Smoothness and decay properties of the limiting Quicksort density function

A local limit theorem for Quicksort key comparisons via multi-round smoothing

As proved by Régnier and Rösler, the number of key comparisons required by the randomized sorting algorithm QuickSort to sort a list of $n$ distinct items (keys) satisfies a global distributional limit theorem. Fill and Janson proved results about the limiting distribution and the rate of convergence, and used these to prove a result part way towards a corresponding local limit theorem. In this paper we use a multi-round smoothing technique to prove the full local limit theorem.

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…

Determinantal Point Processes in the Flat Limit

Determinantal point processes (DPPs) are repulsive point processes where the interaction between points depends on the determinant of a positive-semi definite matrix. In this paper, we study the limiting process of L-ensembles based on kernel matrices, when the kernel function becomes flat (so that every point interacts with every other point, in a sense). We show that these limiting processes are best described in the formalism of extended L-ensembles and partial projection DPPs, and the exact limit depends mostly on the smoothness of the kernel function. In some cases, the limiting process i…

Rapid Convergence of the Unadjusted Langevin Algorithm: Isoperimetry Suffices

We study the Unadjusted Langevin Algorithm (ULA) for sampling from a probability distribution $ν= e^{-f}$ on $\mathbb{R}^n$. We prove a convergence guarantee in Kullback-Leibler (KL) divergence assuming $ν$ satisfies a log-Sobolev inequality and the Hessian of $f$ is bounded. Notably, we do not assume convexity or bounds on higher derivatives. We also prove convergence guarantees in Rényi divergence of order $q > 1$ assuming the limit of ULA satisfies either the log-Sobolev or Poincaré inequality. We also prove a bound on the bias of the limiting distribution of ULA assuming third-order smooth…

Central limit theorem for Fourier transform and periodogram of random fields

In this paper we show that the limiting distribution of the real and the imaginary part of the double Fourier transform of a stationary random field is almost surely an independent vector with Gaussian marginal distributions, whose variance is, up to a constant, the field's spectral density. The dependence structure of the random field is general and we do not impose any restrictions on the speed of convergence to zero of the covariances, or smoothness of the spectral density. The only condition required is that the variables are adapted to a commuting filtration and are regular in some sense.…

The Gap Dimension and Uniform Laws of Large Numbers for Ergodic Processes

Let F be a family of Borel measurable functions on a complete separable metric space. The gap (or fat-shattering) dimension of F is a combinatorial quantity that measures the extent to which functions f in F can separate finite sets of points at a predefined resolution gamma > 0. We establish a connection between the gap dimension of F and the uniform convergence of its sample averages under ergodic sampling. In particular, we show that if the gap dimension of F at resolution gamma > 0 is finite, then for every ergodic process the sample averages of functions in F are eventually within 10 gamm…

关于分数噪声驱动随机微分方程的（非）平稳密度

On the (Non-)Stationary Density of Fractional-Driven Stochastic Differential Equations

本文利用 Wiener-Liouville 桥表示证明任意 Hurst 参数加性分数噪声 SDE 的平稳与非平稳密度具有 Gaussian 界及参数联合光滑性，并移除分数平均原理中的特设假设。

宽稳定神经网络：样本正则性、函数收敛与贝叶斯反问题

Wide stable neural networks: Sample regularity, functional convergence and Bayesian inverse problems

本文研究权重服从α-稳定分布的全连接随机神经网络在大宽度下的样本正则性与函数收敛，证明其无限宽度极限的样本函数属于分数阶Sobolev-Slobodeckij型拟Banach空间，并将该收敛结果应用于保边缘贝叶斯反问题的函数后验分析。