Let $B=(B_{t})_{t\in\mathbb{R}}$ be a two-sided standard Brownian motion. An unbiased shift of $B$ is a random time $T$, which is a measurable function of $B$, such that $(B_{T+t}-B_{T})_{t\in\mathbb{R}}$ is a Brownian motion independent of $B_{T}$. We characterise unbiased shifts in terms of allocation rules balancing mixtures of local times of $B$. For any probability distribution $\nu$ on $\mathbb{R}$ we construct a stopping time $T\ge0$ with the above properties such that $B_{T}$ has distribution $\nu$. We also study moment and minimality properties of unbiased shifts. A crucial ingredient of our approach is a new theorem on the existence of allocation rules balancing stationary diffuse random measures on $\mathbb{R}$. Another new result is an analogue for diffuse random measures on $\mathbb{R}$ of the cycle-stationarity characterisation of Palm versions of stationary simple point processes.

In this paper we show that on bounded degree graphs and general trees, the cover time of the simple random walk is asymptotically equal to the product of the number of edges and the square of the expected supremum of the Gaussian free field on the graph, assuming that the maximal hitting time is significantly smaller than the cover time. Previously, this was only proved for regular trees and the 2D lattice. Furthermore, for general trees, we derive exponential concentration for the cover time, which implies that the standard deviation of the cover time is bounded by the geometric mean of the cover time and the maximal hitting time.

We characterize the asymptotic independence between blocks consisting of multiple Wiener–Itô integrals. As a consequence of this characterization, we derive the celebrated fourth moment theorem of Nualart and Peccati, its multidimensional extension and other related results on the multivariate convergence of multiple Wiener–Itô integrals, that involve Gaussian and non Gaussian limits. We give applications to the study of the asymptotic behavior of functions of short and long-range dependent stationary Gaussian time series and establish the asymptotic independence for discrete non-Gaussian chaoses.

We characterize nondecreasing weight functions for which the associated one-dimensional vertex reinforced random walk (VRRW) localizes on $4$ sites. A phase transition appears for weights of order $n\log\log n$: for weights growing faster than this rate, the VRRW localizes almost surely on, at most, $4$ sites, whereas for weights growing slower, the VRRW cannot localize on less than $5$ sites. When $w$ is of order $n\log\log n$, the VRRW localizes almost surely on either $4$ or $5$ sites, both events happening with positive probability.

We solve a conjecture raised by Evans in 1991 on the characterization of the positively correlated squared Gaussian vectors. We extend this characterization from squared Gaussian vectors to permanental vectors. As side results, we obtain several equivalent formulations of the property of infinite divisibility for squared Gaussian processes.

In this paper, we study the long-term asymptotics for the quenched moment

\[\mathbb{E}_{x}\exp \biggl\{\int_{0}^{t}V(B_{s})\,ds\biggr\}\]

consisting of a $d$-dimensional Brownian motion $\{B_{s};s\ge0\}$ and a generalized Gaussian field $V$. The major progress made in this paper includes: Solution to an open problem posted by Carmona and Molchanov [Probab. Theory Related Fields 102 (1995) 433–453], the quenched laws for Brownian motions in Newtonian-type potentials and in the potentials driven by white noise or by fractional white noise.

Let $\mathbf{x} _{1},\ldots,\mathbf{x}_{n}$ be a random sample from a $p$-dimensional population distribution, where $p=p_{n}\to\infty$ and $\log p=o(n^{\beta})$ for some $0<\beta\leq1$, and let $L_{n}$ be the coherence of the sample correlation matrix. In this paper it is proved that $\sqrt{n/\log p}L_{n}\to2$ in probability if and only if $Ee^{t_{0}|x_{11}|^{\alpha}}<\infty$ for some $t_{0}>0$, where $\alpha$ satisfies $\beta=\alpha/(4-\alpha)$. Asymptotic distributions of $L_{n}$ are also proved under the same sufficient condition. Similar results remain valid for $m$-coherence when the variables of the population are $m$ dependent. The proofs are based on self-normalized moderate deviations, the Stein–Chen method and a newly developed randomized concentration inequality.

