In this paper we first present a multidimensional version of the characterization of the conditional independence in terms of a factorization property proved by Alabert et al. in the scalar case. As an application, we prove that the solution of a particular two-dimensional linear stochastic differential equation with boundary condition, considered by Ocone and Pardoux, is not a Markov field.

We establish some strong limit theorems for the longest excursion lengths of a Bessel process of dimension d ∈ (0,2). In the special case d=1, we recover and improve some well-known results for Wiener processes, and solve an open problem raised. The proof relies on exact distributions evaluated by Pitman and Yor and on a careful analysis of the Bessel sample paths.

We consider discrete penalization schemes for reflecting stochastic differential equations. The convergence results obtained by Liu are generalized and refined. We also compare the penalization schemes with a more well-known recursive projection scheme.

Stationary and long-run distributions of a class of Markov chains with continuous state space S ⊂ R^d are studied. Applications to random search algorithms for global optimization are presented.

For a fixed integer $r\ge 1$, let $Z_{\mathrm{rn}}$ be the $r$th largest of $\{X_1,X_2,...,X_n\}$, where $X_1,X_2,...$ is a sequence of i.i.d. random variables with the common distribution fuction $F\left(x\right)$. We prove that $P\{Z_{\mathrm{rn}}\le u_n,$ i.o.}=$0$ or $1$ accordingly as the series ${\sum }_{n=1}^{\mathrm{\infty }}exp[-n\{1-F\left(u_n\right)\left\}\right]\left[n\right\{1-F\left(u_n\right)\}{]}^r/n<\mathrm{\infty }$ or $=\mathrm{\infty }$ for any real sequence $\left\{u_n\right\}$ such that ${lim}_{n\to \mathrm{\infty }}n\{1-F(u_n\left)\right\}=+\mathrm{\infty }$. This weakens the condition added on the sequence $\left[n\right\{1-F\left(u_n\right)\left\}\right]$ by Wang and Tomkins and generalizes the results of Klass to the case when $r\ge 1$.

We consider the problem of stationary distribution function estimation at a given point by the observations of an ergodic diffusion process on the interval [0,T] as T→∞. First we introduce a lower (minimax) bound on the risk of all estimators and then we prove that the empirical distribution function attains this bound. Hence this estimator is asymptotically efficient in the sense of the given bound.

Let W_t(0≤t<∞) denote a Brownian motion process which has zero drift during the time interval [0,ν) and drift θ during the time interval [ν,∞), where θ and ν are unknown. The process W is observed sequentially. The general goal is to find a stopping time T of W that 'detects' the unknown time point ν as soon and as reliably as possible on the basis of this information. Here stopping always means deciding that a change in the drift has already occurred. We discuss two particular loss structures in a Bayesian framework. Our first Bayes risk is closely connected to that of the Bayes tests of power one of Lerche. The second Bayes risk generalizes the disruption problem of Shiryayev to the case of unknown θ.

Let $\left(Y_n\right)$ be a sequence of independent random variables with common distribution $F$ and define the iteration $X_0=x_0$, $X_n:=X_{n-1}\vee (\alpha X_{n-1}+Y_n)$, $\alpha \in [0,1)$. We denote by $D\left({\Phi }_{\gamma }\right)$ the domain of maximal attraction of ${\Phi }_{\gamma }$, the extreme value distribution of the first type. Greenwood and Hooghiemstra showed in 1991 that for $F\in D\left({\Phi }_{\gamma }\right)$ there exist norming constants $a_n>0$ and $b_n\in R$ such that $a_n^{-1}\{X_n-b_n/(1-\alpha \left)\right\}$ has a non-degenerate (distributional) limit. In this paper we show that the same is true for $F\in D\left({\Psi }_{\gamma }\right)\cup D\left(\Lambda \right)$, the type II and type III domains. The method of proof is entirely different from the method in the aforementioned paper. After a proof of tightness of the involved sequences we apply (modify) a result of Donnelly concerning weak convergence of Markov chains with an entrance boundary.

Acknowledgement of refereeing work BJ vol. 03

Annual index BJ vol. 03

Keyword Index BJ vol. 03