Let $p^\ast$ be the maximum of $p$ and $q$ where $1 < p < \infty$ and $1/p + 1/q = 1$. If $d = (d_1, d_2, \cdots)$ is a martingale difference sequence in real $L^p(0, 1), \varepsilon = (\varepsilon_1, \varepsilon_2, \cdots)$ is a sequence of numbers in $\{-1, 1\}$, and $n$ is a positive integer, then $\|\sum^n_{k=1} \varepsilon_kd_k\|_p \leq (p^\ast - 1) \|\sum^n_{k=1} d_k\|_p$ and the constant $p^\ast - 1$ is best possible. Furthermore, strict inequality holds if and only if $p \neq 2$ and $\|\sum^n_{k=1} d_k\|_p > 0$. This improves an earlier inequality of the author by giving the best constant and conditions for equality. The inequality holds with the same constant if $\varepsilon$ is replaced by a real-valued predictable sequence uniformly bounded in absolute value by 1, thus yielding a similar inequality for stochastic integrals. The underlying method rests on finding an upper or a lower solution to a novel boundary value problem, a problem with no solution (the upper is not equal to the lower solution) except in the special case $p = 2$. The inequality above, in combination with the work of Ando, Dor, Maurey, Odell, Olevskii, Pelczynski, and Rosenthal, implies that the unconditional constant of a monotone basis of $L^p(0, 1)$ is $p^\ast - 1$. The paper also contains a number of other sharp inequalities for martingale transforms and stochastic integrals. Along with other applications, these provide answers to some questions that arise naturally in the study of the optimal control of martingales.

Aldous recently introduced the notion of synonymity of stochastic processes, a notion of equivalence for processes on a stochastic basis which generalizes the notion of "having the same distribution". We show that generalized martingale properties, such as the semimartingale property, are preserved under synonymity, and that synonymous semimartingales have decompositions with the same distribution law. A variation of our method yields a relatively elementary proof of the theorem of Stricker that semimartingale remains a semimartingale with respect to any subfiltration to which it is adapted.

Consider a random time $\tau$ determined by the evolution of a Markov chain $X$ in discrete time and with discrete state space. Assuming that the pre-$\tau$ and post-$\tau$ processes are conditionally independent given $X_{\tau-1}$ and $0 < \tau < \infty$, it is shown that: (i) the pre-$\tau$ process reversed is Markov and in natural duality to $X$ if and only if $\tau$ is almost surely equal to a modified cooptional time; (ii) the pre-$\tau$ process itself is Markov and an $h$-transform of $X$ if and only if $\tau$ is almost surely equal to a cooptional time with, in general, the possible starts for the pre-$\tau$ process restricted. Also, a result is presented characterizing those $\tau$ for which the reversed pre-$\tau$ process is Markov in natural duality to $X$, without the assumption of conditional independence.

We study the relation between distributions of random closed sets and their hitting functions $T$, defined by $T(B) = P\{\varphi \cap B \neq \varnothing\}$ for Borel sets $B$. In particular, a sequence of random sets converges in distribution iff the corresponding sequence of hitting functions converges on some sufficiently large class of bounded Borel sets. This class may be chosen to be countable.

If $(X, \mathscr{F}, P)$ is a probability space then a pseudo-metric $\delta$ can be defined on the sub-$\sigma$-fields of $\mathscr{F}$ by $\delta(\mathscr{A, B}) = \sup_{A \in \mathscr{A}}\inf_{B \in \mathscr{B}}P(A \Delta B) \vee \sup_{B \in \mathscr{B}}\inf_{A \in \mathscr{A}}P(A \Delta B).$ Boylan, Neveu, and Rogge, among others, have considered equiconvergence of conditional expectations of uniformly bounded measurable functions given sub-$\sigma$-fields $\{\mathscr{F}_n:1 \leq n \leq \infty\}$ in probability and in $L_p, 1 \leq p < \infty$, as $\delta(\mathscr{F}_n, \mathscr{F}_\infty) \rightarrow 0$. This paper proves the corresponding almost sure equiconvergence results when $\mathscr{F}_n \uparrow \mathscr{F}_\infty$ or $\mathscr{F}_n \downarrow \mathscr{F}_\infty$. A sharp uniform bound for the rate of convergence is given. A consequence is that if $\mathscr{F}_n \uparrow \mathscr{F}_\infty$ or $\mathscr{F}_n \downarrow \mathscr{F}_\infty$ then the sequence of conditional expectations given $\mathscr{F}_n$ converges uniformly for all uniformly bounded measurable functions to the conditional expectation given $\mathscr{F}_\infty$ if and only if $\delta(\mathscr{F}_n, \mathscr{F}_\infty) \rightarrow 0$.

In the particular model of a zero range jump process in equilibrium, the asymptotic covariances of the--spatially and temporally rescaled--particle number field are computed. The main tool in that computation is a general theorem, whose validity is established for the given class of processes, which states that the asymptotic behaviour of the covariances of any field corresponding to a local function is determined by a suitable projection of that field on the linear (here one-dimensional) space spanned by the fields of conserved quantities (here: the particle number).

