We use the concept of unimodular random graph to show that the branching simple random walk on $\mathbb{Z}^{d}$ indexed by a critical geometric Galton-Watson tree conditioned to survive is recurrent if and only if $d \leq 4$.

We study a percolation process on the planted binary tree,where clusters freeze as soon as they become larger than some fixed parameter N. We show that as N goes to infinity, the processconverges in some sense to the frozen percolation process introducedby Aldous. In particular, our results show that the asymptotic behaviour differs substantially from that on the square lattice, on which a similar process has been studied recently by van den Berg, de Lima and Nolin.

Let $K_n$ be the number of types in the sample $\left\{1,\ldots, n\right\}$ of a $\Xi$-coalescent $\Pi=(\Pi_t)_{t\geq0}$ with mutation and mutation rate $r>0$. Let $\Pi^{(n)}$ be the restriction of $\Pi$ to the sample. It is shown that $M_n/n$, the fraction of external branches of $\Pi^{(n)}$ which are affected by at least one mutation, converges almost surely and in $L^p$ ($p\geq 1$) to $M:=\int^{\infty}_0 re^{-rt}S_t dt$, where $S_t$ is the fraction of singleton blocks of $\Pi_t$. Since for coalescents without proper frequencies, the effects of mutations on non-external branches is neglectible for the asymptotics of $K_n/n$, it is shown that $K_n/n\rightarrow M$ for $n\rightarrow\infty$ in $L^p$ $(p\geq 1)$. For simple coalescents, this convergence is shown to hold almost surely. The almost sure results are based on a combination of the Kingman correspondence for random partitions and strong laws of large numbers for weighted i.i.d. or exchangeable random variables.

Motivated by a generalization of Sturm-Lott-Villani theory to discrete spaces and by a conjecture stated by Shepp and Olkin about the entropy of sums of Bernoulli random variables, we prove the concavity in $t$ of the entropy of the convolution of a probability measure $a$, which has the law of a sum of independent Bernoulli variables, by the binomial measure of parameters $n\geq 1$ and $t$.

Ge and Štefankovič have recently introduced a Markov chain which, if rapidly mixing, would provide an efficientprocedure for sampling independent sets in a bipartite graph. Such a procedure would be a breakthrough because it would give an efficient randomised algorithm for approximately counting independent sets in a bipartite graph, which would in turn imply the existence of efficient approximation algorithms for a number of significant counting problems whose computational complexity is so far unresolved. Their Markov chain is based on a novel two-variable graph polynomial which, when specialised to a bipartite graph, and evaluated at the point (1/2,1), givesthe number of independent sets in the graph. The Markov chain is promising, in the sense that it overcomes the most obvious barrier to rapid mixing. However, we show here, by exhibiting a sequence of counterexamples, that its mixing timeis exponential in the size of the input when the input is chosen from a particular infinite family of bipartite graphs.

Consider the random walk $S_n=\xi_1+\cdots+\xi_n$ with independent and identically distributed increments and negative mean $\mathbf E\xi=-m<0$. Let $M=\sup_{0\le i} S_i$ be the supremum of the random walk. In this note we present derivation of asymptotics for $\mathbf P(M>x), x\to\infty$ for long-tailed distributions. This derivation is based on the martingale arguments and does not require any prior knowledge of the theory of long-tailed distributions. In addition the same approach allows to obtain asymptotics for $\mathbf P(M_\tau>x)$, where $M_\tau=\max_{0\le i<\tau}S_i$ and $\tau=\min\{n\ge 1: S_n\le 0 \}$.

In this paper we consider an equilibrium last-passage percolation model on an environment given by a compound two-dimensional Poisson process. We prove an $\mathbb{L}^2$-formula relating the initial measure with the last-passage percolation time. This formula turns out to be a useful tool to analyze the fluctuations of the last-passage times along non-characteristic directions.

We give a new representation of fractional Brownian motion with Hurst parameter H<1/2 using stochastic partial differential equations. This representation allows us to use the Markov property and time reversal, tools which are not usually available for fractional Brownian motion. We then give simple proofs that fractional Brownian motion does not hit points in the critical dimension, and that it does not have double points in the critical dimension. These facts were already known, but our proofs are quite simple and use some ideas of Lévy.

