A comprehensive description is given of the limiting behaviour of normalised pseudo-MLEs of the coefficients in a discrete-time autoregressive process, with nonstochastic regressors, for all cases: stationary, unit root and explosive situations. The residuals are assumed to be independent and...

Let Xi be iidrv's and Sn=X1+X2+...+Xn. When EX21<+[infinity], by the law of the iterated logarithm (Sn-[alpha]n)/(n log n)1/2-->0 a.s. for some constants [alpha]n. Thus the r.v. Y=supn[greater-or-equal, slanted]1[Sn-[alpha]n-([delta]n log n)1/2]+ is a.s.finite when [delta]0. We prove a rate of convergence theorem related to the classical results of Baum and Katz, and apply...</+[infinity],>

A real-variable proof of a functional generalised law of the iterated logarithm due to Kesten, Kuelbs and Zinn is given, and extended to a trimmed case.

A local limit theorem is given for independent noninteger random variables under a condition which is more general than one previously given, and which reduces, in the case of identically distributed random variables, to a well-known result.

We show that if Xi is a stationary sequence for which Sn/Bn converges to a finite non zero random variable of constant sign, where Sn=X1+X2+...+Xn and Bn is a sequence of constants, then Bn is regularly varying with index 1. If in addition [Sigma]P(X1Bn is finite, then EX1 is finite, and if in...