Recent results on the asymptotic distribution of winding and crossing numbers are presented as part of a larger framework of asymptotic laws for planar Brownian motion. The approach is via random time changes, martingale calculus, and excursion theory.

The asymptotic distribution as $t \rightarrow \infty$ is obtained for the number of jumps of a symmetric Cauchy process across level $x$ up to time $t$, jointly as $x$ varies. This result is related to the asymptotic joint distribution of windings of a planar Brownian motion about several points.

We consider Brownian motion $X$ on a rotationally symmetric manifold $M_g = (\mathbb{R}^n, ds^2), ds^2 = dr^2 + g(r)^2 d\theta^2$. An integral test is presented which gives a necessary and sufficient condition for the nontriviality of the invariant $\sigma$-field of $X$, hence for the existence of nonconstant bounded harmonic functions on $M_g$. Conditions on the sectional curvatures are given which imply the convergence or the divergence of the test integral.

Two characterisations are given of the finite-dimensional laws of Brownian motion indexed by an arbitrary class of subsets of the $d$-dimensional unit cube. There are associated conditions for convergence of finite-dimensional laws of a sequence of set-indexed additive processes. These conditions have a more explicit form in the case of set-indexed partial-sum processes based on mixing random variables.

A uniform bound is found for the variance of a partial-sum set-indexed process under a mixing condition. Sufficient conditions are given for a sequence of partial-sum set-indexed processes to converge to Brownian motion. The requisite tightness follows from hypotheses on the metric entropy of the class of sets and moment and mixing conditions on the summands. The proof uses a construction of Bass [2]. Convergence of finite-dimensional laws in this context is studied in [16].

Let $X(T)$ be the solution to a stochastic differential equation whose coefficients are homogeneous of degree 1 (e.g., a linear S.D.E.). Under mild conditions, it is shown that limits like $\lim_{T\rightarrow\infty} \frac{1}{T} \log P(|X(T)|/|X(0)| \geq R)$ exist and a formula is provided for their computation. The techniques developed apply to a broad class of situations besides the one treated here.

If $x \neq x'$ are two points of $\mathbb{R}^d, d \geq 2$, and if $X$ is a Brownian motion in $\mathbb{R}^d$ started at $x$, then by reflecting $X$ in the hyperplane $L \equiv \{y: |y - x| = |y - x'|\}$ we obtain a Brownian motion $X'$ started at $x'$, which couples with $X$ when $X$ first hits $L$. This paper deduces a number of well-known results from this observation, and goes on to consider the analogous construction for a diffusion $X$ in $\mathbb{R}^d$ which is the solution of an s.d.e. driven by a Brownian motion $B$; the essential idea is the reflection of the increments of $B$ in a suitable (time-varying) hyperplane. A completely different coupling construction is given for diffusions with radial symmetry.

A simple construction of maximal coupling is given by way of a distributional coupling concept.

Let $F$ and $G$ be two continuous distribution functions that cross at a finite number of points $-\infty \leq t_1 < \cdots < t_k \leq \infty$. We study the limiting behavior of the number of times the empirical distribution function $G_n$ crosses $F$ and the number of times $G_n$ crosses $F_n$. It is shown that these variables can be represented, as $n \rightarrow \infty$, as the sum of $k$ independent geometric random variables whose distributions depend on $F$ and $G$ only through $F'(t_i)/G'(t_i), i = 1, \ldots, k$. The technique involves approximating $F_n(t)$ and $G_n(t)$ locally by Poisson processes and using renewal-theoretic arguments. The implication of the results to an algorithm for determining stochastic dominance in finance is discussed.

We investigate the almost sure convergence of a kernel-type conditional empirical distribution function both in sup-norm and weighted sup-norms. As an application we get a strong law for the Nadaraya-Watson estimate of a regression function $m(\mathbf{x}) = \mathbb{E}(Y\mid \mathbf{X} = \mathbf{x})$ under a weak moment condition on $Y$.

Let $\{X_{nk}: k \in \mathbb{N}, n \in \mathbb{N}\}$ be a double array of random variables adapted to the sequence of discrete filtrations $\{\{\mathscr{F}_{nk}: k \in \mathbb{N} \cup \{0\}\}: n \in \mathbb{N}\}$. It is proved that for every weak limit theorem for sums of independent random variables there exists an analogous limit theorem which is valid for the system $(\{X_{nk}\}, \{\mathscr{F}_{nk}\})$ and obtained by conditioning expectations with respect to the past. Functional extensions and connections with the Martingale Invariance Principle are discussed.

Let $X$ be a Banach space valued random variable satisfying the central limit theorem and $\xi$ be a real valued random variable, independent of $X$. If $\xi$ is in the Lorentz space $L_{2,1}$, the product $\xi X$ satisfies the central limit theorem. We show that this condition on $\xi$ cannot be improved, characterizing $L_{2,1}$ as the space of all random variables $\xi$ such that the preceding implication holds for all vector valued $X$ satisfying the central limit theorem. In particular, there exist independent random variables $X$ and $\xi$ both satisfying the central limit theorem such that $\xi X$ does not.

