In this paper, we obtain a moderate deviation principle for a class of point processes, i.e. linear Hawkes processes.

This paper studies moderate deviation behaviors of the generalized method of moments and generalized empirical likelihood estimators for generalized estimating equations, where the number of equations can be larger than the number of unknown parameters. We consider two cases for the data...

Let X1,X2,… be independent identically distributed (i.i.d.) random variables with EXk=0, V arXk=1. Suppose that φ(t)≔logEetXk<∞ for all t>−σ0 and some σ00. Let Sk=X1+⋯+Xk and S0=0. We are interested in the limiting distribution of the multiscale scan statisticMn=max0≤i<j≤nSj−Sij−i. We prove that for an appropriate normalizing sequence an, the random variable Mn2−an converges to the Gumbel extreme-value law exp{−e−cx}. The behavior of Mn depends strongly on the distribution of the Xk’s. We distinguish between four cases. In the superlogarithmic case we assume that φ(t)<t2/2 for every t>0. In this case, we show...</j≤nsj−sij−i.></∞>

In this paper, we consider the large and moderate deviation principles for modified Engel continued fractions which is a representation of real numbers in number theory.

We derive a large deviation result for the log-likelihood ratio for testing simple hypotheses in locally stationary Gaussian processes. This result allows us to find explicitly the rates of exponential decay of the error probabilities of type I and type II for Neyman–Pearson tests....

An asymptotic theory is given for autoregressive time series with weakly dependent innovations and a root of the form rho_{n} = 1+c/n^{alpha}, involving moderate deviations from unity when alpha in (0,1) and c in R are constant parameters. The limit theory combines a functional law to a...

A limit theory is established for autoregressive time series that smooths the transition between local and moderate deviations from unity and provides a transitional form that links conventional unit root distributions and the standard normal. Edgeworth expansions of the limit theory are given....