We study certain singularities of the Brownian sheet which have the property that they propagate parallel to the coordinate axis. This is used to give an intuitive explanation of the fact that the Brownian sheet does not satisfy Levy's sharp Markov property.

The aim of this paper is to give upper and lower bounds for the probability density at $(u - z)$ of the position at time $(x, y) (x, y, z, u \in R^+)$ of a standard Wiener process with two-dimensional parameter $(x, y)$ with the requirement that it did not reach the barrier $u$ in the "past" $\{(x', y'): 0 \leq x' \leq x, 0 \leq y' \leq y\}$. The fundamental tools are Kolmogorov forward inequalities for the density and certain bounds for the behaviour of $p$ near the border.

Let $\{Y(t, \omega) = (X_1(t, \omega), \cdots, X_d(t, \omega)); 0 \leq t \leq 1\}$ be a $d$-dimensional Gaussian process whose components are independent copies of a Gaussian process with index $\alpha$; that is, $E\lbrack X(t, \omega)\rbrack = 0, X(0, \omega) = 0$, and $E\lbrack(X(t, \omega) - X(s, \omega))^2\rbrack = \sigma^2(|t - s|)$, where $\sigma(t) = t^\alpha, 0 < \alpha < 1$. Let $h(t)$ be a positive, non-increasing, continuous function and set $q = \sup\big\{r \geq 0; \int_{+0} e^{-rh^2(t)/2} dt/t = + \infty\big\}.$ Then, as an application of a version of Strassen's $\log \log$ law, we have \begin{equation*}\begin{split}\lim \sup_{t\downarrow 0} t^{-1}m(\{0 \leq s \leq t;\|Y(s, \omega)\| > \sigma(s)h(s)\}) \\ &= \sup_{x\in B}m(\{0 \leq s \leq 1;\|x(s)\| \geq \sigma(s)/ \sqrt{q}\}), \quad\text{a.s.},\end{split}\end{equation*} where $\| \|$ denotes the usual Euclidean norm, $m(\Gamma)$ denotes the Lebesgue measure of a linear set $\Gamma$, and $B$ is the unit ball of the direct sum of $d$ copies of the reproducing kernel Hilbert space with the kernel $R(s, t) = (\sigma^2(t) + \sigma^2(s) - \sigma^2(|t - s|))/2$. In case of the $d$-dimensional Brownian motion, Strassen [7] had proved that the right-hand side of the above formula is equal to $1 - \exp\{-4(q - 1)\}$ if $q \geq 1$, and 0 if $q \leq 1$. As a corollary, $\sigma(t)h(t)$ is an approximate upper function as introduced by D. German [2] if and only if $q \leq 1$. Especially, if $\lim_{t\downarrow0}h(t)/ \sqrt{2 \log \log 1/t} = c, \sigma(t)h(t)$ is an approximate upper function if and only if $c \geq 1$.

Motivated by the problem of establishing laws of the iterated logarithm for least squares estimates in regression models and for partial sums of linear processes, we prove a general $\log \log$ law for weighted sums of the form $\sum^\infty_{i=-\infty} a_{ni}\varepsilon_i$, where the $\varepsilon_i$ are independent random variables with zero means and a common variance $\sigma^2$, and $\{a_{ni}: n \geq 1, -\infty < i < \infty\}$ is a double array of constants such that $\sum^\infty_{i=-\infty} a^2_{ni} < \infty$ for every $n$. Besides applying the general theorem to least squares estimates and linear processes, we also use it to improve earlier results in the literature concerning weighted sums of the form $\sum^n_{i=1} f(i/n)\varepsilon_i$.

Implicitly defined (and easily approximated) universal constants $1.1 < a_n < 1.6, n = 2,3, \cdots$, are found so that if $X_1, X_2, \cdots$ are i.i.d. non-negative random variables and if $T_n$ is the set of stop rules for $X_1, \cdots, X_n$, then $E(\max\{X_1, \cdots, X_n\}) \leq a_n \sup\{EX_t: t \in T_n\}$, and the bound $a_n$ is best possible. Similar universal constants $0 < b_n < \frac{1}{4}$ are found so that if the $\{X_i\}$ are i.i.d. random variables taking values only in $\lbrack a, b\rbrack$, then $E(\max\{X_1, \cdots, X_n\}) \leq \sup\{EX_t: t \in T_n\} + b_n(b - a)$, where again the bound $b_n$ is best possible. In both situations, extremal distributions for which equality is attained (or nearly attained) are given in implicit form.

If $\mu_n, \nu$ are probability measures on a separable Banach space, $j_n \rightarrow \infty$ and $\mu^{jn}_n \rightarrow_w \nu$ (so $\nu$ is necessarily infinitely divisible), then it is possible to construct two row-wise independent triangular arrays $\{X_{nj}\}, \{Y_{nj}\}$ such that $\mathscr{L}(X_{nj}) = \mu_n, \mathscr{L}(Y_{nj}) = \nu^{1/jn}$ and $\max_{k \leq jn} \|S_{nk} - T_{nk}\|\rightarrow_\mathrm{P} 0$, where $S_{nk}$ and $T_{nk}$ are the respective partial row sums. Several refinements are proved. These results are applied to establish the weak convergence of the distributions of certain functionals of the partial row sums, improving well-known results of Skorohod. As concrete applications, we prove an arc-sine law for triangular arrays generalizing the Erdos-Kac law and an arc-sine law for strictly stable processes generalizing P. Levy's law for Brownian Motion.