The coupled branching process $(\eta^\mu_t)$ is a Markov process on $(\mathbb{N})^S (S = \mathbb{Z}^d)$ with initial distribution $\mu$ and the following time evolution: At rate $b\eta(x)$ a particle is born at site $x$, which moves instantaneously to a site $y$ chosen with probability $q(x, y)$. All particles at a site die at rate $pd$, individual particles die independent from each other at rate $(1 - p)d$. Furthermore, all particles perform independent continuous time random walks with kernel $p(x, y)$. We consider here the case $b = d$ and the symmetrized kernels $\hat p, \hat q$ are transient. We show that the measures $\mathscr{L}(\eta^\mu_t(\cdot + \lbrack\alpha \sqrt{tx}\rbrack)), (\alpha \in \mathbb{R}^+, x \in \mathbb{R}^d)$ converge weakly for $t \rightarrow \infty$ to $\nu_{\tau(a,x)}$. Here $\nu_\rho$ is the invariant measure of the process with: $E^{\nu_\rho}(\eta(x)) = \rho$ and which is also extremal in the set of all translationinvariant invariant measures of the process. The density profile $\tau(\alpha, x)$ is calculated explicitly; it is governed by the diffusion equation.

Known results on the asymptotic behavior of the probability that the empirical distribution $\hat P_n$ of an i.i.d. sample $X_1, \cdots, X_n$ belongs to a given convex set $\Pi$ of probability measures, and new results on that of the joint distribution of $X_1, \cdots, X_n$ under the condition $\hat P_n \in \Pi$ are obtained simultaneously, using an information-theoretic identity. The main theorem involves the concept of asymptotic quasi-independence introduced in the paper. In the particular case when $\hat P_n \in \Pi$ is the event that the sample mean of a $V$-valued statistic $\psi$ is in a given convex subset of $V$, a locally convex topological vector space, the limiting conditional distribution of (either) $X_i$ is characterized as a member of the exponential family determined by $\psi$ through the unconditional distribution $P_X$, while $X_1, \cdots, X_n$ are conditionally asymptotically quasi-independent.

A central limit theorem and remainder term estimates are given for the distribution of the sum of scores based on the occupancy numbers resulting from the random allocation of $N$ balls to $n$ boxes. The proof involves bivariate characteristic functions, exploiting the equivalence of multinomial and conditioned Poisson variables. The results are shown to include the statistics for the empty cell test, the chi-squared test and the likelihood ratio test.

Consider a stationary $m$-dependent sequence of random indicator variables. If $m > 1$, assume further that any two nonzero values are separated by at least $m - 1$ zeros. This paper studies the sequence of the lengths of the successive intervals between the nonzero values of the original sequence, and it is shown that, provided a technical condition holds, these lengths converge in distribution (and their moments converge exponentially fast) in all cases but one.

We compute the joint density of Brownian motion, its local time at the origin, and its occupation time of $\lbrack 0, \infty)$. Two derivations of the main result are offered; one is computational, whereas the other uses some of the deep properties of Brownian local time. We use the result to compute the transition probabilities of the optimal process in a stochastic control problem.

For a standard Brownian motion $\omega(t)$ in $R^p, p \geq 3$, let $t_a(\omega)$ be the last exit time from the ball $B(0, a)$ of radius $a$ centered at the origin and let $F(a, t, \omega)$ be the oscillation in the neighbourhood of sphere $S(0, a)$. The distribution of the functional $B_f(a, \omega) = \int^{+\infty}_0 1_{B(0,a)}(\omega(t))f\big(\frac{F(a, t, \omega)}{F(a, +\infty, \omega)}\big) dt,$ where $f: (0, 1) \rightarrow R^+$ is an arbitrary bounded measurable function, coincides with the limiting distribution, when $n \rightarrow +\infty$, of the weighted sojourn time $\frac{T_f(a\sqrt n, \omega)}{n} = \sum^{+\infty}_{k=0} 1_{B(0,a\sqrt n)}(S_k(\omega))f\big(\frac{n(a\sqrt n, k, \omega)}{n(a\sqrt n, +\infty, \omega)}\big)\big/n$ for a standard random walk in $Z^p$ where $n(b, k, \omega)$ denote the number of crossing $S(0, b)$ during the first $k$ steps. We give explicit formulas, in terms of Laplace transform, for the joint distribution of $B_f(a, \omega), F(a, +\infty, \omega)$ and $t_a(\omega)$.