Let $E$ denote a separable Banach space and let $X_i, i \in \mathbb{N}$, be a sequence of i.i.d. $E$-valued random vectors having finite third moment such that the central limit theorem holds. We prove that the convergence rate in the central limit theorem is $O(n^{-1/2})$ for regions $\{x \in E: F(x) < r\}$ which are defined by means of a smooth real valued function $F$ on $E$, provided that the limiting distribution of the gradient of $F$ fulfills a variance condition. Using this result we prove that the rate of convergence in the functional limit theorem for empirical processes is of order $O(n^{-1/2})$.

Let $X^1, X^2,\ldots$ be independent copies of a sequence $X = (X_1, X_2, \ldots)$ of independent symmetric random variables. Let $M$ be a symmetric multilinear form of rank $s$ on $\mathbb{R}^\mathbb{N}$ whose components $a_{i_1,\ldots, i_s}$ relative to the standard basis of $\mathbb{R}^\mathbb{N}$ satisfy $a_{i_1,\ldots, i_s} = 0$ for all but finitely many multi-indices and whenever two indices agree. If $\phi$ is nondecreasing, convex, $\phi(0) = 0$ and $\phi$ satisfies a $\Delta_2$ growth condition then $E\phi(|M(X,\ldots, X)|) \leq cE\phi(|M(X^1,\ldots, X^s)|),$ where $c$ depends only on $\phi$ and $s$.

We study convergence of multilinear forms $\sum f(n_1,\ldots, n_k) X_{n_1} \cdots X_{n_k}$ in symmetric independent random variables. We show that the multilinear form converges if and only if its "tetrahedronal" part and "diagonal" parts of different orders converge simultaneously. For "tetrahedronal" forms a.s. and $L_0$ convergence are equivalent. Moreover, they are equivalent to $L_p$ convergence provided $(X_k)$ satisfies a Marcinkiewicz-Paley-Zygmund condition for $p \geq 2$.

Let $X_{1,n} \leqq \cdots \leqq X_{n,n}$ be the order statistics of $n$ independent and identically distributed positive random variables with common distribution function $F$ satisfying $1 - F(x) = x^{-\alpha}L^\ast(x), x > 0$, where $\alpha$ is any positive number and $L^\ast$ is any function slowly varying at infinity. We give a complete description of the asymptotic distribution of the sum of the top $k_n$ extreme values $X_{n+1-k_n,n}, X_{n+2-k_n,n}, \ldots, X_{n,n}$ for any sequence of positive integers $k_n$ such that $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$.

The notion of regular variation of functions is generalized by defining $l(t) = \lim \inf_{x\rightarrow\infty} L(tx)/L(x), t > 0$, for any positive nondecreasing function $L$. It is shown that $l$ must obey one of: (i) $l(t) = +\infty$ for every $t > 1$; (ii) $l(t) > 1$ for every $t > 1$ and $l(t) \downarrow 1$ as $t \downarrow 1R$; or (iii) $l(t) = 1$ for some $t > 1$. Each of these classes is characterized in terms of the convergence or divergence of the integral $I(r, \delta) = \int_1^{\infty}\exp \{rL(\delta x) - L(x)\} dL(x)$ for $r \geq 1$, $\delta < 1$. Let $X_1, X_2, \ldots$ be i.i.d. random variables with distribution function $F$. Define $\mu_n = F^{-1}(1 - n^{-1}), M_n = \max(X_1,\ldots, X_n)$, and $L(x) = -\log(1 - F(x)). \{M_n\}$ is almost surely stable $\operatorname{iff} M_n/\mu_n \rightarrow 1$ a.s., and this is known to be equivalent to the convergence of $I(1, \delta)$ for every $\delta < 1$. Necessary and sufficient conditions for $\sum^\infty_{n=1} n^\alpha P\lbrack|(M_n/\mu_n) - 1| > \varepsilon\rbrack < \infty$ are presented, where $\alpha \geq -1$. In particular, that series converges $\operatorname{iff} I(\alpha + 2,(1 + \varepsilon)^{-1}) < \infty$. Moreover, the series $\sum^\infty_{n=1} n^\alpha P\lbrack|(M_n/\mu_n) - 1| > \varepsilon\rbrack$ converges for all $\varepsilon > 0$ and some $\alpha > -1 \operatorname{iff}$ it converges for every $\alpha > -1$ and every $\varepsilon > 0$.

The domains of attraction of the univariate extreme value distributions are characterized using inverse cumulative hazard functions. The results are much simpler than those using cumulative distribution functions. We also characterize the differentiable domains of attraction. A particularly simple characterization is given for the twice differentiable domain of attraction.

