The generalized Dickman distribution D θ with parameter θ > 0 is the unique solution to the distributional equality W = d W * , wherewith W non-negative with probability one, U ∼ U [0, 1] independent of W , and = d denoting equality in distribution. Members of this family appear in number theory, stochastic geometry, perpetuities and the study of algorithms. We obtain bounds in Wasserstein type distances between D θ and the distribution of0 MSC 2010 subject classifications: Primary 60F05, 60E99, 91B16 0

We study the total 𝛼-powered length of the rooted edges in a random minimal directed spanning tree -first introduced in Bhatt and Roy ( 2004) -on a Poisson process with intensity s ≥ 1 on the unit cube [0, 1] 𝑑 for 𝑑 ≥ 3. While a Dickman limit was proved in Penrose and Wade (2004) in the case of 𝑑 = 2, in dimensions three and higher, Bai, Lee and Penrose (2006) showed a Gaussian central limit theorem when 𝛼 = 1, with a rate of convergence of the order (log s) −(𝑑−2)∕4 (log log s) (𝑑+1)∕2 . In this article, we extend these results and prove a central limit theorem in any dimension 𝑑 ≥ 3 for any 𝛼 > 0. Moreover, making use of recent results in Stein's method for region-stabilizing functionals, we provide presumably optimal non-asymptotic bounds of the order (log s) −(𝑑−2)∕2 on the Wasserstein and the Kolmogorov distances between the distribution of the total 𝛼-powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable.

Strong embeddings, that is, couplings between a partial sum process of a sequence of random variables and a Brownian motion, have found numerous applications in probability and statistics. We extend Chatterjee's novel use of Stein's method for {−1, +1} valued variables to a general class of discrete distributions, and provide log n rates for the coupling of partial sums of independent variables to a Brownian motion, and results for coupling sums of suitably standardized exchangeable variables to a Brownian bridge.Since the expectation of g(X * ) and g(U X ) agree for any g ∈ C c , with = d denoting distributional equivalence, we obtainSmoothing X by adding an independent random variable Y having the X -zero bias distribution, we obtain the following result which will be used for constructing Stein coefficients for sums. Lemma 2.4. If X is a mean zero random variable with finite non-zero variance, and Y is an independent variable with the X -zero bias distribution, thenfor all Lipschitz functions f and a.e. derivative f ′ for which these expectations exist.

We consider the Gaussian approximation for functionals of a Poisson process that are expressible as sums of region-stabilizing (determined by the points of the process within some specified regions) score functions and provide a bound on the rate of convergence in the Wasserstein and the Kolmogorov distances. While such results have previously been shown in Lachièze-Rey, Schulte and Yukich (2019), we extend the applicability by relaxing some conditions assumed there and provide further insight into the results. This is achieved by working with stabilization regions that may differ from balls of random radii commonly used in the literature concerning stabilizing functionals. We also allow for non-diffuse intensity measures and unbounded scores, which are useful in some applications. As our main application, we consider the Gaussian approximation of number of minimal points in a homogeneous Poisson process in [0, 1] d with d ≥ 2, and provide a presumably optimal rate of convergence.

We consider Gaussian approximation in a variant of the classical Johnson-Mehl birth-growth model with random growth speed. Seeds appear randomly in R d at random times and start growing instantaneously in all directions with a random speed. The location, birth time and growth speed of the seeds are given by a Poisson process. Under suitable conditions on the random growth speed and a weight function h : R d → [0, ∞), we provide sufficient conditions for a Gaussian convergence of the sum of the weights at the exposed points, which are those seeds in the model that are not covered at the time of their birth. Moreover, using recent results on stabilization regions, we provide non-asymptotic bounds on the distance between the normalized sum of weights and a standard Gaussian random variable in the Wasserstein and Kolmogorov metrics.