Let $T_n$ denote the length of the shortest closed path connecting $n$ random points uniformly distributed over the unit square. We prove that for some number $K$, we have, for all $t \geq 0$, $P(|T_n - E(T_n)| \geq t) \leq K \exp(-t^2/K).$

Convergence to the Poisson distribution, for the number of occurrences of dependent events, can often be established by computing only first and second moments, but not higher ones. This remarkable result is due to Chen (1975). The method also provides an upper bound on the total variation distance to the Poisson distribution, and succeeds in cases where third and higher moments blow up. This paper presents Chen's results in a form that is easy to use and gives a multivariable extension, which gives an upper bound on the total variation distance between a sequence of dependent indicator functions and a Poisson process with the same intensity. A corollary of this is an upper bound on the total variation distance between a sequence of dependent indicator variables and the process having the same marginals but independent coordinates.

Start by randomly coloring each site of the one-dimensional integer lattice with any of $N$ colors, labeled $0, 1, \ldots, N - 1$. Consider the following simple continuous time Markovian evolution. At exponential rate 1, the color $\xi(y)$ at any site $y$ randomly chooses a neighboring site $x \in \{y - 1, y + 1\}$ and paints $x$ with its color provided $\xi(y) - \xi(x) = 1 \operatorname{mod} N$. Call this interacting process the cyclic particle system on $N$ colors. We show that there is a qualitative change in behavior between the systems with $N \leq 4$ and those with $N \geq 5$. Specifically, if $N \geq 5$ we show that the process fixates. That is, each site is painted a final color with probability 1. For $N \leq 4$, on the other hand, we show that every site changes color at arbitrarily large times with probability 1.

We consider noninteracting systems of infinite particles each of which follows an irreducible, null recurrent Markov process and prove a large deviation principle for the empirical density. The expected occupation time (up to time $N$) of this Markov process, named as $h(N)$, plays an essential role in our result. We impose on $h(N)$ a regularly varying property as $N \rightarrow \infty$, which a large class of transition probabilities does satisfy. Some features of our result are: (a) The large deviation tails decay like $\exp\lbrack - Nh^{-1}(N)I(\cdot)\rbrack$, more slowly than the known $\exp\lbrack - NI(\cdot) \rbrack$ type of decay in transient situations. (b) Our rate function $I(\lambda(\cdot))$ equals infinity unless $\lambda(\cdot)$ is an invariant distribution. (c) Our rate function is explicit and is rather insensitive to the underlying Markov process. For instance, if we randomized the time steps of a Markov chain by exponential waiting time of mean 1, the resultant system obeys exactly the same large deviation principle.

A stationary $l^2$-valued Ornstein-Uhlenbeck process given formally by $dX(t) = - AX(t) dt + \sqrt{2a} dB(t)$, where $A$ is a positive self-adjoint constant operator on $l^2$ and $B(t)$ is a cylindrical Brownian motion on $l^2$, is considered. An upper bound on $P(\sup_{t \in \lbrack 0, T \rbrack}\|X(t)\| > x)$ is established and the asymptotics for the given bound, as $x \rightarrow \infty$, is derived.

We consider a lattice system of linear interacting diffusion processes with infinitely many invariant distributions. We first prove a nonstandard central limit theorem and identify the equation of the fluctuation field. We then derive dimension dependent large deviation results for the empirical mean.

The Brydges-Spencer lace expansion is used to prove that the scaling limit of the finite-dimensional distributions of self-avoiding random walk in the $d$-dimensional cubic lattice $\mathbb{Z}^d$ is Gaussian, if $d$ is sufficiently large. It is also shown that the critical exponent $\gamma$ for the number of self-avoiding walks is equal to 1, if $d$ is sufficiently large.

Let $S_n, n \in \mathbb{N}$, be a recurrent random walk on $\mathbb{Z}^2 (S_0 = 0)$ and $\xi(\alpha), \alpha \in \mathbb{Z}^2$, be i.i.d. $\mathbb{R}$-valued centered random variables. It is shown that $\sum^n_{i = 1}\xi(S_i)/ \sqrt{n \log n}$ satisfies a central limit theorem. A functional version is presented.

We study an inhomogeneous branching Brownian motion in which individual particles execute standard Brownian movements and reproduce at rates depending on their locations. The rate of reproduction for a particle located at $x$ is $\beta(x) = b + \beta_0(x)$, where $\beta_0(x)$ is a nonnegative, continuous, integrable function. Let $M(t)$ be the position of the rightmost particle at time $t$; then as $t \rightarrow \infty, M(t) - \operatorname{med}(M(t))$ converges in law to a location mixture of extreme value distributions. We determine $\operatorname{med}(M(t))$ to within a constant $+ o(1)$. The rate at which $\operatorname{med}(M(t)) \rightarrow \infty$ depends on the largest eigenvalue $\lambda$ of a differential operator involving $\beta(x)$; the cases $\lambda < 2, \lambda = 2$ and $\lambda > 2$ are qualitatively different.