Let $\{X_n\}$ be a sequence of independent random variables and $\{a_n\}$ a square summable, positive nonincreasing sequence of real numbers such that $\sum a_n X_n$ is a random variable. We show that the condition $\lim_{n\rightarrow\infty} a^2_n \log(a_n)/\sum^\infty_{k=n+1} a^2_k = 0$ implies that the distribution measure $F(dx) = P(\sum a_n X_n \in dx)$ has an infinitely differentiable density for every range-splitting sequence $\{X_n\}$. The class of range-splitting sequences includes all non-trivial i.i.d. sequences with mean 0 and finite second moments. Consequences and examples are discussed.

There exist exponents of an operator stable measure which commute with every operator in the measure's symmetry group. These exponents together with a new norm lead to some simplifications in the representation of the Levy measure.

In the processes under consideration, an interval of length $L$ splits with probability (or exponential rate) proportional to $L^\alpha, \alpha \in \lbrack -\infty, \infty\rbrack$, and when it splits, it splits into two intervals of length $LV$ and $L(1 - V)$ where $V$ has d.f. $F$ on (0, 1). When $\alpha = 1$ and $F(x) = x$, the split points are i.i.d. uniform on (0, 1) and when $\alpha = \infty$ (a longest interval is always split), the model is a splitting process invented by Kakutani. In both these cases, the empirical distribution of the split points converges almost surely to the uniform distribution on (0, 1). On the other hand, when $\alpha = 0$, the model is a binary cascade and the empirical distribution of the split points converges almost surely to a random, continuous, singular distribution. In this paper, we show what happens in the other cases. Can the reader guess at what point the character of the limiting behavior changes?

We consider the asymptotic distribution properties of $f$-expansion digits. In particular, if $x = 1/\varphi_0(x) - 1/\varphi_1(x) - \cdots$ etc., then $\frac{1}{n} \sum^{n-1}_{k=0} \varphi_k \rightarrow 3 \text{in measure}.$

A new method is used to study the optimal stopping set corrected for discreteness introduced by Chernoff and studied by Chernoff and Petkau. The discrete boundary is asymptotically the optimal boundary for a Wiener process translated downward by a constant amount. This amount is shown to be an "excess over the boundary" term, and this method yields it as a simple integral involving the characteristic function of the random walk.

Suppose $\mathscr{V}$ and $\mathscr{F}$ are sets of subsets of $X$, for some fixed $X$. We apply Konig's lemma from infinitary combinatorics to prove that if $\mathscr{V}$ and $\mathscr{F}$ satisfy some simple closure properties, and $T$ is a Choquet capacity on $X$, then there is a probability measure on $\mathscr{F}$ such that for every $V \in \mathscr{F}, \{F \in \mathscr{F}: F \cap V \neq \varnothing\}$ is measurable with probability $T(V)$. This extends the well-known case when $\mathscr{F}$ and $\mathscr{V}$ are the closed (respectively, open) subsets of a second countable Hausdorff space $X$. The result enables us to define a general notion of "random measurable set"; for example, we can build a point process with Poisson distribution on any infinite (possibly nontopological) measure space.

This paper deals with predictable representation and time changed processes. Let $(M^i)_{i\geq 0}$ be a sequence of independent local martingales. Suppose that each $M^i$ has the property of predictable representation with respect to its natural filtration. Suppose also that $(A^i)_{i\geq 1}$ is a sequence of continuous, increasing, $(\mathscr{F}^{M^0}_t)$ adapted processes. We study sufficient conditions in order that $M = M^0 + \sum M^i_{A^i}$ be a local martingale and $M$ have the property of predictable representation with respect to the filtration $(\mathscr{F}^{M^0}_t) \vee (\mathscr{F}^{M^1_{A^1}}_t \vee (\mathscr{F}^{M^2_{A^2}}_t \vee \cdots$. Such problems arise in the modeling of a security market with continuous trading [1].

If the coordinate random variables $\{X_t\}$ on either $C\lbrack 0, \infty)$ or $D\lbrack 0, \infty)$ form a martingale, then for every stopping time $\tau$ which is everywhere finite, $E(X_\tau)$, if defined, equals $E(X_0)$. This version of the optional sampling theorem is not covered by Doob's classical result [1].

A two-parameter Wiener-measurable strong martingale is constructed for which no time change by means of stopping domains into the Wiener sheet exists.

A large class of Markov processes satisfying Hunt's hypothesis (H) is displayed. In particular, if $\Phi$ is a Levy-Khinchin exponent, then the Levy process with exponent $\Phi^\alpha (0 < \alpha < 1)$ satisfies (H). That is, every semipolar set is polar.

Let $X_n, n \geq 1$, be i..d. random variables with common distribution function $F(x)$ and $\gamma_n, n \geq 1$, be a sequence of constants such that $\gamma_n/n$ is nondecreasing in $n$. Set $S_n = X_1 + \cdots + X_n$. The main theorem of this paper gives an integral test which determines the infinite limit points of $\{S_n/\gamma_n\}$. This result extends and combines Feller's (1946) strong law of large numbers (SLLN) and the results Kesten (1970) and Erickson (1973).

The Proofs of Theorems 1 and 2 of [11] contain several gaps which will be corrected in this note.