Let $p(x, y)$ be an arbitrary random walk on $Z^d$. Let $\xi_t$ be the system of coalescing random walks based on $p$, starting with all sites occupied, and let $\eta_t$ be the corresponding system of annihilating random walks. The spatial rescalings $P(0 \in \xi_t)^{1/d}\xi_t$ for $t \geqq 0$ form a tight family of point processes on $R^d$. Any limiting point process as $t \rightarrow\infty$ has Lesbegue measure as its intensity, and has no multiple points. When $p$ is simple random walk on $Z^d$ these rescalings converge in distribution, to the simple Poisson point process for $d \geq 2$, and to a non-Poisson limit for $d = 1$. For a large class of $p$, we prove that $P(0 \in \eta_t)/P(0 \in \xi_t) \rightarrow 1/2$ as $t \rightarrow\infty$. A generalization of this result, proved for nearest neighbor random walks on $Z^1$, and for all multidimensional $p$, implies that the limiting point process for rescalings $P(0 \in \xi_t)^{1/d}\eta_t$ of the system of annihilating random walks is the one half thinning of the limiting point process for the corresponding coalescing system.

The length of the longest monotone increasing subsequence of a random sample of size $n$ is known to have expected value asymptotic to $2n^{1/2}$. We prove that it is possible to make sequential choices which give an increasing subsequence of expected length asymptotic to $(2n)^{1/2}$. Moreover, this rate of increase is proved to be asymptotically best possible.

Let $\{X_i\}$ be a sequence of independent, identically distributed non-degenerate random variables taking values in $\mathbb{R}^d$ and $S_n = \sum^n_{i = 1} X_i, M_n = \max_{1\leqq i \leqq n} |S_i|$. Define for $x > 0, G(x) = P\{| X_1 | > x\}, K(x) = x^{-2}E(| X_1 |^2 1\{| X_1 | \leq x\}), M(x) = x^{-1} |E(X_1 1\{| X_1 | \leq x\})|,$ and $h(x) = G(x) + K(x) + M(x)$. Then if $\beta = \sup \{\alpha: \lim \sup x^\alpha h(x) = 0\}, \delta = \sup \{\alpha: \lim \inf x^\alpha h(x) = 0\}$, it is proved that $n^{-1/\alpha}M_n \rightarrow 0$ for $\alpha < \beta, \rightarrow \infty$ for $\alpha > \delta$, while the $\lim \inf$ is 0 and the $\lim \sup$ is $\infty$ for $\beta < \alpha < \delta$. Some alternative characterizations of the indices $\beta, \delta$ are obtained as well as the analogous results for Levy processes.

The optional sampling theorem for martingales indexed by a partially ordered set is true if the index set is directed. However, the corresponding result for submartingales indexed by a partially ordered set is not true in general. In this paper we completely characterize the class of stopping times for which the optional sampling theorem is true for all uniformly bounded submartingales indexed by countable partially ordered sets. By assuming a conditional independence property, we show that when the index set is $R^2$ the optional sampling theorem is true for all uniformly bounded, right continuous submartingales and all stopping times. This conditional independence property is satisfied in cases where the submartingales and stopping times are measurable with respect to the two-parameter Wiener process. A counterexample shows that the optional sampling result is false for $R^n$ when $n > 2$ even if the conditional independence property is satisfied.

Asymptotic behavior of large deviations of empirical distribution functions (df's) is considered. Borovkov (1967) and Hoadley (1967) obtained results for functionals continuous in the sup norm topology on the set of df's. Groeneboom, Oosterhoff, and Ruymgaart (1979) extended this to functionals continuous in a stronger $\tau$-topology. This result is now extended to functionals that are $\tau$-continuous only on a particular useful subset of df's. Applications to the Anderson-Darling statistic and linear combinations of order statistics are considered. We begin by correcting the work of Abrahamson (1967); from this the role of the key weight function $\psi(t) = -\log t(1 - t)$ is discovered. It is then exploited to the end indicated above, and it is considered as a weight function in tests of fit.

