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The local number variance \(\sigma^2(R)\) associated with a spherical sampling window of radius \(R\) enables a classification of many-particle systems in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) according to the degree to which large-scale density fluctuations are suppressed, resulting in a demarcation between hyperuniform and nonhyperuniform phyla. To more completely characterize density fluctuations, we carry out an extensive study of higher-order moments or cumulants, including the skewness \(\gamma_1(R)\), excess kurtosis \(\gamma_2(R)\), and the corresponding probability distribution function \(P[N(R)]\) of a large family of models across the first three space dimensions, including both hyperuniform and nonhyperuniform systems with varying degrees of short- and long-range order. To carry out this comprehensive program, we derive new theoretical results that apply to general point processes, and we conduct high-precision numerical studies. Specifically, we derive explicit closed-form integral expressions for \(\gamma_1(R)\) and \(\gamma_2(R)\) that encode structural information up to three-body and four-body correlation functions, respectively. We also derive rigorous bounds on \(\gamma_1(R)\), \(\gamma_2(R)\), and \(P[N(R)]\) for general point processes and corresponding exact results for general packings of identical spheres. High-quality simulation data for \(\gamma_1(R)\), \(\gamma_2(R)\), and \(P[N(R)]\) are generated for each model. We also ascertain the proximity of \(P[N(R)]\) to the normal distribution via a novel Gaussian â€śdistanceâ€ť metric \(l_2(R)\). Among all models, the convergence to a central limit theorem (CLT) is generally fastest for the disordered hyperuniform processes in two or higher dimensions such that \(\gamma_1(R)\sim l_2(R)\sim R^{-(d+1)/2}\) for large \(R\). The convergence to a CLT is slower for standard nonhyperuniform models and slowest for the â€śantihyperuniformâ€ť model studied here. We prove that one-dimensional hyperuniform systems of class I or any \(d\)-dimensional lattice cannot obey a CLT. Remarkably, we discover a type of universality in that, for all of our models that obey a CLT, the gamma distribution provides a good approximation to \(P[N(R)]\) across all dimensions for intermediate to large values of \(R\), enabling us to estimate the large-\(R\) scalings of \(\gamma_1(R)\), \(\gamma_2(R)\), and \(l_2(R)\). For any \(d\)-dimensional model that â€śdecorrelatesâ€ť or â€ścorrelatesâ€ť with \(d\), we elucidate why \(P[N(R)]\) increasingly moves toward or away from Gaussian-like behavior, respectively. Our work sheds light on the fundamental importance of higher-order structural information to fully characterize density fluctuations in many-body systems across length scales and dimensions, and thus has broad implications for condensed matter physics, engineering, mathematics, and biology.