Approximating the limiting Quicksort distribution

A characterization of the set of fixed points of the Quicksort transformation

The limiting distribution μof the normalized number of key comparisons required by the Quicksort sorting algorithm is known to be the unique fixed point of a certain distributional transformation T -- unique, that is, subject to the constraints of zero mean and finite variance. We show that a distribution is a fixed point of T if and only if it is the convolution of μwith a Cauchy distribution of arbitrary center and scale. In particular, therefore, μis the unique fixed point of T having zero mean.

Distributional convergence for the number of symbol comparisons used by QuickSort

Most previous studies of the sorting algorithm QuickSort have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the n independent and identically distributed (i.i.d.) keys are each represented as a sequence of symbols from a probabilistic source and that QuickSort operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild "tameness" condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first center…

Convergence of an Adaptive Approximation Scheme for the Wiener Process

The problem of approximating/tracking the value of a Wiener process is considered. The discretization points are placed at times when the value of the process differs from the approximation by some amount, here denoted by eta. It is found that the limiting difference, as eta goes to 0, between the approximation and the value of the process normalized with eta converges in distribution to a triangularly distributed random variable.

Limiting spectral distribution for large sample correlation matrices

In this paper, we consider the empirical spectral distribution of the sample correlation matrix and investigate its asymptotic behavior under mild assumptions on the data's distribution, when dimension and sample size increase at the same rate. First, we give a characterization for the limiting spectral distribution to follow a Marchenko-Pastur law assuming that the underlying data matrix consists of i.i.d. entries. Subsequently, we provide the limiting spectral distribution of the sample correlation matrix when allowing for a dependence structure within the columns of the data matrix. In cont…

Large Portfolio Asymptotics for Loss From Default

We prove a law of large numbers for the loss from default and use it for approximating the distribution of the loss from default in large, potentially heterogenous portfolios. The density of the limiting measure is shown to solve a non-linear SPDE, and the moments of the limiting measure are shown to satisfy an infinite system of SDEs. The solution to this system leads to %the solution to the SPDE through an inverse moment problem, and to the distribution of the limiting portfolio loss, which we propose as an approximation to the loss distribution for a large portfolio. Numerical tests illustr…

Stein's method for the half-normal distribution with applications to limit theorems related to the simple symmetric random walk

We develop Stein's method for the half-normal distribution and apply it to derive rates of convergence in distributional limit theorems for three statistics of the simple symmetric random walk: the maximum value, the number of returns to the origin and the number of sign changes up to a given time $n$. We obtain explicit error bounds with the optimal rate $n^{-1/2}$ for both the Kolmogorov and the Wasserstein metric. In order to apply Stein's method, we compare the characterizing operator of the limiting half-normal distribution with suitable characterizations of the discrete approximating dis…

Banach 空间中半线性随机发展方程的离散到连续极限

Discrete-to-continuum limits of semilinear stochastic evolution equations in Banach spaces

本文建立 Banach 空间中半线性随机发展方程温和解在共同状态空间上的离散到连续收敛框架，并用于点云图离散的流形 Whittle-Matérn 型随机 SPDE。

BSΔEs and BSDEs with non-Lipschitz drivers: Comparison, convergence and robustness

We provide existence results and comparison principles for solutions of backward stochastic difference equations (BS$Δ$Es) and then prove convergence of these to solutions of backward stochastic differential equations (BSDEs) when the mesh size of the time-discretizaton goes to zero. The BS$Δ$Es and BSDEs are governed by drivers $f^N(t,ω,y,z)$ and $f(t,ω,y,z),$ respectively. The new feature of this paper is that they may be non-Lipschitz in z. For the convergence results it is assumed that the BS$Δ$Es are based on d-dimensional random walks $W^N$ approximating the d-dimensional Brownian motion…