We show that one can smoothly renorm the Hilbert space $H$ such that the rate of convergence in the central limit theorem becomes very bad. More precisely, let us fix a sequence $\xi_n \rightarrow 0$ and $\varepsilon > 0$. We can then construct a norm $N(\cdot)$ on the Hilbert space, and a bounded random variable $X$ on $H$ with the following properties: (a) The norm $N(\cdot)$ is $(1 + \varepsilon)$ equivalent to the usual norm. It is infinitely many times differentiable, and each differential is bounded on the unit sphere. (b) If $(X_i)$ denotes independent copies of $X$, and if $\gamma$ is the Gaussian measure with the same covariance as $X$, then the inequality $\operatorname{Sup}_{t>0}|P\{N(n^{-1/2} \sum^n_{i=1} X_i) \leq t\} - \gamma\{x; N(x) \leq t\}| \geq \xi_n$ occurs for infinitely many $n$.

In this paper we extend the Kolmogorov strong law of large numbers to random variables taking their values in a 2-uniformly smooth Banach space $(B, \| \|)$. In our result, the convergence of the classical series of variances is replaced by the convergence of the series having general term $\sup\{Ef^2(X_n)/n^2: \|f\|_{B'} \leq 1\}.$

In this work we construct general additive processes on the nuclear spaces, and prove Khintchin's formula and Paul Levy's decomposition for these processes. As applications, we construct some Ornstein-Uhlenbeck processes with jumps and solve some (stochastic) partial differential equations obtained from the transformations of these processes by a random diffeomorphism corresponding to a finite dimensional diffusion process.

Inequalities are derived for the quality of exponential approximations to NBUE (new better than used) and NWUE (new worse than used) distributions.

For $X_1, X_2, \cdots$ i.i.d. with finite mean and $Y_n = \max(X_1, \cdots, X_n) - cn, c$ positive, a number of authors have considered the problem of determining an optimal stopping rule for the reward sequence $Y_n$. The optimal stopping rule can be given explicitly in this case; however, in general its use requires complete knowledge of the distribution of the $X_i$. This paper examines the problem of approximating the optimal expected reward when only partial information about the distribution is available. Specifically, if the $X_i$ are known to be exponentially distributed with unknown mean, stopping rules designed to approximate the optimal rule (which can be used only when the mean is known) are proposed. Under certain conditions the difference between the expected reward using the proposed stopping rules and the optimal expected reward vanishes as $c$ approaches zero.

This article tries to unify best choice problems under total ignorance of both the candidates, quality distribution and the distribution of the number of candidates. The result is what we shall call the $e^{-1}$-law because of the multiple role which is played by $e^{-1}$, and this in a more general context as only in the solution of the classical secretary problem. The unification is possible whenever best choice problems can be redefined as continuous time decision problems on conditionally independent arrivals. We shall also give several examples to illustrate how the approach and its implications compare with other models.

In this paper a finite-dimensional diffusion is observed in the presence of an additive Brownian motion. A large deviations result is obtained for the conditional probability distribution of the diffusion given the observations as the noise variances go to zero.

A convexity argument is used to establish a representation formula from which one can derive the asymptotics of large deviations of sums of i.i.d. random variables on $\mathbb{R}^d$. This simplifies a proof in [4] which relied on fixed point and probabilistic arguments.

Let $X, X_1, X_2, \cdots$ be i.i.d. nonconstant mean zero random variables and put $S_n = X_1 + \cdots + X_n$. Let $K(y) > 0$ satisfy $yE\{|X/K(y)|^2 \wedge |X/K(y)|\} = 1$ (for $y > 0$). Then let $a_n = (\log \log n)K(n/\log \log n)$ and $L = \lim \sup_{n\rightarrow\infty}S_n/a_n.$ It is known that $L$ is finite iff $P(X_n > a_n \text{i.o.}) = 0$. When $L < \infty$, it is also known that $1 \leq L \leq 1.5$ and that it is possible for $L$ to equal one. In this paper we construct an example for which $L = 1.5$.

We construct a strictly stationary associated sequence $(X_n)_{n \in \mathbb{N}}$ with $EX_n = 0, 0 < EX^2_n < \infty$ such that $K(R) = \operatorname{Cov}(X_1, X_1) + \sum^R_{j=2} \operatorname{Cov}(X_1, X_j) \sim \log R$ as $R \rightarrow \infty$, but $\sum^n_{j=1} X_j/(nK(n))^{1/2}$ does not converge to $\mathscr{N}(0, 1)$ in distribution. This is a counterexample to a conjecture of Newman and Wright (1981).

It is proved that the spectral densities of arrays of weakly stationary processes fulfilling a certain regularity condition converge uniformly to a continuous spectral density.

It is shown that if $B(t), t \geq 0$, is a Wiener process, $U$ is an independent random variable uniformly distributed on (0, 1), and $\varepsilon$ is a constant, then the distribution of $B(t) + \varepsilon \sqrt{(t - U)^+}, 0 \leq t \leq 1$, is absolutely continuous with respect to Wiener measure on $C\lbrack 0, 1\rbrack$ if $0 < \varepsilon < 2$, and singular with respect to this measure if $\varepsilon > \sqrt 8$.