Let $G$ be a finite group. Choose a set $S$ of size $k$ uniformly from $G$ and consider a lazy random walk on the corresponding Cayley graph. We show that for almost all choices of $S$ given $k = 2a\, \log_2 |G|$, $a\gt1$, this walk mixes in under $m = 2a \,\log\frac{a}{a-1} \log |G|$ steps. A similar result was obtained earlier by Alon and Roichman and also by Dou and Hildebrand using a different techniques. We also prove that when sets are of size $k = \log_2 |G| + O(\log \log |G|)$, $m = O(\log^3 |G|)$ steps suffice for mixing of the corresponding symmetric lazy random walk. Finally, when $G$ is abelian we obtain better bounds in both cases.

It is shown that value functions for controlled degenerate diffusion processes can be approximated with error of order $h^{1/3}$ by using policies which are constant on intervals $[kh^{2},(k+1)h^{2})$.

We present an upper bound on the mixing rate of the equilibrium state of a dynamical system defined by the one-sided shift and a non Hölder potential of summable variations. The bound follows from an estimation of the relaxation speed of chains with complete connections with summable decay, which is obtained via a explicit coupling between pairs of chains with different histories.

Genetic models incorporating resampling and migration are now fairly well-understood. Problems arise in the analysis, if both selection and mutation are incorporated. This paper addresses some aspects of this problem, in particular the analysis of the long-time behaviour before the equilibrium is reached (quasi-equilibrium, which is the time range of interest in most applications).

Recently, Naiman and Wynn introduced the concept of an abstract tube in order to obtain improved inclusion-exclusion identities and inequalities that involve much fewer terms than their classical counterparts. In this paper, we introduce a particular class of abstract tubes which plays an important role with respect to chromatic polynomials and network reliability. The inclusion-exclusion identities and inequalities associated with this class simultaneously generalize several well-known results such as Whitney's broken circuit theorem, Shier's expression for the reliability of a network as an alternating sum over chains in a semilattice and Narushima's inclusion-exclusion identity for posets. Moreover, we show that under some restrictive assumptions a polynomial time inclusion-exclusion algorithm can be devised, which generalizes an important result of Provan and Ball on network reliability.

We extend the definition of Walsh's martingale measure stochastic integral so as to be able to solve stochastic partial differential equations whose Green's function is not a function but a Schwartz distribution. This is the case for the wave equation in dimensions greater than two. Even when the integrand is a distribution, the value of our stochastic integral process is a real-valued martingale. We use this extended integral to recover necessary and sufficient conditions under which the linear wave equation driven by spatially homogeneous Gaussian noise has a process solution, and this in any spatial dimension. Under this condition, the non-linear three dimensional wave equation has a global solution. The same methods apply to the damped wave equation, to the heat equation and to various parabolic equations.

We study the weak convergence for the row sums of a general triangular array of empirical processes indexed by a manageable class of functions converging to an arbitrary limit. As particular cases, we consider random series processes and normalized sums of i.i.d. random processes with Gaussian and stable limits. An application to linear regression is presented. In this application, the limit of the row sum of a triangular array of empirical process is the mixture of a Gaussian process with a random series process.

We prove moderate deviations principles for:

• 1. unbounded additive functionals of the form $S_n = \sum_{j=1}^{n} g(X^{(p)}_{j-1})$, where $(X_n)_{n\in N}$ is a stable $R^d$-valued functional autoregressive model of order $p$ with white noise and stationary distribution $\mu$, and $g$ is an $R^q$-valued Lipschitz function of order $(r,s)$;

• 2. the error of the least squares estimator (LSE) of the matrix $\theta$ in an $R^d$-valued regression model $X_n = \theta^t \phi_{n-1} + \epsilon_n$, where $(\epsilon_n)$ is a generalized gaussian noise.

In this paper we contribute to the investigation of the fractal nature of the intersection local time measure on the intersection of independent Brownian paths. We particularly point out the difference in the small scale behaviour of the intersection local times in three-dimensional space and in the plane by studying almost sure limit theorems motivated by the notion of average densities introduced by Bedford and Fisher. We show that in 3-space the intersection local time measure of two paths has an average density of order two with respect to the gauge function $\varphi(r)=r$, but in the plane, for the intersection local time measure of p Brownian paths, the average density of order two fails to converge. The average density of order three, however, exists for the gauge function $\varphi_p(r)=r^2[\log(1/r)]^p$. We also prove refined versions of the above results, which describe more precisely the fluctuations of the volume of small balls around these gauge functions by identifying the density distributions, or lacunarity distributions, of the intersection local times.

Let $T(x,r)$ denote the total occupation measure of the ball of radius $r$ centered at $x$ for a transient symmetric stable processes of index $b \lt d$ in $R^d$ and $K(b,d)$ denote the norm of the convolution with its 0-potential density, considered as an operator on $L^2(B(0,1),dx)$. We prove that as $r$ approaches 0, almost surely $\sup_{|x| \leq 1} T(x,r)/(r^b|\log r|) \to b K(b,d)$. Furthermore, for any $a \in (0,b/K(b,d))$, the Hausdorff dimension of the set of ``thick points'' $x$ for which $\limsup_{r \to 0} T(x,r)/(r^b |\log r|)=a$, is almost surely $b-a/K(b,d)$; this is the correct scaling to obtain a nondegenerate ``multifractal spectrum'' for transient stable occupation measure. The liminf scaling of $T(x,r)$ is quite different: we exhibit positive, finite, non-random $c(b,d), C(b,d)$, such that almost surely $c(b,d) \lt \sup_x \liminf_{r \to 0} T(x,r)/r^b \lt C(b,d)$.

