An asymptotic result is given for the least absolute deviations (LAD) estimation of autoregressive time series with a root of the form ρn=1+c/kn, where kn increases to infinity at a rate slower than n. For c<0, a nkn rate of convergence and asymptotic normality for the serial correlation coefficient are provided. While in the case of c>0, the serial correlation coefficient is shown to have a Cauchy limit distribution with...</0,>

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

Let {X,Xn,n≥1} be a sequence of i.i.d. random variables, and set Sn=X1+⋯+Xn. For 1<p≤3/2, if EX=0, EX2=1 and E[|X|2p(log+|X|)−p]<∞, we prove that limε↘0ε1/2∑n≥2np−2P(|Sn|≥(2p−2+ε)nlogn)=2p−1, thus extending a result by Spătaru (2004). For 1<p≤3/2 and δ>0, we establish conditions for the convergence of the related series ∑n≥2np−2P(|Sn|≥(2p−2)nlogn+δn1/2loglogn(2p−2)nlogn), and derive its precise asymptotic as δ↘1/2.