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The classical Marcinkiewicz theorem states that if an entire characteristicfunction $\Psi$ of a non-degenerate real-valued random variable $X$ is of theform $\exp(P(u))$ for some polynomial $P$, then $X$ has to be a Gaussian. Inthis work, we obtain a broad, quantitative extension of this framework inseveral directions, establish central limit theorems (CLTs) with explicit ratesof convergence, and demonstrate Gaussian fluctuations in continuous spinsystems and general classes of point processes. In particular, we obtainquantitative decay estimates on the Kolmogorov-Smirnov distance between $X$ anda Gaussian under the condition that $\Psi$ does not vanish only on a boundeddisk. This leads to quantitative CLTs applicable to very general and possiblystrongly dependent random systems. In spite of the general applicability, ourrates for the CLT match the classic Berry-Esseen bounds for independent sums upto a log factor. We implement this programme for two important classes ofmodels in probability and statistical physics. First, we extend to the settingof continuous spins a popular paradigm for obtaining CLTs for discrete spinsystems that is based on the theory of Lee-Yang zeros, focussing in particularon the XY model, Heisenberg ferromagnets and generalised Ising models.Secondly, we establish Gaussian fluctuations for linear statistics of so-called$\alpha$-determinantal processes for $\alpha \in \mathbb{R}$ (including theusual determinantal, Poisson and permanental processes) under very generalconditions, including in particular higher dimensional settings wherestructural alternatives such as random matrix techniques are not available. Ourapplications demonstrate the significance of having to control thecharacteristic function only on a (small) disk, and lead to CLTs which, to thebest of our knowledge, are not known in generality.

Quantitative Marcinkiewicz's theorem and central limit theorems: applications to spin systems and point processes