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Gaussian approximation for sums of region-stabilizing scores Bhattacharjee, Chinmoy ; in Electronic Journal of Probability (2022), 27 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 ... [more ▼] 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 $d \geq 2$, and provide a presumably optimal rate of convergence. [less ▲] Detailed reference viewed: 60 (6 UL)Large degrees in scale-free inhomogeneous random graphs Bhattacharjee, Chinmoy ; in Annals of Applied Probability (2022), 32(1), 696-720 We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that ... [more ▼] We consider a class of scale-free inhomogeneous random graphs, which includes some long-range percolation models. We study the maximum degree in such graphs in a growing observation window and show that its limiting distribution is Frechet. We achieve this by proving convergence of the underlying point process of the degrees to a certain Poisson process. Estimating the index of the power-law tail for the typical degree distribution is an important question in statistics. We prove consistency of the Hill estimator for the inverse of the tail exponent of the typical degree distribution. [less ▲] Detailed reference viewed: 67 (4 UL)Dickman Approximation of weighted random sums in the Kolmogorov distance Bhattacharjee, Chinmoy ; E-print/Working paper (2022) Detailed reference viewed: 13 (0 UL)Gaussian approximation in random minimal directed spanning trees Bhattacharjee, Chinmoy in Random Structures and Algorithms (2021) We study the total $\alpha$-powered length of the rooted edges in a random minimal directed spanning tree - first introduced in \cite{BR04} - on a Poisson process with intensity $s \ge 1$ on the unit cube ... [more ▼] We study the total $\alpha$-powered length of the rooted edges in a random minimal directed spanning tree - first introduced in \cite{BR04} - on a Poisson process with intensity $s \ge 1$ on the unit cube $[0,1]^d$ for $d \ge 3$. While a Dickman limit was proved in \cite{PW04} in the case of $d=2$, in dimensions three and higher, \cite{BLP06} showed a Gaussian central limit theorem when $\alpha=1$, with a rate of convergence of the order $(\log s)^{-(d-2)/4} (\log \log s)^{(d+1)/2}$. In this paper, we extend these results and prove a central limit theorem in any dimension $d \ge 3$ for any $\alpha>0$. 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 $\alpha$-powered length of rooted edges, suitably normalized, and that of a standard Gaussian random variable. [less ▲] Detailed reference viewed: 61 (8 UL)Central limit theorem for a birth-growth model with Poisson arrivals and random growth speed. Bhattacharjee, Chinmoy ; ; E-print/Working paper (2021) Detailed reference viewed: 49 (1 UL) |
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