We study the limiting spectral measure of large symmetric random matrices of linear algebraic structure.

For Hankel and Toeplitz matrices generated by i.i.d. random variables {X_k} of unit variance, and for symmetric Markov matrices generated by i.i.d. random variables {X_ij}_j>i of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, nonrandom, symmetric distributions γ_H, γ_M and γ_T of unbounded support. The moments of γ_H and γ_T are the sum of volumes of solids related to Eulerian numbers, whereas γ_M has a bounded smooth density given by the free convolution of the semicircle and normal densities.

For symmetric Markov matrices generated by i.i.d. random variables {X_ij}_j>i of mean m and finite variance, scaling the eigenvalues by n we prove the almost sure, weak convergence of the spectral measures to the atomic measure at −m. If m=0, and the fourth moment is finite, we prove that the spectral norm of M_n scaled by $\sqrt{2n\log n}$ converges almost surely to 1.

We study random exponential sums of the form ∑_k=1ⁿX_kexp{i(λ_k⁽¹⁾t₁+⋯+λ_k^(s)t_s)}, where {X_n} is a sequence of random variables and {λ_n⁽ⁱ⁾:1≤i≤s} are sequences of real numbers. We obtain uniform estimates (on compact sets) of such sums, for independent centered {X_n} or bounded {X_n} satisfying some mixing conditions. These results generalize recent results of Weber [Math. Inequal. Appl. 3 (2000) 443–457] and Fan and Schneider [Ann. Inst. H. Poincaré Probab. Statist. 39 (2003) 193–216] in several directions. As applications we derive conditions for uniform convergence of these sums on compact sets. We also obtain random ergodic theorems for finitely many commuting measure-preserving point transformations of a probability space. Finally, we show how some of our results allow to derive the Wiener–Wintner property (introduced by Assani [Ergodic Theory Dynam. Systems 23 (2003) 1637–1654]) for certain functions on certain dynamical systems.

Several classical results on boundary crossing probabilities of Brownian motion and random walks are extended to asymptotically Gaussian random fields, which include sums of i.i.d. random variables with multidimensional indices, multivariate empirical processes, and scan statistics in change-point and signal detection as special cases. Some key ingredients in these extensions are moderate deviation approximations to marginal tail probabilities and weak convergence of the conditional distributions of certain “clumps” around high-level crossings. We also discuss how these results are related to the Poisson clumping heuristic and tube formulas of Gaussian random fields, and describe their applications to laws of the iterated logarithm in the form of the Kolmogorov–Erdős–Feller integral tests.

In this paper we consider probabilistic analogues of some classical integral geometric formulae: Weyl–Steiner tube formulae and the Chern–Federer kinematic fundamental formula. The probabilistic building blocks are smooth, real-valued random fields built up from i.i.d. copies of centered, unit-variance smooth Gaussian fields on a manifold M. Specifically, we consider random fields of the form f_p=F(y₁(p),…,y_k(p)) for F∈C²(ℝ^k;ℝ) and (y₁,…,y_k) a vector of C² i.i.d. centered, unit-variance Gaussian fields.

The analogue of the Weyl–Steiner formula for such Gaussian-related fields involves a power series expansion for the Gaussian, rather than Lebesgue, volume of tubes: that is, power series expansions related to the marginal distribution of the field f. The formal expansions of the Gaussian volume of a tube are of independent geometric interest.

As in the classical Weyl–Steiner formulae, the coefficients in these expansions show up in a kinematic formula for the expected Euler characteristic, χ, of the excursion sets M∩f⁻¹[u,+∞)=M∩y⁻¹(F⁻¹[u,+∞)) of the field f.

The motivation for studying the expected Euler characteristic comes from the well-known approximation $\mathbb{P}[\sup_{p\in M}f(p)\geq u]\simeq\mathbb{E}[\chi(f^{-1}[u,+\infty))]$.

We study the two-dimensional fractional Brownian motion with Hurst parameter H>½. In particular, we show, using stochastic calculus, that this process admits a skew-product decomposition and deduce from this representation some asymptotic properties of the motion.