We obtain functional central limit theorems for both discrete time expressions of the form $1/\sqrt{N}\sum_{n=1}^{[Nt]}(F(X(q_{1}(n)),\ldots,X(q_{\ell}(n)))-\bar{F})$ and similar expressions in the continuous time where the sum is replaced by an integral. Here $X(n)$, $n\geq0$ is a sufficiently fast mixing vector process with some moment conditions and stationarity properties, $F$ is a continuous function with polynomial growth and certain regularity properties, $\bar{F}=\int F\,d(\mu\times\cdots\times\mu)$, $\mu$ is the distribution of $X(0)$ and $q_{i}(n)=in$ for $i\le k\leq\ell$ while for $i>k$ they are positive functions taking on integer values on integers with some growth conditions which are satisfied, for instance, when $q_{i}$’s are polynomials of increasing degrees. These results decisively generalize [Probab. Theory Related Fields 148 (2010) 71–106], whose method was only applicable to the case $k=2$ under substantially more restrictive moment and mixing conditions and which could not be extended to convergence of processes and to the corresponding continuous time case. As in [Probab. Theory Related Fields 148 (2010) 71–106], our results hold true when $X_{i}(n)=T^{n}f_{i}$, where $T$ is a mixing subshift of finite type, a hyperbolic diffeomorphism or an expanding transformation taken with a Gibbs invariant measure, as well as in the case when $X_{i}(n)=f_{i}({\Upsilon }_{n})$, where ${\Upsilon}_{n}$ is a Markov chain satisfying the Doeblin condition considered as a stationary process with respect to its invariant measure. Moreover, our relaxed mixing conditions yield applications to other types of dynamical systems and Markov processes, for instance, where a spectral gap can be established. The continuous time version holds true when, for instance, $X_{i}(t)=f_{i}(\xi_{t})$, where $\xi_{t}$ is a nondegenerate continuous time Markov chain with a finite state space or a nondegenerate diffusion on a compact manifold. A partial motivation for such limit theorems is due to a series of papers dealing with nonconventional ergodic averages.

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with infinitely divisible weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation. We obtain an explicit characterization of the structure of these measures, which reflects the constraints imposed by the continuous setting. In particular, we show that the continuous equation enjoys some specific properties that do not appear in the discrete star equation. To that purpose, we define a Lévy multiplicative chaos that generalizes the already existing constructions.

We study large random dissections of polygons. We consider random dissections of a regular polygon with $n$ sides, which are chosen according to Boltzmann weights in the domain of attraction of a stable law of index $\theta\in(1,2]$. As $n$ goes to infinity, we prove that these random dissections converge in distribution toward a random compact set, called the random stable lamination. If $\theta=2$, we recover Aldous’ Brownian triangulation. However, if $\theta\in(1,2)$, large faces remain in the limit and a different random compact set appears. We show that the random stable lamination can be coded by the continuous-time height function associated to the normalized excursion of a strictly stable spectrally positive Lévy process of index $\theta$. Using this coding, we establish that the Hausdorff dimension of the stable random lamination is almost surely $2-1/\theta$.

In this paper, we obtain sufficient conditions in terms of projective criteria under which the partial sums of a stationary process with values in $\mathcal{H}$ (a real and separable Hilbert space) admits an approximation, in $\mathbb{L}^{p}(\mathcal{H})$, $p>1$, by a martingale with stationary differences, and we then estimate the error of approximation in $\mathbb{L}^{p}(\mathcal{H})$. The results are exploited to further investigate the behavior of the partial sums. In particular we obtain new projective conditions concerning the Marcinkiewicz–Zygmund theorem, the moderate deviations principle and the rates in the central limit theorem in terms of Wasserstein distances. The conditions are well suited for a large variety of examples, including linear processes or various kinds of weak dependent or mixing processes. In addition, our approach suits well to investigate the quenched central limit theorem and its invariance principle via martingale approximation, and allows us to show that they hold under the so-called Maxwell–Woodroofe condition that is known to be optimal.

The celebrated results of Komlós, Major and Tusnády [Z. Wahrsch. Verw. Gebiete 32 (1975) 111–131; Z. Wahrsch. Verw. Gebiete 34 (1976) 33–58] give optimal Wiener approximation for the partial sums of i.i.d. random variables and provide a powerful tool in probability and statistics. In this paper we extend KMT approximation for a large class of dependent stationary processes, solving a long standing open problem in probability theory. Under the framework of stationary causal processes and functional dependence measures of Wu [Proc. Natl. Acad. Sci. USA 102 (2005) 14150–14154], we show that, under natural moment conditions, the partial sum processes can be approximated by Wiener process with an optimal rate. Our dependence conditions are mild and easily verifiable. The results are applied to ergodic sums, as well as to nonlinear time series and Volterra processes, an important class of nonlinear processes.

Unique existence of solutions to porous media equations driven by continuous linear multiplicative space–time rough signals is proven for initial data in $L^{1}(\mathcal{O})$ on bounded domains $\mathcal{O} $. The generation of a continuous, order-preserving random dynamical system on $L^{1}(\mathcal{O})$ and the existence of a random attractor for stochastic porous media equations perturbed by linear multiplicative noise in space and time is obtained. The random attractor is shown to be compact and attracting in $L^{\infty}(\mathcal{O})$ norm. Uniform $L^{\infty}$ bounds and uniform space–time continuity of the solutions is shown. General noise including fractional Brownian motion for all Hurst parameters is treated and a pathwise Wong–Zakai result for driving noise given by a continuous semimartingale is obtained. For fast diffusion equations driven by continuous linear multiplicative space–time rough signals, existence of solutions is proven for initial data in $L^{m+1}(\mathcal{O})$.