Parabolic and hyperbolic stochastic partial differential equations in one-dimensional space have been proposed as models for the term structure of interest rates. The solution to these equations is reviewed, and their sample path properties are studied. In the parabolic case the sample paths essentially are H\"older continuous of order $\frac{1}{2}$ in space and $\frac{1}{4}$ in time, and in the hyperbolic case the sample paths essentially are H\"older continuous of order $\frac{1}{2}$ simultaneously in time and space. Parametric likelihood inference given an observation at discrete lattice points in time and space is also considered. The associated infinite-dimensional state-space model is described, and a finite-dimensional approximation is proposed. Conditions are presented under which the resulting approximate maximum likelihood estimator is asymptotically efficient when the number of observations in time increases to infinity at a fixed time step. The asymptotic distribution of the approximative likelihood ratio test for a parabolic equation against the hyperbolic alternative is found to be a truncated chi-square. Moreover, explicit moment estimators are derived which can be used as a starting point for a numerical optimization of the likelihood function.

Contact processes -- and, more generally, interacting particle processes -- can serve as models for a large variety of statistical problems, especially if we allow some simple modifications that do not essentially complicate the mathematical treatment of these processes. In a forthcoming paper we shall begin a statistical study of the supercritical contact process $xi_t ^{\{0\}$ that starts with a single infected site at the origin and is conditioned on survival. There we shall consider the simplest statistical problem imaginable, that is, to find an estimator of the parameter of the process based on observing the set of infected sites at a single time $t$. We shall show that this estimator is consistent as $t \rightarrow \infty$ and establish its limit distribution after proper normalization. First, however, we must push some known properties of the contact process a little further. The present paper is devoted to these matters. In particular, we study the convex hull of the set of infected sites for the conditional $xi_t ^{\{0\}$ process as well as its spatial correlation. We find that under some restrictions this correlation decays faster than any negative power of the distance.

We address the problem of estimating the value of a linear functional $local asymptotic normalitygle f,x \rangle$ from random noisy observations of $y=Ax$ in Hilbert scales. Both the white noise and density observation models are considered. We propose an estimation procedure that adapts to unknown smoothness of $x$, of $f$, and of the noise covariance operator. It is shown that accuracy of this adaptive estimator is worse only by a logarithmic factor than one could achieve in the case of known smoothness. As an illustrative example, the problem of deconvolving a bivariate density with singular support is considered.

Asymptotic expansions of long-memory sequences indexed by piecewise differentiable functionals are investigated, and upper bounds of outer expectations of these functionals are given. These results differ strikingly from the classical theories of empirical processes of independent random variables. Our results go beyond earlier ones by allowing wider classes of function as well as by presenting sharper bounds, and thus provide a more versatile approach for related statistical inferences. A complete characterization of empirical processes for the class of indicator functions is presented, and an application to $M$-estimation is discussed.

For $j=1,\rm dots,J$, let $K_j:\mathbb{R}\to\mathbb{R}$ be measurable bounded functions and $X_{n,j} = \int_\mathbb{R}a_j(n-c_jx)M(\rm dx)$, $n\ge 1$, be $\alphapha$-stable moving averages where $\alphapha\in(0,2)$, $c_j>0$ for $j=1,\rm dots,J$, and $M(\rm dx)$ is an $\alphapha$-stable random measure on $\mathbb{R}$ with the Lebesgue control measure and skewness intensity $Berry--Esseen boundsta\in[-1,1]$. We provide conditions on the functions $a_j$ and $K_j$, $j=1,\rm dots,J$, for the normalized partial sums vector $ N_j^{-1/2} \sum_{n=1}^{N_j} (K_j(X_{j,n})-\rm EK_j (X_{j,n}))$, $j=1,\rm dots,J$, to be asymptotically normal as $N_j\to\infty$. This extends a result established by Tailen Hsing in the context of causal moving averages with discrete-time stable innovations. We also consider the case of moving averages with innovations that are in the stable domain of attraction.

We present an extended definition of the second-order stationarity concept. This is based on the theory of harmonic analysis for semigroups with involution. It provides a spectral representation for a wide class of processes which are non-stationary in the usual weak sense, and allows miscellaneous spectral representation results to be unified. Many applications are given to illustrate the concept. Most of these are already known, %as symmetric, locally %stationary, stationary reducible by space transformation, %multiplicative-stationary processes or processes with independent %increments. but some are new, such as the multiplicative-symmetric processes. We are less concerned with proving fundamental results than with opening up a new field of investigation for spectral representation of non-stationary processes.

We define a negative multinomial distribution on $\mathbb{N}^{n}_0$, where $\mathbb{N}_0$ is the set of non-negative integers, by its probability generating function which will be of the form $(A(a_{1},\rm dots,a_{n})/$ $A(a_{1}z_{2},\rm dots,a_{n}z_{n}))^{\lambdaambda}$, where \[A(\mathbf{z})=\sum\lambdaimits_{T\subset\lambdaeft\{1,2,\rm dots,n\right\}}a_{T}\prod\lambdaimits_{i\in T}z_{i},\] \(a_{\rm emptyset}\neq0\), and \(\lambdaambda\) is a positive number. Finding couples \(\lambdaeft( A,\lambdaambda\right)\) for which we obtain a probability generating function is a difficult problem. We establish necessary and sufficient conditions on the coefficients of \(A\) for which we obtain a probability generating function for any positive number \(\lambdaambda\). In consequence, we obtain all infinitely divisible negative multinomial distributions on $\mathbb{N}^{n}_0$.

We propose a new analytic formula for evaluating the derivatives of a matrix exponential. In contrast to the diagonalization method, eigenvalues and eigenvectors do not appear explicitly in the derivation, although we show that a necessary and sufficient condition for the validity of the formula is that the matrix has distinct eigenvalues. The new formula expresses the derivatives of a matrix exponential in terms of minors, polynomials, the exponential of the matrix as well as matrix inversion, and hence is algebraically more manageable. For sparse matrices, the formula can be further simplified. Two examples are discussed in some detail. For the companion matrix of a continuous-time autoregress\-ive moving average process, the derivatives of the exponential of the companion matrix can be computed recursively. The second example concerns the exponential of the tri\-diagonal transition intensity matrix of a finite-state-space continuous-time Markov chain whose instantaneous transitions must be between adjacent states. We present a numerical study to show that the new method may yield numerically more accurate results than the diagonalization method, at the expense of a slight increase in computation.