We consider a multidimensional diffusion process $X$ whose drift and diffusion coefficients depend respectively on a parameter $\lambda$ and $\theta$. This process is observed at $n+1$ equally spaced times $0,{\Delta }_n,2{\Delta }_n,\dots ,n{\Delta }_n$, and $T_n=n{\Delta }_n$ denotes the length of the `observation window'. We are interested in estimating $\lambda$ and/or $\theta$. Under suitable smoothness and identifiability conditions, we exhibit estimators ${\hat{\lambda }}_n$ and ${\hat{\theta }}_n$, such that the variables $\sqrt{n}.({\hat{\theta }}_n-\theta )$ and $\sqrt{T_n}({\hat{\lambda }}_n-\lambda )$ are tight for ${\Delta }_n\to 0$ and $T_n\to \mathrm{\infty }$. When $\lambda$ is known, we can even drop the assumption that $T_n\to \mathrm{\infty }$. These results hold without any kind of ergodicity or even recurrence assumption on the diffusion process.

This paper examines the Gaussian maximum likelihood estimator (GMLE) in the context of a general form of spatial autoregressive and moving average (ARMA) processes with finite second moment. The ARMA processes are supposed to be causal and invertible under the half-plane unilateral order, but not necessarily Gaussian. We show that the GMLE is consistent. Subject to a modification to confine the edge effect, it is also asymptotically distribution-free in the sense that the limit distribution is normal, unbiased and has variance depending only on the autocorrelation function. This is an analogue of Hannan's classic result for time series in the context of spatial processes.

We introduce a class of stochastic differential equations driven by fractional Brownian motion which allow for a constructive method in order to obtain stationary solutions. This leads to a substantial extension of the fractional Ornstein-Uhlenbeck processes. Structural properties of this class of new models are investigated, and their stationary densities are explicitly given.

We investigate the L^α-convergence, 1≤α≤2, of the class of bisexual branching processes with population-size dependent mating, suitably normalized, to a finite limit W such that P(W>0)>0. Through different probabilistic approaches, we provide some necessary and sufficient conditions for such convergence. In particular we establish, by analogy with the classical Kesten and Stigum result for Bienaymé-Galton-Watson processes, a logarithmic criterion for L¹-convergence.

A new method for monotone estimation of a regression function is proposed, which is potentially attractive to users of conventional smoothing methods. The main idea of the new approach is to construct a density estimate from the estimated values $\hat{m}(i/N)$ ($i=1,\dots ,N$) of the regression function and to use these `data' for the calculation of an estimate of the inverse of the regression function. The final estimate is then obtained by a numerical inversion. Compared to the currently available techniques for monotone estimation the new method does not require constrained optimization. We prove asymptotic normality of the new estimate and compare the asymptotic properties with the unconstrained estimate. In particular, it is shown that for kernel estimates or local polynomials the bandwidths in the procedure can be chosen such that the monotone estimate is first-order asymptotically equivalent to the unconstrained estimate. We also illustrate the performance of the new procedure by means of a simulation study.

A saddlepoint approximation is developed for the likelihood ratio test statistic in non-regular exponential models. For the one-parameter case, it is proved that the saddlepoint correction has a relative error of order $O\left(n^{-1}\right)$, assuming that the canonical statistic has four finite moments. The results are applied in reliability theory and survival analysis, testing for exponentiality. Note that we are using the saddlepoint method for statistics without moment generating function.

We present a general and quite simple upper bound for the total variation distance $d_{\mathrm{TV}}$ between any stochastic process $(X_i{)}_{i\in \Gamma }$ defined over a countable space $\Gamma$, and a compound Poisson process on $\Gamma .$ This result is sufficient for proving weak convergence for any functional of the process $(X_i{)}_{i\in \Gamma }$ when the real-valued $X_i$ are rarely non-zero and locally dependent. Our result is established after introducing and employing a generalization of the basic coupling inequality. Finally, two simple examples of application are presented in order to illustrate the applicability of our results.

The aim of this paper is to define the entropy rate of a semi-Markov process with a Borel state space by extending the strong asymptotic equirepartition property (also called the ergodic theorem of information theory or Shannon-McMillan-Breiman theorem) to this class of non-stationary processes. The mean asymptotic equirepartition property (also called the Shannon-McMillan theorem) is also proven to hold. The relative entropy rate between two semi-Markov processes is defined. All earlier results concerning entropy for semi-Markov processes, jump Markov processes and Markov chains thus appear as special cases. Two invariance principles are established for entropy, one for the central limit theorem and the other for the law of the iterated logarithm.

A distribution $F$ on $(-\mathrm{\infty },\mathrm{\infty })$ is said to belong to the class $S\left(\gamma \right)$ for some $\gamma \ge 0$ if ${lim}_{x\to \mathrm{\infty }}\overline{F}(x-u)/\overline{F}\left(x\right)=e^{\gamma u}$ holds for all $u$ and ${lim}_{x\to \mathrm{\infty }}\overline{F^{*2}}\left(x\right)\overline{F}\left(x\right)=2m_F$ exists and is finite. Let $X$ and $Y$ be two independent random variables, where $X$ has a distribution in the class $S\left(\gamma \right)$ and $Y$ is non-negative with an endpoint $\hat{y}=sup\{y:P(Y\le y\left)<1\right\}\in (0,\mathrm{\infty })$. We prove that the product $\mathrm{XY}$ has a distribution in the class $S(\gamma /\hat{y})$. We further apply this result to investigate the tail probabilities of Poisson shot noise processes and certain stochastic equations with random coefficients.

We prove a functional non-central limit theorem for scaled Markov processes generated by pseudo-differential operators of periodic variable order. Two different situations occur. If the measure of the set where the order function attains its minimum α_o is positive with respect to the invariant measure, the limit turns out to be an α_o -stable Lévy process. In the other case the scaled sequence converges in probability to the zero function. The large deviation for this convergence is typical of processes having heavy-tail increments. It turns out that only a finite number of large jumps can be recovered on large scales. We also apply the results in order to understand the recurrence and transience of periodic stable-like processes.