Let $X_1, X_2,\cdots$ be an i.i.d. sequence with $EX_1 = 0, EX^2_1 = 1, Ee^{tX_1} < \infty (|t| < t_0)$, and partial sums $S_n = X_1 + \cdots + X_n$. Starting from some analogous results for the Wiener process, this paper studies the almost sure limiting behaviour of $\max_{0 \leq n \leq N - a_N} a^{-1/2}_N (S_{n + a_N} - S_n)$ as $N \rightarrow \infty$ under various conditions on the integer sequence $a_N$. Improvements of the Erdos-Renyi law of large numbers for partial sums are obtained as well as strong invariance principle-type versions via the Komlos-Major-Tusnady approximation. An appearing gap between these two results is also going to be closed.

We study Banach-space-valued martingale transforms and, in particular, characterize those Banach spaces for which the classical theorems of the real-valued case carry over. For example, if $B$ is a Banach space and $1 < p < \infty$, then there exists a positive real number $c_p$ such that $\|\epsilon_1d_1 + \cdots + \epsilon_n d_n \|_p \leq c_p \|d_1 + \cdots + d_n \|_p$ for all $B$-valued martingale difference sequences $d = (d_1, d_2,\cdots)$ and all numbers $\epsilon_1, \epsilon_2,\cdots$ in $\{-1, 1\}$ if and only if there is a symmetric biconvex function $\zeta$ on $B \times B$ satisfying $\zeta(0, 0) > 0$ and $\zeta(x, y) \leq | x + y | \text{if} | x | \leq 1 \leq | y |$.

Let $\{X(t), t \in \lbrack 0, \infty)\}$ be a subordinator whose Levy spectral function $H(x)$ satisfies the inequality $c_1x^{-\alpha} \leq - H(x) \leq c_2x^{-\alpha},$ for all $x > 0$, for a $\alpha \in (0, 1)$ and for certain constants $c_1$ and $c_2, 0 < c_1 \leq c_2 < \infty$. In this paper we obtain (in the $M_1$ topology) the set of all almost sure limit functions of the sequence $(n^{-1/\alpha}X(nt))^{\frac{1}{\log \log n}}, t \in \lbrack 0, 1\rbrack, n \geq 3.$

The technique of raw time-change is applied to give another proof that the Knight-Pittenger procedure of deleting excursions of a strong Markov process from a set $A$ which meet a disjoint set $B$ yields a strong Markov process. A natural filtration is associated with the new process, and generalizations are given. Under natural hypotheses, the debuts of a class of nonadapted homogeneous sets are shown to be killing times of a strong Markov process. These are generalized (i.e. raw) terminal times. Let $A_t$ be an increasing nonadapted continuous process, and let $T_t$ be its right continuous inverse satisfying a hypothesis which ensures that the collection of $\sigma$-fields $\mathscr{F}_{T(t)}$ is increasing. The optional times of $\mathscr{F}_{T(t)}$ are characterized in terms of killing operators and the points of increase of $A$, and it is shown that $\mathscr{F}_{T(t)} = \mathscr{F}_{T(t+)}$.

This note proves a stronger result than Woodroofe's Theorem (1977) on uniform integrability which is a key tool to obtain second order approximations in certain sequential point and interval estimation problems. The conditions for the stronger conclusion are also weaker than the conditions used by Woodroofe (1977) for his theorem.

This paper deals with a wide class of point processes which are subsumed under the name of $z$-processes. These processes are generalizations, in the sense that the initial distribution of the vehicles are not necessarily stationary Poisson, of point processes occurring in a traffic model of Renyi (1964). Using the Laplace functional, we derive the distributions of various $z$-processes when the initial process is stationary Poisson and prove a weak convergence result to the doubly stochastic Poisson process when the initial process is not necessarily Poisson distributed.

In general the median of a random variable is not uniquely determined. This leads to difficulties in estimation and test theory. This paper suggests a limit process for choosing exactly one median. The choice uses the whole distribution of the random variable in a natural and reasonable way.

Assuming points to be polar, a time changed diffusion has zero or an infinity of jumps in any predictable set.