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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: 57 (3 UL) Approximation of Fractional Local Times: Zero Energy and DerivativesJaramillo Gil, Arturo ; Nourdin, Ivan ; Peccati, Giovanni in Annals of Applied Probability (2021), In pressDetailed reference viewed: 64 (4 UL) Edgeworth expansion for Euler approximation of continuous diffusion processesPodolskij, Mark ; Veliyev, Bezirgen; Yoshida, Nakahiroin Annals of Applied Probability (2020), 30(4), 19712003Detailed reference viewed: 66 (4 UL) Gaussian Phase Transitions and Conic Intrinsic Volumes: Steining the Steiner FormulaGoldstein, Larry; Nourdin, Ivan ; Peccati, Giovanni in Annals of Applied Probability (2017), 27(1), 1-47Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical ... [more ▼]Intrinsic volumes of convex sets are natural geometric quantities that also play important roles in applications, such as linear inverse problems with convex constraints, and constrained statistical inference. It is a well-known fact that, given a closed convex cone $C\subset \mathbb{R}^d$, its conic intrinsic volumes determine a probability measure on the finite set $\{0,1,...d\}$, customarily denoted by $\mathcal{L}(V_C)$. The aim of the present paper is to provide a Berry-Esseen bound for the normal approximation of ${\cal L}(V_C)$, implying a general quantitative central limit theorem (CLT) for sequences of (correctly normalised) discrete probability measures of the type $\mathcal{L}(V_{C_n})$, $n\geq 1$. This bound shows that, in the high-dimensional limit, most conic intrinsic volumes encountered in applications can be approximated by a suitable Gaussian distribution. Our approach is based on a variety of techniques, namely: (1) Steiner formulae for closed convex cones, (2) Stein's method and second order Poincar\'e inequality, (3) concentration estimates, and (4) Fourier analysis. Our results explicitly connect the sharp phase transitions, observed in many regularised linear inverse problems with convex constraints, with the asymptotic Gaussian fluctuations of the intrinsic volumes of the associated descent cones. In particular, our findings complete and further illuminate the recent breakthrough discoveries by Amelunxen, Lotz, McCoy and Tropp (2014) and McCoy and Tropp (2014) about the concentration of conic intrinsic volumes and its connection with threshold phenomena. As an additional outgrowth of our work we develop total variation bounds for normal approximations of the lengths of projections of Gaussian vectors on closed convex sets. [less ▲]Detailed reference viewed: 186 (10 UL) New Kolmogorov bounds for functionals of binomial point processesPeccati, Giovanni ; Lachièze-Rey, Raphaelin Annals of Applied Probability (2017), 27(4), 1992-20131Detailed reference viewed: 108 (4 UL) Linear and quadratic functionals of random hazard rates: an asymptotic analysisPeccati, Giovanni ; Prünster, Igorin Annals of Applied Probability (2008), 18(5), 1910--1943Detailed reference viewed: 147 (0 UL) 1