This is an Erratum for ECP volume 14 paper number 6

We consider continuous-time random interlacements on a transient weighted graph. We prove an identity in law relating the field of occupation times of random interlacements at level u to the Gaussian free field on the weighted graph. This identity is closely linked to the generalized second Ray-Knight theorem, and uniquely determines the law of occupation times of random interlacements at level u.<br /><br />

We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $\mathbb{Z}^d$ in the spirit of Donsker-Varadhan [DV75-83]. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomized negative Laplace operator in the domain.

We derive a formula for the moments and the free cumulants of the multiplication of $k$ free random variables in terms of $k$-equal and $k$-divisible non-crossing partitions. This leads to a new simple proof for the bounds of the right-edge of the support of the free multiplicative convolution $\mu^{\boxtimes k}$, given by Kargin, which show that the support grows at most linearly with $k$. Moreover, this combinatorial approach generalize the results of Kargin since we do not require the convolved measures to be identical. We also give further applications, such as a new proof of the limit theorem of Sakuma and Yoshida.

Using the linear programming approach to stochastic control introduced by Buckdahn, Goreac, and Quincampoix, and by Goreac and Serea, we provide a semigroup property for some set of probability measures leading to dynamic programming principles for stochastic control problems. An abstract principle is provided for general bounded costs. Linearized versions are obtained under further (semi)continuity assumptions.

We study the Laha-Lukacs property of the free Meixner laws (processes). We prove that some families of free Meixner distribution have the linear regression function. We also show that this families have the property of quadratic conditional variances.

This work provides exponential tail inequalities for sums of random matrices that depend only on intrinsic dimensions rather than explicit matrix dimensions. These tail inequalities are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the explicit dimensions are large or infinite. Some applications to covariance estimation and approximate matrix multiplication are given to illustrate the utility of the new bounds.

Let $(B_t \colon t \ge 0)$ be a planar Brownian motion and define gauge functions $\phi_\alpha(s)=\log(1/s)^{-\alpha}$ for $\alpha>0$. If $\alpha<1$ we show that almost surely there exists a point $x$ in the plane such that ${\mathcal H}^{\phi_\alpha}(\{t \ge 0 \colon B_t=x\})>0$,but if $\alpha>1$ almost surely ${\mathcal H}^{\phi_\alpha} (\{t \ge 0 \colon B_t=x\})=0$ simultaneously for all $x\in{\mathbb R}^2$. This resolves a longstanding open problem posed by S.J. Taylor in 1986.

We give a criterion for functionals of partitions to converge to a universal limit under a class of measures that "behaves like" the Ewens measure. Various limit theorems for the Ewens measure, most notably the Poisson-Dirichlet limit for the longest parts, the functional central limit theorem for the number of parts, and the Erdos-Turan limit for the product of parts, extend to these asymptotically Ewens measures as easy corollaries. Our major contributions are: (1) extending the classes of measures for which these limit theorems hold; (2) characterising universality by an intuitive and easily-checked criterion; and (3) providing a new and much shorter proof of the limit theorems by taking advantage of the Feller coupling.

We study the free energy of a particle in (arbitrary) high-dimensional Gaussian random potentials with isotropic increments. We prove a computable saddle point variational representation in terms of a Parisi-type functional for the free energy in the infinite-dimensional limit. The proofs are based on the techniques developed in the course of the rigorous analysis of the Sherrington-Kirkpatrick model with vector spins.

This is an erratum for "Maximal arithemtic progressions in random subsets" appeared in ECP volume 12 paper number 35 (2007), 365-376

We investigate a Coulomb gas in a potential satisfying a weaker growth assumption than usual and establish a large deviation principle for its empirical measure. As a consequence the empirical measure is seen to converge towards a non-random limiting measure, characterized by a variational principle from logarithmic potential theory, which may not have compact support. The proof of the large deviation upper bound is based on a compactification procedure which may be of help for further large deviation principles.

We show that the number of cycles in a random permutation chosen according to generalized Ewens measure is normally distributed and compute asymptotic estimates for the mean and variance.

Upper bounds on characteristic functions are derived in terms of the entropic distance to the class of normal distributions.