A special class of stationary one-dependent two-valued stochastic processes is defined. We associate to each member of this class two parameter values, whereby different members receive different parameter values. For any given values of the parameters, we show how to determine whether: 1. a process exists having the given parameter values, and if so, 2. this process can be obtained as a two-block factor from an independent process. This determines a two-parameter subfamily of the class of stationary one-dependent two-valued stochastic processes which are not two-block factors of independent processes.

In this article we prove that for each $t > 0$, almost surely $\partial C(t)$, the boundary of the convex hull of two dimensional Brownian motion up to time $t$, is a $C^1$ curve in the plane. We also prove that if $\eta$ is a modulus of continuity such that $x\eta(x)$ is convex and $\int^1_0\eta(x) dx/x < \infty$ then for each $t > 0$, almost surely $\partial C(t)$ is not a $C^{1, \eta}$ curve in the plane.

In this paper, coupling methods for diffusion processes are studied mainly to obtain upper bound estimates in two different probability metrics. We use the martingale approach and explore the construction of explicit coupling operators which are sometimes optimal. The paper presents some criteria for the success of coupling and for the finiteness of the moments of the coupling times. Rates of convergence in various metrics are also studied.

We study the asymptotic growth rates of discrete-time stochastic processes $(X_n)$, where the first two conditional moments of the process depend only on the present state. Such processes satisfy a stochastic difference equation $X_{n + 1} = X_n + g(X_n) + R_{n + 1}$, where $g$ is a positive function and $(R_n)$ is a martingale difference sequence. It is known that a large class of such processes diverges with positive probability, and when properly normalized converges almost surely or converges in distribution to a normal or a lognormal distribution. Here we find a class of processes that when properly normalized converges in distribution to a generalized gamma distribution. Applications of this result to state dependent random walks and population size-dependent branching processes yield new results and reprove some of the known results.

For each closed set $F \subseteq \lbrack 0, 1\rbrack, \dim X(F + t) = \min(1, 2 \dim F)$ for almost all $t > 0. (X$ is one-dimensional Brownian motion). For each closed set $F \subseteq \lbrack 0, 1 \rbrack$ of dimension greater than $1/2, m(X(F + t)) > 0$ for almost all $t > 0$. These statements are true outside a single null-set in the sample space.

By iterating a martingale representation result a homogeneous chaos expansion is obtained. Using the martingale representation, the integration-by-parts formula of the Malliavin calculus is derived using properties of stochastic flows. The infinite-dimensional calculus of variations is not required.

In this paper we obtain necessary and sufficient conditions for the reversibility of the diffusion property, assuming the existence of a density at every time $t$. The proofs are based on techniques of the stochastic calculus of variations.

Let $x_1, \ldots, x_n$ be independent random variables with uniform distribution over $\lbrack 0, 1\rbrack$, defined on a rich enough probability space $\Omega$. Denoting by $\hat{\mathbb{F}}_n$ the empirical distribution function associated with these observations and by $\alpha_n$ the empirical Brownian bridge $\alpha_n(t) = \sqrt n(\hat{\mathbb{F}}_n(t) - t)$, Komlos, Major and Tusnady (KMT) showed in 1975 that a Brownian bridge $\mathbb{B}^0$ (depending on $n$) may be constructed on $\Omega$ in such a way that the uniform deviation $\|\alpha_n - \mathbb{B}^0\|_\infty$ between $\alpha_n$ and $\mathbb{B}^0$ is of order of $\log(n)/\sqrt n$ in probability. In this paper, we prove that a Poisson bridge $\mathbb{L}^0_n$ may be constructed on $\Omega$ (note that this construction is not the usual one) in such a way that the uniform deviations between any two of the three processes $\alpha_n, \mathbb{L}^0_n$ and $\mathbb{B}^0$ are of order of $\log(n)/\sqrt n$ in probability. Moreover, we give explicit exponential bounds for the error terms, intended for asymptotic as well as nonasymptotic use.

Consider a sequence $X_1, X_2, \ldots$, of i.i.d. random variables. For each integer $m \geq 1$ let $S_m$ denote the $m$th partial sum of these random variables and set $S_0 = 0$. Assuming that $EX_1 \geq 0$ and the moment generating function $\phi$ of $X_1$ exists in a right neighborhood of 0 the Erdos-Renyi strong law of large numbers states that whenever $k(n)$ is a sequence of positive integers such that $\log n/k(n) \sim c$ as $n \rightarrow \infty$, where $0 < c < \infty$ then $\max\{(S_{m + k(n)} - S_m)/(\gamma(c)k(n)): 0 \leq m \leq n - k(n)\}$ converges almost surely to 1, where $\gamma(c)$ is a constant depending on $c$ and $\phi$. An extended version of this strong law is presented which shows that it remains true in a slightly altered form when $\log n/k(n) \rightarrow \infty$.