We consider discrete-time random perturbations of monotone cellular automata (CA) in two dimensions. Under general conditions, we prove the existence of half-space velocities, and then establish the validity of the Wulff construction for asymptotic shapes arising from finite initial seeds. Such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases. In many cases, exact stability is observed. That is, for small perturbations, the shapes of the deterministic and random processes agree exactly. We give a complete characterization of such cases, and show that they are prevalent among threshold growth CA with box neighborhood. We also design a nontrivial family of CA in which the shape is exactly computable for all values of its probability parameter.

Let $\mathscr{T}_{n}(x)$ denote the time of first visit of a point x on the lattice torus ℤ_n²=ℤ²/nℤ² by the simple random walk. The size of the set of α, n-late points $ℒ_{n}(\alpha )=\{x\in \mathbb {Z}_{n}^{2}: \mathscr{T}_{n}(x)\geq \alpha \frac{4}{\pi}(n\log n)^{2}\}$ is approximately n^2(1−α), for α∈(0,1) [ℒ_n(α) is empty if α>1 and n is large enough]. These sets have interesting clustering and fractal properties: we show that for β∈(0,1), a disc of radius n^β centered at nonrandom x typically contains about n^{2β(1−α/β2)} points from ℒ_n(α) (and is empty if $\beta <\sqrt{ \alpha }$), whereas choosing the center x of the disc uniformly in ℒ_n(α) boosts the typical number of α,n-late points in it to n^2β(1−α). We also estimate the typical number of pairs of α, n-late points within distance n^β of each other; this typical number can be significantly smaller than the expected number of such pairs, calculated by Brummelhuis and Hilhorst [Phys. A 176 (1991) 387–408]. On the other hand, our results show that the number of ordered pairs of late points within distance n^β of each other is larger than what one might predict by multiplying the total number of late points, by the number of late points in a disc of radius n^β centered at a typical late point.

It is shown in this paper that the transition kernel corresponding to a spatially inhomogeneous random walk on Z^d admits upper and lower Gaussian estimates.

Bellman equations of ergodic type related to risk-sensitive control are considered. We treat the case that the nonlinear term is positive quadratic form on first-order partial derivatives of solution, which includes linear exponential quadratic Gaussian control problem. In this paper we prove that the equation in general has multiple solutions. We shall specify the set of all the classical solutions and classify the solutions by a global behavior of the diffusion process associated with the given solution. The solution associated with ergodic diffusion process plays particular role. We shall also prove the uniqueness of such solution. Furthermore, the solution which gives us ergodicity is stable under perturbation of coefficients. Finally, we have a representation result for the solution corresponding to the ergodic diffusion.

Large deviation for Markov processes can be studied by Hamilton–Jacobi equation techniques. The method of proof involves three steps: First, we apply a nonlinear transform to generators of the Markov processes, and verify that limit of the transformed generators exists. Such limit induces a Hamilton–Jacobi equation. Second, we show that a strong form of uniqueness (the comparison principle) holds for the limit equation. Finally, we verify an exponential compact containment estimate. The large deviation principle then follows from the above three verifications.

This paper illustrates such a method applied to a class of Hilbert-space-valued small diffusion processes. The examples include stochastically perturbed Allen–Cahn, Cahn–Hilliard PDEs and a one-dimensional quasilinear PDE with a viscosity term. We prove the comparison principle using a variant of the Tataru method. We also discuss different notions of viscosity solution in infinite dimensions in such context.

The probabilistic type spaces in the sense of Harsanyi [Management Sci. 14 (1967/68) 159–182, 320–334, 486–502] are the prevalent models used to describe interactive uncertainty. In this paper we examine the existence of a universal type space when beliefs are described by finitely additive probability measures. We find that in the category of all type spaces that satisfy certain measurability conditions (κ-measurability, for some fixed regular cardinal κ), there is a universal type space (i.e., a terminal object) to which every type space can be mapped in a unique beliefs-preserving way. However, by a probabilistic adaption of the elegant sober-drunk example of Heifetz and Samet [Games Econom. Behav. 22 (1998) 260–273] we show that if all subsets of the spaces are required to be measurable, then there is no universal type space.