symmetries on quantizable compact symplectic manifolds. The local trace formula involves certain scaling asymptotics along the clean fixed locus of the Hami...

Abstract. We generalize the classical work of de Bruijn, Knuth and Rice (giving the asymptotics of the average height of Dyck paths of length n) to the case of p– watermelons with a wall (i.e., to a certain family of p nonintersecting Dyck paths; simple Dyck paths being the special case p = 1.) We work out this asymptotics for the case p = 2 only, since the computations involved are already qui...

We establish the precise asymptotics of the quantization and entropy coding errors for fractional Brownian motion with respect to the supremum norm and L[0, 1]-norm distortions. We show that all moments in the quantization problem lead to the same asymptotics. Using a general principle, we conclude that entropy coding and quantization coincide asymptotically. Under supremum-norm distortion, our...

Abelian differentials on Riemann surfaces can be seen as translation surfaces, which are flat surfaces with cone-type singularities. Closed geodesics for the associated flat metrics form cylinders whose number under a given maximal length was proved by Eskin and Masur to generically have quadratic asymptotics in this length, with a common coefficient constant for the quadratic asymptotics calle...

We discuss in this paper asymptotics of the sample average approximation (SAA) of the optimal value of a minimax stochastic programming problem. The main tool of our analysis is a specific version of the infinite dimensional delta method. As an example, we discuss asymptotics of SAA of risk averse stochastic programs involving the absolute semideviation risk measure. Finally, we briefly discuss...

Hyperbolic systems of second-order differential equations are considered in a domain with conical points at the boundary; in particular, the equations of elastodynamics are discussed. The asymptotics of solutions near conical points is studied. The “hyperbolic character” of the asymptotics shows itself in the properties of the coefficients (stress intensity factors) depending on time. Some form...

We propose a novel technique for analyzing the long-time asymptotics of integrable wave equations in the case where the underlying isospectral problem has purely discrete spectrum. To this end we introduce a natural coupling problem for entire functions which serves as a replacement for the usual Riemann–Hilbert approach, which does not work for these kind of problems. As a prototypical example...

The paper discusses the spectrum of Toeplitz operators in Bargmann spaces. Our Toeplitz operators have real symbols with a variable sign and a compact support. A class of examples is considered where the asymptotics of the eigenvalues of such operators can be computed. These examples show that this asymptotics depends on the geometry of the supports of the positive and negative parts of the sym...