Let $(X, \mathscr{A}, P)$ be a probability space. Let $X_1, X_2,\cdots,$ be independent $X$-valued random variables with distribution $P$. Let $P_n := n^{-1}(\delta_{X_1} + \cdots + \delta_{X_n})$ be the empirical measure and let $\nu_n := n^\frac{1}{2}(P_n - P)$. Given a class $\mathscr{C} \subset \mathscr{a}$, we study the convergence in law of $\nu_n$, as a stochastic process indexed by $\mathscr{C}$, to a certain Gaussian process indexed by $\mathscr{C}$. If convergence holds with respect to the supremum norm $\sup_{C \in \mathscr{C}}|f(C)|$, in a suitable (usually nonseparable) function space, we call $\mathscr{C}$ a Donsker class. For measurability, $X$ may be a complete separable metric space, $\mathscr{a} =$ Borel sets, and $\mathscr{C}$ a suitable collection of closed sets or open sets. Then for the Donsker property it suffices that for some $m$, and every set $F \subset X$ with $m$ elements, $\mathscr{C}$ does not cut all subsets of $F$ (Vapnik-Cervonenkis classes). Another sufficient condition is based on metric entropy with inclusion. If $\mathscr{C}$ is a sequence $\{C_m\}$ independent for $P$, then $\mathscr{C}$ is a Donsker class if and only if for some $r, \sigma_m(P(C_m)(1 - P(C_m)))^r < \infty$.

Let $\{\xi_k, k \geqslant 1\}$ be a sequence of random variables uniformly distributed over $\lbrack 0, 1\rbrack$ and let $\mathscr{F}$ be a class of functions on $\lbrack 0, 1\rbrack$ with $\int^1_0 f(x) dx = 0$. In this paper we give upper and lower bounds for $\sup_{f \in \mathscr{F}}|\sigma_{k \leqslant N}f(\xi_k)|$ for the class of functions of variation bounded by 1 and for the class of functions satisfying a Lipschitz condition.

We introduce a new tool into the study of spin-flip processes, which we call controlled spin-flip systems. For each spin-flip process we define a related class of controlled spin-flip systems. Our main theorem states that bounds on the behavior of a spin-flip process can be obtained by studying the behavior of the related controlled spin-flip system. Since controlled spin-flip systems are in general easier to work with than regular spin-flip processes (they correspond to finite state space Markov processes), our main theorem has applications to some of the important problems concerning spin-flip processes. In particular, we discuss several applications to the uniqueness problem. These include proofs of some new results, as well as new proofs of earlier results.

Let $X$ be a standard Markov process with semigroup $(P_t)$. We show how to compute the infinitesimal generators (weak and strong) of the semigroup $Q_tf(x) = E^x\{m_tf(X_t)\}$ with $m_t = \exp(-\tau_t)$ and $\tau_t$ a right continuous, increasing strong additive functional; the computation is in terms of the infinitesimal operators of $(P_t)$ and the Levy system of the joint process $(X, \tau)$.

Formulas for the Hausdorff dimension of the graph, image, and level sets of Gaussian vector fields are given under general conditions which allow for different local behavior of the components and for dependence among them. Conditions for the field to have a local time, to hit any fixed point, and for the image to have positive Lebesgue measure are given, and relations between these properties are discussed. Applications of the results are given and include a discussion of when differentiable planar fields have critical points at fixed levels.

An ergodic theorem is given for the age process $(I(t), Z(t))$ associated with a (possibly transient) semi-Markov chain $(I_n, X_n)^\infty_{n=0}$ whose sojourn times are not exclusively integer valued. Asymptotically the Markov part $(I(t):$ the state occupied at time $t)$ and the renewal part $(Z(t):$ the age in $I(t)$ at time $t)$ split into independent parts. This yields the following ergodic result for a semiregenerative process $V_t$ with embedded semi-Markov chain $(I_n, X_n)^\infty_{n=0}$: $$\lim_{t\rightarrow \infty}\big| \operatorname{Prob}\{V_t \in A\} - \int_\pi\frac{A_\pi}{\mu_\pi} \operatorname{Prob}\{I(t) = d\pi\}\big| = 0$$ where $\pi$ is in the state space of $I_n, \mu_\pi$ is the mean sojourn time in $\pi$ and $A_\pi$ is the mean time $V_t$ is in a set $A$ during a sojourn in $\pi$.

In this paper the asymptotic distribution of a sequence of random variables $(X_n)_{n \in \mathbf{N}}$, given by the recursion $$X_{n+1} = X_n(1 - a_n^2g(X_n)) + a_n Y_n,$$ is considered, where $(Y_n)$ is a sequence of independent identically distributed random variables, $g : \mathbb{R} \rightarrow \mathbb{R}$ is a positive continuous function, and $(a_n)$ is a sequence of positive numbers, going to zero. One application to the Robbins-Monro process is discussed, in which the function $g$ will not be constant. Here the asymptotic distribution is no longer normal.

Let $(Y_n)$ be a recurrent Markov chain with discrete or continuous state space. A model of a birth and death chain $(Z_n)$ controlled by a random environment $(Y_n)$ is formulated wherein the bivariate process $(Y_n, Z_n)$ is taken to be Markovian and the marginal process $(Z_n)$ is a birth and death chain on the nonnegative integers with absorbing state $z = 0$ when a fixed sequence of environmental states of $(Y_n)$ is specified. In this paper, the property of uniform $\phi$-recurrence of $(Y_n)$ is used to prove that with probability one the sequence $(Z_n)$ does not remain positive or bounded. An example is given to show that uniform $\phi$-recurrence of $(Y_n)$ is required to insure this instability property of $(Z_n)$. Conditions are given for the extinction of the process $(Z_n)$ when (i) $(Z_n)$ possesses homogeneous transition probabilities and $(Y_n)$ possesses an invariant measure on discrete state space, and (ii) $(Z_n)$ possesses nonhomogeneous transition probabilities and $(Y_n)$ has general state space.

A probability measure $P$ on a partially ordered Polish space $E$ is called stochastically smaller than $Q$ (notation: $P \leqslant Q$) if $\int f dP \leqslant \int f dQ$ holds for all bounded increasing measurable $f$. We investigate the question when for a stochastically increasing family $\{P_t, t \in \mathbb{R}\}$ there exists an increasing process $\{X_t, t \in \mathbb{R}\}$ with 1-dimensional marginal distributions $P_t$. A sufficient condition, satisfied, e.g., for $E = \mathbb{R}^\mathbf{N}$, for compact $E$ and for spaces $E$ of Lipschitz-functions, is the compactness of all intervals $\{z \in E: x \leqslant z \leqslant y\}$; but for general countable $E$ such an increasing $E$-valued process $\{X_t\}$ need not exist.