We prove that Vertex Reinforced Random Walk on $\mathbb{Z}$ with weight of order $k^\alpha$, with $\alpha\in [0,1/2)$, is either almost surely recurrent or almost surely transient. This improves a previous result of Volkov who showed that the set of sites which are visited infinitely often was a.s. either empty or infinite.

With the help of the Gauss-Laplace transform for the exit time from a cone of planar Brownian motion, we obtain some infinite divisibility properties for the reciprocal of this exit time.

We propose a method to construct the stochastic integral simultaneously under a non-dominated family of probability measures. Path-by-path, and without referring to a probability measure, we construct a sequence of Lebesgue-Stieltjes integrals whose medial limit coincides with the usual stochastic integral under essentially any probability measure such that the integrator is a semimartingale. This method applies to any predictable integrand.

We consider the following statistical problem: based on an i.i.d. sample of size $n$ of integer valued random variables with common law $\mu$, is it possible to test whether or not the support of $\mu$ is finite as $n$ goes to infinity? This question is in particular connected to a simple case of Tsirelson's equation, for which it is natural to distinguish between two main configurations, the first one leading only to laws with finite support, and the second one including laws with infinite support. We show that it is in fact not possible to discriminate between the two situations, even using a very weak notion of statistical test.

Symmetry properties of the random grid spanned on the set of records from the planar Poisson point process imply bizarre multivariate distributional identities for certain rational functions of independent exponential and uniform random variables.

We consider a version of a classical concentration inequality for sums of independent, isotropic random vectors with a mild restriction on the distribution of the radial part of these vectors. The proof uses a little Fourier analysis, the Laplace asymptotic method and a conditioning idea that traces its roots to some of the original works on concentration inequalities.

We consider a class of real random matrices with dependent entries and show that the limiting empirical spectral distribution is given by the Marchenko-Pastur law. Additionally, we establish a rate of convergence of the expected empirical spectral distribution.

We prove weak convergence of sums over triangular arrays to the compound Poisson limit using Tikhomirov's method. The result is applied to statistical estimation of the threshold parameter in autoregressive models.

We consider the Cheeger constant $\phi(n)$ of the giant component of supercritical bond percolation on $\mathbb{Z}^d/n\mathbb{Z}^d$. We show that the variance of $\phi(n)$ is bounded by $\frac{\xi}{n^d}$, where $\xi$ is a positive constant that depends only on the dimension $d$ and the percolation parameter.

In this paper we derive the exact tail asymptotic behaviour of $S_\infty=\sum_{i=1}^\infty \lambda_i X_iY_i$, where $\lambda_i, i\ge 1,$ are non-negative square summable deflators (weights) and $X_i,Y_i, i\ge1,$ are independent standard Gaussian random variables. Further, we consider the tail asymptotics of $S_{\infty;p}=\sum_{i=1}^\infty\lambda_i X_i|Y_i|^p, p> 1$, and also discuss the influence on the asymptotic results when $\lambda_i$'s are independent random variables.

This paper studies a particle model connected to a Pieri-type formula for the symplectic compact group. Particles evolve independently apart from a blocking and pushing interaction such that they remain in the set of symplectic Gelfand-Tsetlin patterns. We show that if only one row of the patterns is considered, one obtains a symplectic Schur process. The model leads in particular to the particle model with a wall defined by Jon Warren and Peter Windridge and to a matrix model previously studied by the author.

We establish the existence and uniqueness ofstrong solutions to some jump-type stochastic equations under non-Lipschitz conditions. The results improve those of Fu and Li (2010) and Li and Mytnik (2011).

For the fundamental solutions of heat-type equations of order $n$ we give a general stochastic representation in terms of damped oscillations with generalized gamma distributed parameters. By composing the pseudo-process $X_m$ related to the higher-order heat-type equation with positively skewed stable r.v.'s $T^j_{1/3}$, $j=1,2, ..., n$ we obtain genuine r.v.'s whose explicit distribution is given for $n=3$ in terms of Cauchy asymmetric laws. We also prove that $X_3(T^1_{1/3}(...(T^n_{(1/3)}(t))...))$ has a stable asymmetric law.

