A self-contained outline of recent work on the white noise approach to nonlinear filtering is given together with the necessary background of white noise calculus.

We prove central limit theorems, strong laws, large deviation results, and a weak convergence theorem for suitably normalized occupation times of critical binary branching Brownian motions started from Poisson random fields on $R^d, d \geq 2$. The results are strongly dimension dependent. The main result (Theorem 2) asserts that in two dimensions, as opposed to all other dimensions, the average occupation time of a bounded set with positive measure converges in distribution to a nondegenerate limit.

We consider a Markov process $(\eta^\mu_t)_{t \in \mathbb{R}^+}$ on $(\mathbb{N}^S (S = \mathbb{Z}^d)$ with initial distribution $\mu$ and the following time evolution: At rate $b\sum_y q(y, x)\eta(y)$ a particle is born at site $x$; at rate $\tilde{d}\eta(x)$ a particle dies at site $x$. All particles perform independent from each other continuous time random walk with kernel $p(x, y)$ and rate $m$. All particles at a site $x$ die at rate $D(x)$. Here $D(x)$ are random variables taking the values $D_1, D_2 (D_2 \geq D_1 \geq 0)$. We assume $\{D(x)\}_{x \in S}$ to be stationary and ergodic. This paper studies the features of the model for $p(x, y), q(x, y)$ symmetric. We calculate the exponential growth rate $\lambda$ of $\tilde{E}(\eta^\mu_t(x))$ (with $\tilde{E}$ denoting conditional expectation with respect to the environment) and show that $\lambda$ is nonrandom and strictly bigger than $b - \tilde{d} - E(D(x))$, if $D_2 > D_1$. We have $\lambda = b - \tilde{d} - D_1$. Introduce the process $(\hat{\eta}^\mu_t)_{t\in \mathbb{R}^+}$ by setting $\hat{\eta}^\mu_t(x) = (\tilde{E}(\eta^\mu_t(x)))^{-1} \eta^\mu_t(x)$. A critical phenomenon with respect to the parameter $p := D_1(\tilde{d} + ED(x))^{-1}$ occurs in the sense that for $p > p^{(2)}$ the quantity $\tilde{E}(\hat{\eta}^\mu_t(x))^2$ grows exponentially fast, while for $p \leq p^{(2)}, \lambda > 0$ the exponential growth rate of $\tilde{E}(\hat{\eta}^\mu_t(x))^2$ is 0. $p^{(2)}$ is the same as for a system with $D(x) \equiv D_1$ and can be calculated explicitly.

A functional law of the iterated logarithm for time changed Brownian motion is given for stopping times that increase at a geometric rate. This result is applied to various quantities associated with a Galton-Watson process.

The almost sure growth behavior of some time-homogeneous Markov chains is studied. They generalize the ordinary Galton-Watson processes with regard to allowing state-dependent offspring distributions and also to controlling the number of reproducing individuals by a random variable that depends on the state of the process. The main assumption is that the mean offspring per individual is nonincreasing while the state increases. These controlled Galton-Watson processes can be included in a general growth model whose divergence rate is determined. In case of processes that differ from the Galton-Watson process only by the state dependence of the offspring distributions, a necessary and sufficient moment condition for divergence with "natural" rate is obtained generalizing the $(x \log x)$ condition of Galton-Watson processes. In addition, some criteria are given when a state-dependent Galton-Watson process behaves like an ordinary supercritical Galton-Watson process.

A new method of tackling convergence properties of random processes turns out to be applicable to finite mean supercritical age-dependent branching processes. If $\{Z^\phi_t\}$ is a Crump-Mode-Jagers process counted with general characteristics $\phi$, convergence in probability of $\{e^{-\alpha t} Z^\phi_t\}$ follows from convergence in distribution. Under some mild restrictions on $\phi$, norming constants $\{C(t)\}$ are identified such that $\{C^{-1}(t)Z^\phi_t\}$ converges almost surely to a nondegenerate limit.

We show that an $L^1$-bounded Banach-space-valued martingale in the limit $(X_n)$ can be written $X_n = Y_n + Z_n$, where $(Y_n)$ is an $L^1$-bounded martingale and where $(Z_n)$ is a martingale in the limit that goes to zero a.s. in norm. This theorem still holds for a new class that generalizes martingales in the limit. We show that a real-valued $L^1$-bounded pramart $(X_n)$ can be written $X_n = Y_n + Z_n$, where $Y_n$ is an $L^1$-bounded martingale, and $Z_n$ has the following property: For each $\varepsilon > 0$, there is an $m$, an $\Sigma_m$-measurable subset $A$ of $\Omega$, and a supermartingale $(T_n)_{n \geq m}$ on $A$ such that $\int_A T_n dP \leq \varepsilon$ and $|Z_n| \leq T_n$ on $A$ for $n \geq m$.

