This paper presents moment analyses and characterizations of limit distributions for the construction cost of hash tables under the linear probing strategy. Two models are considered, that of full tables and that of sparse tables with a fixed filling ratio strictly smaller than one. For full tables, the construction cost has expectation O(n3/2), the standard deviation is of the same order, and a limit law of the Airy type holds. (The Airy distribution is a semiclassical distribution that is defined in terms of the usual Airy functions or equivalently in terms of Bessel functions of indices − 1 2 3, 3.) For sparse tables, the construction cost has expectation O(n), standard deviation O ( √ n), and a limit law of the Gaussian type. Combinatorial relations with other problems leading to Airy phenomena (like graph connectivity, tree inversions, tree path length, or area under excursions) are also briefly discussed.

. The Airy distribution (of the \area" type) occurs as limit distribution of cumulative parameters in a number of combinatorial structures, like path length in trees, area below walks, displacement in parking sequences, and it is also related to basic graph and polyomino enumeration. We obtain curious explicit evaluations for certain moments of the Airy distribution, including moments of orders 1; 3; 5; &c, as well as + 1 3 ; 5 3 ; 11 3 ; &c. and 7 3 ; 13 3 ; 19 3 ; &c . Our proofs are based on integral transforms of the Laplace and Mellin type and they rely essentially on \non-probabilistic" arguments like analytic continuation. A by-product of this approach is the existence of relations between moments of the Airy distribution, the asymptotic expansion of the Airy function Ai(z) at +1, and power symmetric functions of the zeros k of Ai(z). For probabilists, the Airy distribution considered here is nothing but the distribution of the area under the Brownian excursion. The ...

We find the distribution of the areas under the positive parts of a Brownian motion process and a Brownian bridge process, and compare these distributions with the corresponding areas for the absolute values of these processes.

Abstract. Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength, the system state will eventually leave the domain of attraction of. We analyse the case when, as, the exit location on the boundary is increasingly concentrated near a saddle point of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter, equal to the ratio of the stable and unstable eigenvalues of the linearized deterministic flow at. If then the exit location distribution is generically asymptotic as! " to a Weibull distribution with shape parameter #$ % , on the &'(*) +-,. lengthscale near. If 0/1 it is generically asymptotic to a distribution on the &'(-23+, lengthscale, whose moments we compute. Our treatment employs both matched asymptotic expansions and stochastic analysis. As a byproduct of our treatment, we clarify the limitations of the traditional Eyring formula for the weak-noise exit time asymptotics. Key words. Stochastic exit problem, large fluctuations, large deviations, Wentzell-Freidlin theory, exit location, saddle point avoidance, first passage time, matched asymptotic expansions, singular perturbation theory, stochastic analysis, Ackerberg-O’Malley resonance. AMS subject classifications. 60J60, 35B25, 34E20 1. Introduction. We

this paper. In Section 3 we show how some refinements of the Feynman-Kac formula may be understood in terms of a decomposition of the Brownian path at the time of the last visit to zero before time ` where ` is an exponentially distributed random time independent of the Brownian motion. We also show how D.Williams' decomposition at the maximum of the generic excursion under Ito's measure translates in terms of solutions of a Sturm-Liouville equation. Finally, Section 4 is devoted to explicit computations of the laws of

Abstract. Several Brownian areas are considered in this paper: the Brownian excursion area, the Brownian bridge area, the Brownian motion area, the Brownian meander area, the Brownian double meander area, the positive part of Brownian bridge area, the positive part of Brownian motion area. We are interested in the asymptotics of the right tail of their density function. Inverting a double Laplace transform, we can derive, in a mechanical way, all terms of an asymptotic expansion. We illustrate our technique with the computation of the first four terms. We also obtain asymptotics for the right tail of the distribution function and for the moments. Our main tool is the two-dimensional saddle point method.

Consider a two-dimensional continuous-time dynamical system, with an attracting fixed point S. If the deterministic dynamics are perturbed by white noise (random perturbations) of strength ffl, the system state will eventually leave the domain of attraction W of S. We analyse the case when, as ffl ! 0, the exit location on the boundary @W is increasingly concentrated near a saddle point H of the deterministic dynamics. We show using formal methods that the asymptotic form of the exit location distribution on @W is generically non-Gaussian and asymmetric, and classify the possible limiting distributions. A key role is played by a parameter ¯, equal to the ratio j s (H)j= u (H) of the stable and unstable eigenvalues of the linearized deterministic flow at H . If ¯ ! 1 then the exit location distribution is generically asymptotic as ffl ! 0 to a Weibull distributionwith shape parameter 2=¯, on the O(ffl ¯=2 ) lengthscale near H . If ¯ ? 1 it is generically asymptotic to a distribution o...

The distribution function of the integral of the absolute value of the Brownian motion was expressed by L.Takács in the form of various series. In the present paper we determine the exact tail asymptotics of this distribution function. The proposed method is applicable to a variety of other Wiener functionals as well.

We analyze the Shell Sort algorithm under the usual random permutation model. Using empirical distribution functions, we recover Louchard's result that the running time of the 1-stage of (2; 1)-Shell Sort has a limiting distribution given by the area under the absolute Brownian bridge. The analysis extends to (h; 1)-Shell Sort where we nd a limiting distribution given by the sum of areas under correlated absolute Brownian bridges. A variation of (h; 1)-Shell Sort which is slightly more ecient is presented and its asymptotic behavior analyzed. 1 Research supported in part by National Science Foundation grant ?? 2 Research supported in part by National Science Foundation grant DMS-Research supported in part by National Science Foundation grant DMS-9532039 and NIAID grant 2R01 AI291968-04. AMS 1980 subject classications. Primary: 62E17; secondary 65D20. Key words and phrases. Brownian bridge, empirical process, keys, permutation, sorting. 1 1. Introduction Shell Sort is an algor...

This paper provides an alternative approach to the structural credit risk mod-els. The first-passage-time approach extends the original Merton [Journal of Fi-nance 29, 449-470] model by accounting for the fact that the default may occur not only at the debt’s maturity, but also prior to this date. Default happens when the firm value process crosses an exhaust barrier. In contrast, this paper defines default as the first time the firm value process crosses a barrier, and the area un-der the barrier is greater than the exogenous level. This technique is used to price risky debt as an example.