For a random process $X$ consider the random vector defined by the values of $X$ at times $0 \lt U_{n,1} \lt ... \lt U_{n,n} \lt 1$ and the minimal values of $X$ on each of the intervals between consecutive pairs of these times, where the $U_{n,i}$ are the order statistics of $n$ independent uniform $(0,1)$ variables, independent of $X$. The joint law of this random vector is explicitly described when $X$ is a Brownian motion. Corresponding results for Brownian bridge, excursion, and meander are deduced by appropriate conditioning. These descriptions yield numerous new identities involving the laws of these processes, and simplified proofs of various known results, including Aldous's characterization of the random tree constructed by sampling the excursion at $n$ independent uniform times, Vervaat's transformation of Brownian bridge into Brownian excursion, and Denisov's decomposition of the Brownian motion at the time of its minimum into two independent Brownian meanders. Other consequences of the sampling formulae are Brownian representions of various special functions, including Bessel polynomials, some hypergeometric polynomials, and the Hermite function. Various combinatorial identities involving random partitions and generalized Stirling numbers are also obtained.

Consider a countable collection of particles located on a countable group, performing a critical branching random walk where the branching rate of a particle is given by a random medium fluctuating both in space and time. Here we study the case where the time-space random medium (called catalyst) is also a critical branching random walk evolving autonomously while the local branching rate of the reactant process is proportional to the number of catalytic particles present at a site. The catalyst process and the reactant process typically have different underlying motions.

In this paper we investigate stopped partial sums for weak dependent sequences. In particular, the results are used to obtain new maximal inequalities for strongly mixing sequences and related almost sure results.

A common technique in the theory of stochastic process is to replace a discrete time coordinate by a continuous randomized time, defined by an independent Poisson or other process. Once the analysis is complete on this poissonized process, translating the results back to the original setting may be nontrivial. It is shown here that, under fairly general conditions, if the process $S_n$ and the time change $\phi_n$ both converge, when normalized by the same constant, to limit processes combined process $S_n(\phi_n(t))$ converges, when properly normalized, to a sum of the limit of the orginal process, and the limit of the time change multiplied by the derivative of $E S_n$. It is also shown that earlier results on the fine structure of the maxima are preserved by these time changes.

Let $M_d$ be the maximum of a standard Bessel bridge of dimension $d$. A series formula for $P(M_d \le a)$ due to Gikhman and Kiefer for $d = 1,2, \ldots$ is shown to be valid for all real $d \gt 0$. Various other characterizations of the distribution of $M_d$ are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of $M_d$ is described both as $d$ tends to infinity and as $d$ tends to zero.

The limit process of aggregational models---(i) sum of random coefficient AR(1) processes with independent Brownian motion (BM) inputs and (ii) sum of AR(1) processes with random coefficients of Gamma distribution and with input of common BM's,---proves to be Gaussian and stationary and its transfer function is the mixture of transfer functions of Ornstein--Uhlenbeck (OU) processes by Gamma distribution. It is called Gamma-mixed Ornstein--Uhlenbeck process ($\Gamma\mathsf{MOU}$). For independent Poisson alternating $0$-$1$ reward processes with proper random intensity it is shown that the standardized sum of the processes converges to the standardized $\Gamma\mathsf{MOU}$ process. The $\Gamma\mathsf{MOU}$ process has various interesting properties and it is a new candidate for the successful modelling of several Gaussian stationary data with long-range dependence. Possible applications and problems are also considered.

We establish a large deviations type evaluation for the family of integral functionals.

For a quasi-regular (symmetric) Dirichlet space $( {\cal E}, {\cal F})$ and an associated symmetric standard process $(X_t, P_x)$, we show that, for $u in {\cal F}$, the additive functional $u^*(X_t) - u^*(X_0)$ is a semimartingale if and only if there exists an ${\cal E}$-nest $\{F_n\}$ and positive constants $C_n$ such that $ \vert {\cal E}(u,v)\vert \leq C_n \Vert v\Vert_\infty, v \in {\cal F}_{F_n,b}.$ In particular, a signed measure resulting from the inequality will be automatically smooth. One of the variants of this assertion is applied to the distorted Brownian motion on a closed subset of $R^d$, giving stochastic characterizations of BV functions and Caccioppoli sets.

The generator of a Borel right processis extended so that it maps functions to smooth measures. This extension may be defined either probabilistically using martingales or analytically in terms of certain kernels on the state space of the process. Then the associated Schrödinger equation with a (signed) measure serving as potential may be interpreted as an equation between measures. In this context general existence and uniqueness theorems for solutions are established. These are then specialized to obtain more concrete results in special situations.