In the study of recurrent sets for transient Levy processes on the real line, we present two main results. As long as the process has a "well-behaved" (bounded in a particular way) kernel, a set is recurrent for the process if and only if the sum of the capacities of pieces of the set is infinite. In the second result, we show that a simple condition on the Levy measure guarantees that the process has a "well-behaved" kernel. Finally, the results are applied to subordinators in order to construct examples of recurrent sets including a recurrent set with finite Lebesgue measure.

For a Harris recurrent Markov chain with invariant initial distribution $\pi$, we consider the return times $\tau_\varepsilon$ to state sets $A_\varepsilon$ with $0 < \pi(A_\varepsilon) \rightarrow 0$ as $\varepsilon \rightarrow 0$ and show that, provided the probability of early returns to $A_\varepsilon$ approaches 0, the $\tau_\varepsilon$, multiplied by suitable scaling factors, are asymptotically exponentially distributed.

The optimal sampling theorem (OST) is not necessarily true for supermartingales indexed by a partially ordered set. However, if the index set satisfies a mild separability condition, without necessarily being directed or countable, we prove the OS inequality for a class of supermartingales that extends the concept of $S$-processes defined by Cairoli on the plane $\mathbb{R}^2_+$. Under a further restriction on these processes we obtain the OS equation, thus extending the corresponding result for martingales to the case of nondirected index sets. We then introduce strong martingales and strong supermartingales for separable partially ordered index sets, and show that these processes again satisfy the OST. By defining stopping domains as well as the value of a process for a stopping domain, we show that the strong (super)martingales are precisely those processes which satisfy the OST for all bounded stopping domains. This extends a result of Cairoli-Walsh and Wong-Zakai on $\mathbb{R}^2_+$.

We give a generalization of the result of Erdos and Renyi on the length $R_n$ of the longest head run in the first $n$ tosses of a coin. Consider two independent sequences, $X_1 X_2\cdots X_m$ and $Y_1Y_2\cdots Y_n$. Suppose that $X_1, X_2,\cdots$ are i.i.d. $\mu$, and $Y_1, Y_2,\cdots$ are i.i.d. $\nu$, where $\mu$ and $\nu$ are possibly different distributions on a common finite alphabet $S$. Let $p \equiv P(X_1 = Y_1) \in (0, 1)$. The length of the longest matching consecutive subsequence is $M_{m,n} \equiv \max \{k: X_{i+r} = Y_{j+r}$ for $r = 1$ to $k$, for some $0 \leq i \leq m - k, 0 \leq j \leq n - k\}$. For $m$ and $n \rightarrow \infty$ with $\log(m)/\log(mn) \rightarrow \lambda \in (0,1)$, our result is that there is a constant $K \equiv K(\mu, \nu, \lambda) \in (0, 1\rbrack$ such that $P(\lim M_{m,n}/\log_{1/p}(mn) = K) = 1$. The proof uses large deviation methods. The constant $K$ is determined from a variational formula involving the Kullback-Liebler distance or relative entropy. A simple necessary and sufficient condition for $K = 1$ is given. For the case $m = n (\lambda = 1/2)$ and $\mu = \nu, K = 1$. The set of $(\mu, \nu, \lambda)$ for which $K = 1$ has nonempty interior. The boundary of this set is the location of a phase transition. The results generalize to more than two sequences and to Markov chains. A strong law of large numbers is given for the proportion of letters within the longest matching word; the limiting proportion exhibits critical behavior, similar to that of $K$.

Let $\mathscr{P}$ be a homogeneous Poisson process in $\mathbb{R}^k$. At the points of $\mathscr{P}$, centre $k$-dimensional spheres whose radii are independent and identically distributed. It is shown that there exists a positive critical intensity for the formation of clumps whose mean size is infinite, if and only if sphere content has finite variance. It is also proved that under a strictly weaker condition than existence of finite variance, there exists a positive critical intensity for the formation of clumps whose size is infinite with positive probability. Therefore these two critical intensities need not be the same. Continuum percolation in the case of general random sets, not just spheres, is studied, and bounds are obtained for a critical intensity.

The problem of estimating $\operatorname{Prob}(X(B) \leq x)$ for large regions $B \subset \mathbb{R}^d$ when $X$ is a random measure is solved under a condition of positive dependence and summable correlations. Several applications are given and, in cases in which the applications have been examined previously, it is shown that the results are true under milder moment conditions than known before.

