An upper bound is given for the behaviour of the radial part of a $\Gamma$-martingale, generalizing previous work of the author on the radial part of Riemannian Brownian motion. This upper bound is applied to establish an integral curvature condition to determine when $\Gamma$-martingales cannot "implode" in finite intrinsic time, answering a question of Emery and generalizing work of Hsu on the $C_0$-diffusion property of Brownian motion.

We consider Markov processes on the positive integers for which the origin is an absorbing state. Quasi-stationary distributions (qsd's) are described as fixed points of a transformation $\Phi$ in the space of probability measures. Under the assumption that the absorption time at the origin, $R,$ of the process starting from state $x$ goes to infinity in probability as $x \rightarrow \infty$, we show that the existence of a $\operatorname{qsd}$ is equivalent to $E_xe^{\lambda R} < \infty$ for some positive $\lambda$ and $x$. We also prove that a subsequence of $\Phi^n\delta_x$ converges to a minimal $\operatorname{qsd}$. For a birth and death process we prove that $\Phi^n\delta_x$ converges along the full sequence to the minimal $\operatorname{qsd}$. The method is based on the study of the renewal process with interarrival times distributed as the absorption time of the Markov process with a given initial measure $\mu$. The key tool is the fact that the residual time in that renewal process has as stationary distribution the distribution of the absorption time of $\Phi\mu$.

Let $X$ and $Y$ be two continuous-time martingales. If quadratic variation of $X$ minus that of $Y$ is a nondecreasing and nonnegative function of time, we say that $Y$ is differentially subordinate to $X$ and prove that $\|Y\|_p \leq (p^\ast - 1)\|X\|_p$ for $1 < p < \infty$, where $p^\ast = p \vee q$ and $q$ is the conjugate of $p$. This inequality contains Burkholder's $L^p$-inequality for stochastic integrals, which implies that the above inequality is sharp. We also extend his concept of strong differential subordination and several other of his inequalities, and sharpen an inequality of Banuelos.

Let $M$ be a locally square integrable martingale with predictable quadratic variance $\langle M\rangle$ and let $\Delta M = M - M_-$ be the jump process of $M$. In this paper, under the various restrictions on $\Delta M$, the different increasing rates of $M$ in terms of $\langle M\rangle$ are obtained. For stochastic integrals $X = B \cdot M$ of the predictable process $B$ with respect to $M$, the a.s. asymptotic behavior of $X$ is also discussed under restrictions on the rates of increase of $B$ and the restrictions on the conditional distributions of $\Delta M$ or on the conditional moments of $\Delta M$. This is applied to some simple examples to determine the convergence rates of estimators in statistics.

We consider two-dimensional reflected Brownian motions in sharp thorns pointed downward with horizontal vectors of reflection. We present a decomposition of the process into a Brownian motion and a process which has bounded variation away from the tip of the thorn. The construction is based on a new Skorohod-type lemma.

Let $X$ be an $\mathbb{R}^d$-valued special semimartingale on a probability space $(\Omega, \mathscr{F}, (\mathscr{F}_t)_{0\leq t \leq T},P)$ with decomposition $X = X_0 + M + A$ and $\Theta$ the space of all predictable, $X$-integrable processes $\theta$ such that $\int\theta dX$ is in the space $\mathscr{J}^2$ of semimartingales. If $H$ is a random variable in $\mathscr{L}^2$, we prove, under additional assumptions on the process $X$, that $H$ can be written as the sum of an $\mathscr{F}_0$-measurable random variable $H_0$, a stochastic integral of $X$ and a martingale part orthogonal to $M$. Moreover, this decomposition is unique and the function mapping $H$ with its decomposition is continuous with respect to the $\mathscr{L}^2$-norm. Finally, we deduce from this continuity that the subspace of $\mathscr{L}^2$ generated by $\int\theta dX$, where $\theta\in \Theta$, is closed in $\mathscr{L}^2$, and we give some applications of this result to financial mathematics.

In this paper the $L^1$-stochastic integral and the mixed stochastic integral of a process $Y$ with respect to a process $X$ is defined in a way that extends Riemann-Stieltjes integration of deterministic functions with respect to $X$. The $L^1$-integral will include the classical Ito integral. However, the concepts of "filtration" and adaptability do not play any role; instead, the $p$-variation of Dolean functions of the processes $X$ and $Y$ is the determining factor.

We show that in between the two critical values for the contact process on trees, there are infinitely many extremal nontranslation invariant stationary distributions. This conclusion is reminiscent of results of Grimmett and Newman for percolation on the product of a tree with the integers where there is an intermediate phase with infinitely many infinite clusters.

The problem of recovering, say, a band-limited weakly stationary process from a set of its irregularly spaced samples is studied. For rather general sampling sequences some sufficient conditions ensuring mean square or pathwise reconstruction are obtained. For the cases of regular samples with either finitely many missing ones and/or finitely many irregular ones, a necessary and sufficient condition is presented. Some elements of the proofs involve classical results on nonharmonic Fourier series as well as more recent results on frames.