We describe scaling limits of recurrent excited random walks (ERWs) on $\mathbb{Z}$ in i.i.d. cookie environments with a bounded number of cookies per site. We allow both positive and negative excitations. It is known that ERW is recurrent if and only if the expected total drift per site, $\delta$, belongs to the interval $[-1,1]$. We show that if $|\delta|<1$ then the diffusively scaled ERW under the averaged measure converges to a $(\delta,-\delta)$-perturbed Brownian motion. In the boundary case, $|\delta|=1$, the space scaling has to be adjusted by an extra logarithmic term, and the weak limit of ERW happens to be a constant multiple of the running maximum of the standard Brownian motion, a transient process.

Let L be the class of limiting laws associated with sequences in the second Wiener chaos. We exhibit a large subset $L_0$ of $L$ satisfying that, for any $F_\infty$ in $L_0$, the convergence of only a finite number of cumulants suffices to imply the convergence in law of any sequence in the second Wiener chaos to $F_\infty$. This result is in the spirit of the seminal paper by Nualart and Peccati, in which the authors discovered the surprising fact that convergence in law for sequences of multiple Wiener-Itô integrals to the Gaussian is equivalent to convergence of just the fourth cumulant. Also, we offer analogues of this result in the case of free Brownian motion and double Wigner integrals, in the context of free probability.

An Erratum is available in ECP volume 17 paper number 54.

This paper proves that if points $Z_1,Z_2,...$ are chosen independently and identically using some measure $\mu$ from the unit circle in the complex plane, with $p_n(z) = (z-Z_1)(z-Z_2)...(z-Z_n)$, then the empirical distribution of the critical points of $p_n$ converges weakly to $\mu$.

Let $X_{1},X_{2},...$ denote independent, identically distributed random variables with common distribution $F$, and $S$ the corresponding random walk with $\rho :=\lim_{n\rightarrow \infty }P(S_{n}>0)$ and $\tau :=\inf \{n\geq 1:S_{n}\leq 0\}$. We assume that $X$ is in the domain of attraction of an $\alpha $-stable law, and that $P(X\in \lbrack x,x+\Delta ))$ is regularly varying at infinity, for fixed $\Delta >0$. Under these conditions, we find an estimate for $P(S_{n}\in \lbrack x,x+\Delta )|\tau >n)$, which holds uniformly as $x/c_{n}\rightarrow \infty $, for a specified norming sequence $c_{n}$. This result is of particular interest as it is related to the bivariate ladder height process $((T_{n},H_{n}),n\geq 0)$, where $T_{r}$ is the $r$th strict increasing ladder time, and $H_{r}=S_{T_{r}}$ the corresponding ladder height. The bivariate renewal mass function $g(n,dx)=\sum_{r=0}^{\infty }P(T_{r}=n,H_{r}\in dx)$ can then be written as $g(n,dx)=P(S_{n}\in dx|\tau >n)P(\tau >n)$, and since the behaviour of $P(\tau >n)$ is known for asymptotically stable random walks, our results can be rephrased as large deviation estimates of $g(n,[x,x+\Delta))$.

The purpose of this article is a set-indexed extension of the well known Ornstein-Uhlenbeck process. The first part is devoted to a stationary definition of the random field and ends up with the proof of a complete characterization by its $L^2$-continuity, stationarity and set-indexed Markov properties. This specific Markov transition system allows to define a general set-indexed Ornstein Uhlenbeck (SIOU) process with any initial probability measure. Finally, in the multiparameter case, the SIOU process is proved to admit a natural integral representation.

We prove that the multi-time particle distributions for annihilating or coalescing Brownian motions, under the maximal entrance law on the real line, are extended Pfaffian point processes. <strong>There is an erratum in</strong> <a href="http://dx.doi.org/10.1214/ECP.v20-4302"><strong>ECP volume 20 paper 46</strong></a>.<br /><br />

The aim of this note is to extend the result of Angel and Holroyd concerning the transience and the recurrence of transfinite rotor-router walks, for random initial configuration of rotors on homogeneous trees. We address the same question on directed covers of finite graphs, which are also called trees with finitely many cone types or periodic trees. Furthermore, we provide an example of a directed cover such that the rotor-router walk can be either recurrent or transient, depending only on the planar embedding of the periodic tree. <strong><a href="http://dx.doi.org/10.1214/ECP.v19-3848">An Erratum is available in ECP volume 19, paper 71, (2014)</a>.</strong>

It is shown that if a planar graph admits no non-constant bounded harmonic function then the trajectories of two independent simple random walks intersect almost surely.