Let $\{Z_n\}$ be a sequence of random variables converging in probability to zero. If the convergence is also in $L^2$, it is common to measure the rate of convergence by the $L^2$ norm of $Z_n$. However, in many interesting cases the variables $Z_n$ do not have finite variance, and then it seems appropriate to study the truncated $L^2$ norm, $\Delta_n = E\lbrack\min(1, Z^2_n)\rbrack$. We put $\Delta_n$ forward as a global measure of the rate of convergence. The paper concentrates on the case where $Z_n$ is a normalised sum of independent and identically distributed random variables, and we derive very precise descriptions of the rate of convergence in this situation.

Let $X_1, X_2, \cdots$ be a sequence of independent identically distributed (i.i.d.) positive random variables in the domain of attraction of a completely asymmetric stable law with characteristic exponent $\alpha \in (0, 1)$, i.e. their common distribution function $G$ is given by $P(X_1 > x) = 1 - G(x) = x^{-\alpha}L(x),$ where $L$ is a slowly varying function at infinity. In this paper we study the set of limit points of $\{n^{-1/\alpha}(X_1 + \cdots + X_n): n = 1,2, \cdots\}$. The sets of limit points that are possible are $\{0\}, \{\infty\}, \lbrack 0, \infty\rbrack$ and $\lbrack b, \infty\rbrack$ for some positive number $b$. In Section 2 we consider the case where $L$ is non-decreasing and in Section 3 the case where $L$ is non-increasing. In both sections we give the conditions in terms of $L$ for each of the limit sets.

A local limit theorem for maxima of i.i.d. random variables is proved. Also it is shown that under the so-called von Mises' conditions the density of the normalized maximum converges to the limit density in $L_p(0 < p \leq \infty)$ provided both the original density and the limit density are in $L_p$. Finally an occupation time result is proved. The methods of proof are different from those used for the corresponding results concerning partial sums.

In this paper we derive a law of the logarithm for the maximal deviation between a kernel density estimator and the true underlying density function. Extensions to higher derivatives are included. The results are applied to get optimal window-widths with respect to almost sure uniform convergence.

Let $T_n = n^{-1} \sum^n_{i=1} c_{in}X_{in}$ be a linear combination of order statistics and put $T^\ast_n = (T_n - E(T_n))/ \sqrt{\operatorname{Var}(T_n)}$. Sufficient conditions on the $c_{in}$ and on the moments of the underlying distribution are established under which the ratio $P(T^\ast_n > x)/(1 - \Phi(x))$ tends to 1, either uniformly in the range $-A \leq x \leq c \sqrt{\ln n}(A \geq 0, c > 0)$ (moderate deviation theorem) or uniformly in the range $-A \leq x \leq o(n^\alpha)(A \geq 0)$ (Cramer type large deviation theorem). The proof relies on Helmers' approximation method and on the corresponding results for U-statistics.

Two related extensions of the stochastic integral are discussed. These extensions allow the integrand to anticipate the Brownian motion, and arise in the study of linear stochastic integral equations. The development is based on the homogeneous chaos expansion of the integrand. Some properties of these extended integrals, and their commutativity with the classical integrals, are derived.

Existence, uniqueness, and a Markov property are proved for the solutions of a hyperbolic equation with a white Gaussian noise driving term. A two-parameter analog of the Stratonovich stochastic integral is introduced and is used to formulate integral versions of the hyperbolic equation. The stochastic calculus associated with the Stratonovich integral formally agrees with ordinary calculus. A class of two-parameter semimartingales is found which is closed under all the operations of a complete stochastic calculus. The class of processes which are solutions to the type of hyperbolic equation studied is closed under smooth state space transformations.

Let $\pi$ be a finite set, $\lambda$ a probability measure on $\pi, 0 < x < 1$ and $a \in \pi$. Let $P(a, x)$ denote the probability that in a random order of $\pi, a$ is the first element (in the order) for which the $\lambda$-accumulated sum exceeds $x$. The main result of the paper is that for every $\varepsilon > 0$ there exist constants $\delta > 0$ and $K > 0$ such that if $\rho = \max_{a\in\pi} \lambda(a) < \delta$ and $\mathrm{K}\rho < x < 1 - \mathrm{K}\rho$ then $\sum_{a\in\pi} |P(a, x) - \lambda(a)| < \varepsilon$. This result implies a new variant of the classical renewal theorem, in which the convergence is uniform on classes of random variables.

A general multitype branching process in which individuals are counted according to some possibly type-dependent characteristic may be defined along the lines laid out by Jagers (1969, 1974) for the single type process. In the critical case, the probability of nonextinction at time $t$ is shown to be $O(t^{-1})$, and, conditioned on nonextinction at time $t$, the totals of the characteristic counts, normalized by $t$, are shown to satisfy an exponential limit law, under weak (essentially, second moment) hypotheses.

It is shown that normal variables are associated if and only if their correlations are nonnegative.

We give a necessary and sufficient condition for the distribution function of $n^{-1/2} \sum^n_{i=1} X_i$, where the $X_i$ are independently identically distributed with $EX_1 = 0, EX^2_1 = 1$ and $E|X_1|^{k+3} < \infty$, to possess an Edgeworth expansion to $k$ terms. The condition is not practicable but clarifies the relation between the existence of an Edgeworth expansion and the smoothness of the distribution function of the sum.

Minimax linear smoothers are considered for the problem of estimating a homogeneous signal field in an additive orthogonal noise field. A minimax game with the quadratic-mean estimation error as an objective function is used to formulate this problem. Uncertainty in signal and noise field spectra is modeled using general nonparametric classes of measures proposed by Huber and Strassen for the problem of minimax hypothesis testing. These classes, which are described in terms of Choquet alternating capacities of order 2, include the conventional models for spectral uncertainty and admit a general solution to the minimax linear smoothing problem.