The best known upper bound for the critical value $\lambda_c$ of the basic one dimensional contact process is 2. Most techniques for finding bounds on critical values have the property that they can be modified in order to obtain improved bounds. This seemed not to be the case for the approach which yielded $\lambda_c \leq 2$ for the basic contact process. In this paper, we propose a technique for generating better bounds in this context. To illustrate its use, we carry out the full program in one case, with the conclusion that $\lambda_c \leq 1.942$.

The hydrodynamic limit appears as a law of large numbers for rescaled density profiles of a large stochastic system. We study the large deviations from this scaling limit for a particular nongradient system, the nongradient version of the Ginzburg-Landau model.

We consider the full weak convergence, in appropriate function spaces, of systems of noninteracting particles undergoing critical branching and following a self-similar spatial motion with stationary increments. The limit processes are measure-valued, and are of the super and historical process type. In the case in which the underlying motion is that of a fractional Brownian motion, we obtain a characterization of the limit process as a kind of stochastic integral against the historical process of a Brownian motion defined on the full real line.

Consider $0 < \alpha < 1$ and the Gaussian process $Y(t)$ on $\mathbb{R}^N$ with covariance $E(Y(t)Y(s)) = |t|^{2\alpha} + |s|^{2\alpha} - |t - s|^{2\alpha}$, where $|t|$ is the Euclidean norm of $t$. Consider independent copies $X^1,\ldots,X^d$ of $Y$ and the process $X(t) = (X^1(t),\ldots,X^d(t))$ valued in $\mathbb{R}^d$. In the transient case $(N < \alpha d)$ we show that a.s. for each compact set $L$ of $\mathbb{R}^N$ with nonempty interior, we have $0 < \mu_\varphi(X(L)) < \infty$, where $\mu_\varphi$ denotes the Hausdorff measure associated with the function $\varphi(\varepsilon) = \varepsilon^{N/\alpha} \log \log(1/\varepsilon)$. This result extends work of A. Goldman in the case $\alpha = 1/2$; the proofs are considerably simpler.

Consider a mean zero random variable $X$, and an independent sequence $(X_n)$ distributed like $X$. We show that the random Fourier series $\sum_{n\geq 1} n^{-1} X_n \exp(2i\pi nt)$ converges uniformly almost surely if and only if $E(|X|\log\log(\max(e^e, |X|))) < \infty$.

Let $W$ be a real-valued, two-parameter Brownian sheet. Let us define $N(t; h)$ to be the total number of bubbles of $W$ in $\lbrack 0, t\rbrack^2$, whose maximum height is greater than $h$. Evidently, $\lim_{h\downarrow 0} N(t; h) = \infty$ and $\lim_{t\uparrow\infty} N(t; h) = \infty$. It is the goal of this paper to provide fairly accurate estimates on $N(t; h)$ both as $t\rightarrow\infty$ and as $h\rightarrow 0$. Loosely speaking, we show that there are of order $h^{-3}$ many such bubbles as $h \downarrow 0$ and $t^3$ many, as $t\uparrow \infty$.

In this paper we present a decoupling inequality that shows that multivariate $U$-statistics can be studied as sums of (conditionally) independent random variables. This result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration. More precisely, we get the following result: Let $\{X_j\}$ be a sequence of independent random variables on a measurable space $(\mathscr{J}, S)$ and let $\{X^{(j)}_i\}, j = 1,\ldots,k$, be $k$ independent copies of $\{X_i\}$. Let $f_{i_1i_2\ldots i_k}$ be families of functions of $k$ variables taking $(S \times \cdots \times S)$ into a Banach space $(B, \|\cdots\|)$. Then, for all $n \geq k \geq 2, t > 0$, there exist numerical constants $C_k$ depending on $k$ only so that $P\bigg(\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(1)}_{i_2}, \ldots, X^{(1)}_{i_k})\big\|\geq t\bigg)$ $\leq C_kP\bigg(C_k\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(2)}_{i_2}, \ldots, X^{(k)}_{i_k})\big\|\geq t\bigg).$ The reverse bound holds if, in addition, the following symmetry condition holds almost surely: $f_{i_1i_2\cdots i_k}(X_{i_1}, X_{i_2},\ldots,X_{i_k}) = f_{i_{\pi(1)}i_{\pi(2)}\cdots i_{\pi(k)}} (X_{i_{\pi(1)}, X_{i_{\pi(2)}}, \ldots,X_{i_{\pi(k)}}),$ for all permutations $\pi$ of $(1,\ldots,k)$.