In this paper, we prove that, in dimension one, the Poincaré inequality is equivalent to a new transport-chi-square inequality linking the square of the quadratic Wasserstein distance with the chi-square pseudo-distance. We also check tensorization of this transport-chi-square inequality.<br /><br />

We consider a finite range symmetric exclusion process on the integer lattice in any dimension. We interpret it as a non-elliptic time-dependent random conductance model by setting conductances equal to one over the edges with end points occupied by particles of the exclusion process and to zero elsewhere. We prove a law of large number and a central limit theorem for the random walk driven by such a dynamical field of conductances using the Kipnis-Varhadan martingale approximation. Unlike the tagged particle in the exclusion process, which is in some sense similar to this model, this random walk is diffusive even in the one-dimensional nearest-neighbor symmetric case.

Large $H$-selfadjoint random matrices are considered. The matrix $H$ is assumed to have one negative eigenvalue, hence the matrix in question has precisely one eigenvalue of nonpositive type. It is showed that this eigenvalue converges in probability to a deterministic limit. The weak limit of distribution of the real eigenvalues is investigated as well.

;We consider a problem of predicting of the ultimate maximum of the process over a finite interval of time. Mathematically, this problem relates to a particular optimal stopping problem. In this paper we discuss exponential Lévy processes. As the Lévy processes, we discuss $\alpha$-stable Lévy processes, $0<\alpha\leq 2$, and generalized hyperbolic Lévy processes. The method of solution uses the representations of these processes as time-changed Brownian motions with drift. Our results generalize results of papers by Toit and Peskir and by Shiryaev and Xu, and Zhou.

We obtain non asymptotic concentration bounds for two kinds of stochastic approximations. We first consider the deviations between the expectation of a given function of an Euler like discretization scheme of some diffusion process at a fixed deterministic time and its empirical mean obtained by the Monte Carlo procedure. We then give some estimates concerning the deviation between the value at a given time-step of a stochastic approximation algorithm and its target. Under suitable assumptions both concentration bounds turn out to be Gaussian. The key tool consists in exploiting accurately the concentration properties of the increments of the schemes. Also, no specific non-degeneracy conditions are assumed.

An Erratum is available in ECP volume 17 paper number 60.

It is well known that upward conditioned Brownian motion is a three-dimensional Bessel process, and that a downward conditioned Bessel process is a Brownian motion. We give a simple proof for this result, which generalizes to any continuous local martingale and clarifies the role of finite versus infinite time in this setting. As a consequence, we can describe the law of regular diffusions that are conditioned upward or downward.

We study the pathwise description of a (sub-)critical continuous-state branching process (CSBP) conditioned to be never extinct, as the solution to a stochastic differential equation driven by Brownian motion and Poisson point measures. The interest of our approach, which relies on applying Girsanov theorem on the SDE that describes the unconditioned CSBP, is that it points out an explicit mechanism to build the immigration term appearing in the conditioned process, by randomly selecting jumps of the original one. These techniques should also be useful to represent more general $h$-transforms of diffusion-jump processes.

We provide a complete characterisation of the class of one-dimensional time-homogeneous diffusions consistent with a given law at an exponentially distributed time using classical results in diffusion theory. To illustrate we characterise the class of diffusions with the same distribution as Brownian motion at an exponentially distributed time.

This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. When the sampling distribution belongs to a maximum domain of attraction, these bounds are checked to be asymptotically tight. When the sampling distribution has a non decreasing hazard rate, we derive an exponential Efron-Stein inequality for order statistics, that is an inequality connecting the logarithmic moment generating function of order statistics with exponential moments of Efron-Stein (jackknife) estimates of variance. This connection is used to derive variance and tail bounds for order statistics of Gaussian samples that are not within the scope of the Gaussian concentration inequality. Proofs are elementary and combine Rényi's representation of order statistics with the entropy approach to concentration of measure popularized by M. Ledoux.

This article proves an exponential probability tail inequality for positive semidefinite quadratic forms in a subgaussian random vector. The bound is analogous to one that holds when the vector has independent Gaussian entries.

We give sharp estimate for the free energy of directed polymers in random environment in dimension 1+1. This estimate was known for a Gaussian environment, we extend it to the case where the law of the environment is infinitely divisible.

