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Register a new userModerate Deviation estimates for Nodal Lengths of Random Spherical Harmonics
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We prove Moderate Deviation estimates for nodal lengths of random spherical harmonics both on the whole sphere and on shrinking spherical domains. Central Limit Theorems for the latter were recently established in Marinucci, Rossi and Wigman (2020) and Todino (2020) respectively. Our proofs are based on the combination of a Moderate Deviation Principle by Schulte and Thale (2016) for sequences of random variables living in a fixed Wiener chaos with a well-known result based on the concept of exponential equivalence.
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We consider the branching process in random environment ${Z_n}_{ngeq 0}$, which is a~population growth process where individuals reproduce independently of each other with the reproduction law randomly picked at each generation. We describe precise asymptotics of upper large deviations, i.e. $mathbb{P}[Z_n > e^{rho n}]$. Moreover in the subcritical case, under the Cramer condition on the mean of the reproduction law, we investigate large deviations-type estimates for the first passage time of the branching process in question and its total population size.
In this paper, we are concerned with SIR epidemics in a random environment on complete graphs, where every edges are assigned with i.i.d. weights. Our main results give large and moderate deviation principles of sample paths of this model.
Taking into account some likeness of moderate deviations (MD) and central limit theorems (CLT), we develop an approach, which made a good showing in CLT, for MD analysis of a family $$ S^kappa_t=frac{1}{t^kappa}int_0^tH(X_s)ds, ttoinfty $$ for an ergodic diffusion process $X_t$ under $0.5<kappa<1$ and appropriate $H$. We mean a decomposition with ``corrector: $$ frac{1}{t^kappa}int_0^tH(X_s)ds={rm corrector}+frac{1}{t^kappa}underbrace{M_t}_{rm martingale}. $$ and show that, as in the CLT analysis, the corrector is negligible but in the MD scale, and the main contribution in the MD brings the family ``$ frac{1}{t^kappa}M_t, ttoinfty. $ Starting from Bayer and Freidlin, cite{BF}, and finishing by Wus papers cite{Wu1}-cite{WuH}, in the MD study Laplaces transform dominates. In the paper, we replace the Laplace technique by one, admitting to give the conditions, providing the MD, in terms of ``drift-diffusion parameters and $H$. However, a verification of these conditions heavily depends on a specificity of a diffusion model. That is why the paper is named ``Examples ....
A Cramer-type moderate deviation theorem quantifies the relative error of the tail probability approximation. It provides theoretical justification when the limiting tail probability can be used to estimate the tail probability under study. Chen Fang and Shao (2013) obtained a general Cramer-type moderate result using Steins method when the limiting was a normal distribution. In this paper, Cramer-type moderate deviation theorems are established for nonnormal approximation under a general Stein identity, which is satisfied via the exchangeable pair approach and Steins coupling. In particular, a Cramer-type moderate deviation theorem is obtained for the general Curie--Weiss model and the imitative monomer-dimer mean-field model.
Large and moderate deviation principles are proved for Engel continued fractions, a new type of continued fraction expansion with non-decreasing partial quotients in number theory.
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