A new version of Kingman's subadditive ergodic theorem is presented, in which the subadditivity and stationarity assumptions are relaxed without weakening the conclusions. This result applies to a number of situations that were not covered by Kingman's original theorem. The proof involves a rather simple reduction to the additive case, where Birkhoff's ergodic theorem can be applied.

Shorter and more elementary proofs of some results of Asmussen and Kurtz are given. We determine first those subsequences for which mean zero is the necessary and sufficient requirement for complete convergence and then give integrability conditions in terms of the growth of the subsequences in the case when a moment of order greater than one exists.

Let $\{X_1, X_2,\cdots\}$ be a stationary process with probability densities $f(X_1, X_2,\cdots, X_n)$ with respect to Lebesgue measure or with respect to a Markov measure with a stationary transition measure. It is shown that the sequence of relative entropy densities $(1/n)\log f(X_1, X_2,\cdots, X_n)$ converges almost surely. This long-conjectured result extends the $L^1$ convergence obtained by Moy, Perez, and Kieffer and generalizes the Shannon-McMillan-Breiman theorem to nondiscrete processes. The heart of the proof is a new martingale inequality which shows that logarithms of densities are $L^1$ dominated.

In this paper it is established that the normalized sample paths of a $\phi$-mixing sequence converge to the Brownian motion, under the Lindeberg's condition and under some stationarity assumptions. No mixing rate is required.

Davydov showed that in a classic central limit theorem of Ibragimov under the strong mixing condition, the combinations of mixing rate and moment assumption were practically as weak as permissible. Here some constructions will be given to show that Davydov's observation still holds under the additional assumption that the growth of $\operatorname{var}(X_1 + \cdots + X_n)$ be asymptotically linear. Our constructions will be similar to a recent one of Herrndorf.

Let $\{\xi_n\}$ be a stationary sequence and $\xi^{(n)}_1 \leq \cdots \leq \xi^{(n)}_n$ be the order statistics of $\xi_1,\cdots, \xi_n$. In this paper the limiting distribution of $\{\xi^{(n)}_{k_n}\}$, where $\{k_n\}$ satisfies $\min(k_n, n - k_n) \rightarrow \infty$, is determined under appropriate conditions. Further results for some special $\{k_n\}$ that satisfy $k_n/n \rightarrow \lambda \in \lbrack 0, 1\rbrack$ are also obtained. These results are applied to discussing the limiting distributions of corresponding order statistics from $m$-dependent stationary sequences and stationary normal sequences.

The problem of the identifiability of the mixing distribution and of the unknown parameters for a continuous mixture of Gaussian distributions is considered. Relevance of the problem under various analytical, statistical, and applicative points of view is stressed. Uniqueness of the mixing distribution and of the mean and variance functions for the mixed Gaussian distribution is proved. Furthermore, their continuous dependence on the mixture itself is proved under suitable topologies. These results also extend to the multidimensional case and to the case of non-Gaussian distributions, and/or signed mixing measure.

We first show by an example that, unlike the domination of hitting distributions, the process with smaller last exit distributions may not be obtainable from the one with bigger last exit distributions by killing and time-change. Then we prove that, under a fairly general condition, the last exit distributions of a transient Hunt process cannot nontrivially dominate those of another transient Hunt process.

For positive integers $s, n$ let $M_s = (1/s)V_sV^T_s$, where $V_s$ is an $n \times s$ matrix composed of i.i.d. $N(0, 1)$ random variables. Assume $n = n(s)$ and $n/s \rightarrow y \in (0, 1)$ as $s \rightarrow \infty$. Then it is shown that the smallest eigenvalue of $M_s$ converges almost surely to $(1 - \sqrt y)^2$ as $s \rightarrow \infty$.

Let $X, X_1, X_2,\cdots$ be i.i.d. random variables and let $M_n = \max_{1\leq j \leq n}X_j$. For each nondecreasing real sequence $\{b_n\}$ such that $P(X > b_n) \rightarrow 0$ and $P(M_n \leq b_n) \rightarrow 0$ we show that $P(M_n \leq b_n i.o.) = 1$ if and only if $\sum_nP(X > b_n)\exp\{- nP(X > b_n)\} = \infty$. The restrictions on the $b_n's$ can be removed.

Is there a distribution in $n$ space that has the same uniform distribution for its projection in all directions?

In this paper we define the concept of a normal fuzzy random variable and we prove the following representation theorem: Every normal fuzzy random variable equals the sum of its expected value and a mean zero random vector.