This is an erratum for paper "<strong>Convergence in law in the second Wiener/Wigner chaos"</strong> appeared in <strong><a href="http://dx.doi.org/10.1214/ECP.v17-2023">ECP volume 17 paper number 36</a></strong>.

Moments formulae for Wiener integrals of a subordinator with exponential moments are obtained in terms of the general Bell polynomials and the moments of the Lévy measure of this subordinator. We also express the Appell and Scheffer polynomials associated to a random variable in terms of the Bell polynomials.

We provide quantitative bounds for the long time behavior of a class of Piecewise Deterministic Markov Processes with state space $\mathbb{R}^d\times E$ where $E$ is a finite set. The continous component evolves according to a smooth vector field that it switched at the jump times of the discrete coordinate. The jump rates may depend on the whole position of the process. Under regularity assumptions on the jump rates and stability conditions for the vector fields we provide explicit exponential upper bounds for the convergence to equilibrium in terms of Wasserstein distances.

In this note we show that percolation on non-amenable Cayley graphs of high girth has a phase of non-uniqueness, i.e., $p_c< p_u$. Furthermore, we show that percolation and self-avoiding walk on such graphs have mean-field critical exponents. In particular, the self-avoiding walk has positive speed.

We consider (near-)critical percolation on the square lattice. Let $\mathcal{M}_{n}$ be the size of the largest open cluster contained in the box $[-n,n]^2$, and let $\pi(n)$ be the probability that there is an open path from $O$ to the boundary of the box. It is well-known that for all $0< a < b$ the probability that $\mathcal{M}_{n}$ is smaller than $a n^2 \pi(n)$ and the probability that $\mathcal{M}_{n}$ is larger than $b n^2 \pi(n)$ are bounded away from $0$ as $n \rightarrow \infty$. It is a natural question, which arises for instance in the study of so-called frozen-percolation processes, if a similar result holds for the probability that $\mathcal{M}_{n}$ is {\em between} $a n^2 \pi(n)$ and $b n^2 \pi(n)$. By a suitable partition of the box, and a careful construction involving the building blocks, we show that the answer to this question is affirmative. The `sublinearity' of $1/\pi(n)$ appears to be essential for the argument.<br />

Let $(X_{i}, \mathcal{F}_{i})_{i\geq 1}$ be a sequence of supermartingale differences and let $S_k=\sum_{i=1}^k X_i$. We give an exponential moment condition under which $\mathbb{P}( \max_{1\leq k \leq n} S_k \geq n)=O(\exp\{-C_1 n^{\alpha}\}),$ $n\rightarrow \infty, $ where $\alpha \in $ is given and $C_{1}>0$ is a constant. We also show that the power $\alpha$ is optimal under the given moment condition.

This is an erratum for paper "Concentration bounds for stochastic approximation" appeared in ECP volume 17 paper number 47

In this paper we describe the pathwise behaviour of the integral functional $\int_0^t f(Y_u)\,du$ for any $t\in[0,\zeta]$, where $\zeta$ is (a possibly infinite) exit time of a one-dimensional diffusion process $Y$ from its state space, $f$ is a nonnegative Borel measurable function and the coefficients of the SDE solved by $Y$ are only required to satisfy weak local integrability conditions. Two proofs of the deterministic characterisation of the convergence of such functionals are given: the problem is reduced in two different ways to certain path properties of Brownian motion where either the Williams theorem and the theory of Bessel processes or the first Ray-Knight theorem can be applied to prove the characterisation. As a simple application of the main results we give a short proof of Feller's test for explosion.<br />

We study the optional projection of a standard Brownian motion on the natural filtration of certain kinds of observation processes. The observation process, $Y$, is defined as a solution of a stochastic differential equation such that it reveals some (possibly noisy) information about the signs of the Brownian motion when $Y$ hits $0$. As such, the associated optional projections are related to Azéma's martingales which are obtained by projecting the Brownian motion onto the filtration generated by observing its signs. <br />

We present a new proof of the extended arc-sine law related to Walsh's Brownian motion, known also as Brownian spider. The main argument mimics the scaling property used previously, in particular by D. Williams, in the 1- dimensional Brownian case, which can be generalized to the multivariate case. A discussion concerning the time spent positive by a skew Bessel process is also presented.