In a recent paper by the author it has been shown that there exists a general law of the iterated logarithm (LIL) in Banach space, which contains the LIL of Ledoux and Talagrand and an LIL for infinite-dimensional random variables in the domain of attraction to a Gaussian law as special cases. We now investigate the corresponding cluster set problem, which we completely solve for random vectors in two-dimensional Euclidean space. Among other things, we show that all cluster sets arising from this generalized LIL must be sets of diameter 2, which are star-shaped and symmetric about the origin, and any closed set of this type occurs as a cluster set for a suitable random vector. Moreover, we show that if the random vectors under consideration have independent components, one only obtains cluster sets from the subclass of all sets, which can be represented as closures of countable unions of standard ellipses.

We consider the concentration of measure for $n$ i.i.d., two-dimensional random variables under the conditioning that they form a record. Under mild conditions, we show that all random variables tend to concentrate, as $n \rightarrow \infty$, around limiting curves, which are the solutions of an appropriate variational problem. We also show that the same phenomenon occurs, without the records conditioning, for the longest increasing subsequence in the sample.

Let $\{\xi_j\}$ be a strictly stationary sequence of random variables with regularly varying tail probabilities. We consider, via point process methods, weak convergence of the partial sums, $S_n = \xi_1 + \cdots + \xi_n$, suitably normalized, when $\{\xi_j\}$ satisfies a mild mixing condition. We first give a characterization of the limit point processes for the sequence of point processes $N_n$ with mass at the points $\{\xi_j/a_n, j = 1,\ldots,n\}$, where $a_n$ is the $1 - n^{-1}$ quantile of the distribution of $|\xi_1|$. Then for $0 < \alpha < 1 (-\alpha$ is the exponent of regular variation), $S_n$ is asymptotically stable if $N_n$ converges weakly, and for $1 \leq \alpha < 2$, the same is true under a condition that is slightly stronger than the weak convergence of $N_n$. We also consider large deviation results for $S_n$. In particular, we show that for any sequence of constants $\{t_n\}$ satisfying $nP\lbrack\xi_1 > t_n\rbrack \rightarrow 0, P\lbrack S_n > t_n\rbrack/(nP\lbrack\xi_1 > t_n\rbrack)$ tends to a constant which can in general be different from 1. Applications of our main results to self-norming sums, $m$-dependent sequences and linear processes are also given.

This paper contains some extension of Kolmogorov's maximal inequality to dependent sequences. Next we derive dependent Marcinkiewicz-Zygmund type strong laws of large numbers from this inequality. In particular, for stationary strongly mixing sequences $(X_i)_{i\in\mathbb{Z}$ with sequence of mixing coefficients $(\alpha_n)_{n\geq 0}$, the Marcinkiewicz-Zygmund SLLN of order $p$ holds if $\int^1_0\lbrack\alpha^{-1}(t)\rbrack^{p-1}Q^p(t)dt < \infty,$ where $\alpha^{-1}$ denotes the inverse function of the mixing rate function $t \rightarrow \alpha_{\lbrack t\rbrack}$ and $Q$ denotes the quantile function of $|X_0|$. The condition is obtained by an interpolation between the condition of Doukhan, Massart and Rio implying the CLT $(p = 2)$ and the integrability of $|X_0|$ implying the usual SLLN $(p = 1)$. Moreover, we prove that this condition cannot be improved for stationary sequences and power-type rates of strong mixing.

It is shown that $\sum^n_{i=1} X_n$ and $\max^n_{i=1}X_i$ are asymptotically independent if $\{X_i\}$ is strongly mixing and $\sum^n_{i=1} X_i$ is asymptotically Gaussian. This generalizes a result of Anderson and Turkman.

A Rosenthal-type inequality for maximum partial sums of $\rho$-mixing sequences is obtained. Applications to the complete convergence and almost sure summability of partial sums are also discussed.

Let $(S_i, i = 1,2,\ldots, n), n > 1$, be a partition of the circle into sets $S_i$ each consisting of union of $\delta(i) < \infty \operatorname{arcs} A_{kl}$. Let $f_t$ be a rotation of length $t$ of the circle and denote Lebesgue measure by $\lambda$. Then every recurrent stochastic matrix $P$ on $S = \{1,\ldots,n\}$ is given according to a theorem of Cohen $(n = 2)$, Alpern and Kalpazidou $(n \geq 2)$ by $p_{ij} = \lambda(S_i \cap f^{-1}_t(S_j))/\lambda(S_i)$ for some choice of rotation $f_t$ and partition $\mathscr{J} = \{S_i\}$. The number $\delta(\mathscr{J}) = \max_i \delta(i)$ is called the length of description of the partition $\mathscr{J}$. Then it turns out that the minimal value of $\delta(\mathscr{J})$, when $\mathscr{J}$ varies, characterizes the matrix $P$. We call this value the rotational dimension of $P$. We prove that for certain recurrent $n \times n$ stochastic matrices the rotational dimension is provided by the number of Betti circuits of the graph of $P$. One preliminary result shows that there are recurrent $n \times n$ stochastic matrices which admit minimal positive circuit decompositions in terms of the Betti circuits of their graph. Finally, a generalization of the rotational dimension for the transition matrix functions is also given.