Let $P$ be the Lebesque measure on the unit cube in $\mathbb{R}^d$ and $Z_n$ be the centered and normalized empirical process associated with $n$ independent observations with common law $P$. Given a collection of Borel sets $\mathscr{J}$ in $\mathbb{R}^d$, it is known since Dudley's work that if $\mathscr{J}$ is not too large (e.g., either $\mathscr{J}$) is a Vapnik-Cervonenkis class (VC class) or $\mathscr{J}$ fulfills a suitable "entropy with bracketing" condition), then $(Z_n)$ may be strongly approximated by some sequence of Brownian bridges indexed by $\mathscr{J}$, uniformly over $\mathscr{J}$ with some rate $b_n$. We apply the one-dimensional dyadic scheme previously used by Komlos, Major and Tusnady (KMT) to get as good rates of approximation as possible in the above general multidimensional situation. The most striking result is that, up to a possible power of $\log(n), b_n$ may be taken as $n^{-1/2d}$ which is the best possible rate, when $\mathscr{J}$ is the class of Euclidean balls (this is the KMT result when $d = 1$ and the lower bounds are due to Beck when $d \geq 2$). We also obtain some related results for the set-indexed partial-sum processes.

Let $X$ be a real-valued random variable with distribution function $F(x)$ and characteristic function $c(t)$. Let $F_n(x)$ be the $n$th empirical distribution function associated with $X$ and let $c_n(t)$ be the characteristic function $F_n(x)$. Necessary and sufficient conditions in terms of $c(t)$ are obtained for $c_n(t) - c(t)$ to obey bounded and compact laws of the iterated logarithm in the Banach space of continuous complex-valued functions on $\lbrack -1, 1 \rbrack$.

Rates of convergence are obtained for Strassen's functional law of the iterated logarithm for polygonal processes under classical conditions.

We consider the set of mappings of the integers $\{1, 2, \ldots, n\}$ into $\{1, 2, \ldots, n\}$ and put a uniform probability measure on this set. Any such mapping can be represented as a directed graph on $n$ labelled vertices. We study the component structure of the associated graphs as $n \rightarrow \infty$. To each mapping we associate a step function on $\lbrack 0, 1 \rbrack$. Each jump in the function equals the number of connected components of a certain size in the graph which represents the map. We normalize these functions and show that the induced measures on $D\lbrack 0, 1 \rbrack$ converge to Wiener measure. This result complements another result by Aldous on random mappings.

A general stability theorem for $B$-valued random variables is obtained which refines a result of Kuelbs and Zinn. Our proof is based on two exponential inequalities for sums of independent $B$-valued r.v.'s essentially due to Yurinskii and appears particularly simple. We then use our theorem to prove strong invariance principles, LIL results and other related stability results for sums of i.i.d. $B$-valued r.v.'s in the domain of attraction of a Gaussian law. Most of these results seem to be still unknown for real-valued r.v.'s.

Convergence properties are obtained for repeated applications of the operator $f \rightarrow |f - E(f)|$, where $E$ denotes conditional expectation. If, for example, $E$ is the integral with respect to a probability measure $P, f \in L^\infty(P)$ and $T(f) = |f - E(f)|$, then $T^n(f)$ converges to 0 in $L^\infty(P)$ and $\Sigma T^n(f)$ converges in $L^1(P)$.

We obtain a generalization of part "if" of the Erdos-Wintner theorem to additive arithmetic functions with values in certain Banach spaces. Some applications in asymptotic density are considered.

Under fairly general assumptions on the underlying distribution function, the bootstrap process, pertaining to the sample $q$-quantile, converges weakly in $D_\mathbb{R}$ to the standard Brownian motion. Furthermore, weak convergence of a smoothed bootstrap quantile estimate is proved which entails that in this particular case the smoothed bootstrap estimate outperforms the nonsmoothed one.

In this note it is shown that the $L^1$-bounded uniform amarts in $l^1$ form a vector lattice. This extends a result of Austin, Edgar and Ionescu Tulcea on real-valued $L^1$-bounded amarts and parallels a result of Ghoussoub on $L^1$-bounded order amarts.

For a certain class of Markov processes, the $\lambda$-capacitary measure $\pi^\lambda_S$ of a semipolar set $S$ has the following property under a mild condition: A subset $B$ of $S$ is polar if and only if $\pi^\lambda_S(B) = 0$.

In 1946 Feller proposed and proved a famous law of iterated logarithm. Unfortunately his proofs were found to be incorrect, although the main result was true. In this paper the author gives a new proof of the main result.

The exact total variation distances are obtained between a binomial distribution with parameters $n$ and $p$ and Poisson distributions with means $np$ and $-n \log(1 - p)$, for small values of $p$. It is shown that the latter distance is smaller for $0 < p < c_n$ and larger for $c_n < p < a'_{n0}$, where as $n \rightarrow \infty, nc_n \rightarrow 1.596 \ldots$ and $na'_{n0} \rightarrow 3.414 \ldots.$