Given a set-valued stochastic process $(V_t)_{t=0}^T$, we say that the martingale selection problem is solvable if there exists an adapted sequence of selectors $\xi_t\in V_t$, admitting an equivalent martingale measure. The aim of this note is to underline the connection between this problem and the problems of asset pricing in general discrete-time market models with portfolio constraints and transaction costs. For the case of relatively open convex sets $V_t(\omega)$ we present effective necessary and sufficient conditions for the solvability of a suitably generalized martingale selection problem. We show that this result allows to obtain computationally feasible formulas for the price bounds of contingent claims. For the case of currency markets we also sketch a new proof of the first fundamental theorem of asset pricing.

Bertoin and Le Gall (2003) introduced a certain probability measure valued Markov process that describes the evolution of a population, such that a sample from this population would exhibit a genealogy given by the so-called $\Lambda$-coalescent, or coalescent with multiple collisions, introduced independently by Pitman (1999) and Sagitov (1999). We show how this process can be extended to the case where lineages can experience mutations. Regenerative compositions enter naturally into this model, which is somewhat surprising, considering a negative result by Möhle (2007).

Suppose $s_n$ is the spectral norm of either the Toeplitz or the Hankel matrix whose entries come from an i.i.d. sequence of random variables with positive mean $\mu$ and finite fourth moment. We show that $n^{-1/2}(s_n-n\mu)$ converges to the normal distribution in either case. This behaviour is in contrast to the known result for the Wigner matrices where $s_n-n\mu$ is itself asymptotically normal.

We present a short probabilistic proof of a weak convergence result for the number of cuts needed to isolate the root of a random recursive tree. The proof is based on a coupling related to a certain random walk.

A Central Limit Theorem for non-commutative random variables is proved using the Lindeberg method. The theorem is a generalization of the Central Limit Theorem for free random variables proved by Voiculescu. The Central Limit Theorem in this paper relies on an assumption which is weaker than freeness.

The central limit theorem for the product of sums of various random variables has been studied in a variety of settings. The purpose of this note is to show that this kind of result is a corollary of the invariance principle.

We show that any element from the ($L^2$-)maximal domain of a jump-type symmetric Dirichlet form can be approximated by test functions under some conditions. This gives us a direct proof of the fact that the test functions is dense in Bessel potential spaces.

We construct a Riemannian manifold where the Kendall-Cranston coupling of two Brownian particle does not maximize the coupling probability.

We show how to recover Euler's formula for $\zeta(2n)$, as well as $L_{\chi_4}(2n+1)$, for any integer $n$, from the knowledge of the density of the product $\mathbb{C}_1,\mathbb{C}_2\ldots,\mathbb{C}_k$, for any $k\geq 1$, where the $\mathbb{C}_i$'s are independent standard Cauchy variables.

We study a branching Brownian motion (BBM) with absorption, in which particles move as Brownian motions with drift $-\rho$, undergo dyadic branching at rate $\beta>0$, and are killed on hitting the origin. In the case $\rho>\sqrt{2\beta}$ the extinction time for this process, $\zeta$, is known to be finite almost surely. The main result of this article is a large-time asymptotic formula for the survival probability $P^x(\zeta>t)$ in the case $\rho>\sqrt{2\beta}$, where $P^x$ is the law of the BBM with absorption started from a single particle at the position $x>0$. We also introduce an additive martingale, $V$, for the BBM with absorption, and then ascertain the convergence properties of $V$. Finally, we use $V$ in a `spine' change of measure and interpret this in terms of `conditioning the BBM to survive forever' when $\rho>\sqrt{2\beta}$, in the sense that it is the large $t$-limit of the conditional probabilities $P^x(A\mid \zeta > t+s)$, for $A\in F_s$.

Consider a sequence of multinomial random vectors with increasing number of equiprobable cells. We show that if the number of trials increases fast enough, the sequence of maxima of the cells after a suitable centering and scaling converges to the Gumbel distribution. While results are available for maxima of triangular arrays of independent random variables with certain types of distribution, such results in a dependent setup is new. We also prove that the maxima of a triangular sequence of appropriate Binomial random variables have the same limit distribution. An auxiliary large deviation result for multinomial distribution with increasing number of equiprobable cells may also be of independent interest.

Let $Z(t,z)$ be a $\mathbb{R}^d$-valued controlled jump diffusion starting from the point $z$ at time $t$. The aim of this paper is to characterize the set $V(t)$ of initial conditions $z$ such that $Z(t,z)$ can be driven into a given target at a given time. We do this by proving that the characteristic function of the complement $V(t)$ satisfies some partial differential equation in the viscosity sense. As an application, we study the problem of hedging in a financial market with a large investor.

We consider a $\psi$-irreducible, discrete-time Markov chain on a general state space with transition kernel $P$. Under suitable conditions on the chain, kernels can be treated as bounded linear operators between spaces of functions or measures and the Drazin inverse of the kernel operator $I - P$ exists. The Drazin inverse provides a unifying framework for objects governing the chain. This framework is applied to derive a computational technique for the asymptotic variance in the central limit theorems of univariate and higher-order partial sums. Higher-order partial sums are treated as univariate sums on a `sliding-window' chain. Our results are demonstrated on a simple AR(1) model and suggest a potential for computational simplification.

By means of a simple conditioning/comparison argument, we derive the chance of a long lifetime for Brownian motion in a horn-shaped domain.

This note answers one question in [1] concerning the connected allocation for the Poisson process in $\mathbb{R}^2$. The proposed solution makes use of the Riemann map from the plane minus the minimal spanning forest of the Poisson point process to the halfplane. A picture of a numerically simulated example is included.

The Laplace transform of the first exit time from a finite interval by a regular spectrally negative $\alpha$-stable Ornstein-Uhlenbeck process is provided in terms of the Wright's generalized hypergeometric function. The Laplace transform of first passage times is also derived for some related processes such as the process killed when it enters the negative half line and the process conditioned to stay positive. The law of the maximum of the associated bridges is computed in terms of the $q$-resolvent density. As a byproduct, we deduce some interesting analytical properties for some Wright's generalized hypergeometric functions.

In this paper we study three self-similar, long-range dependence, Gaussian processes. The first one, with covariance $$ \int^{s\wedge t}_0 u^a [(t-u)^b+(s-u)^b]du, $$ parameters $a>-1$, $-1 < b\leq 1$, $|b|\leq 1+a$, corresponds to fractional Brownian motion for $a=0$, $-1 < b < 1$. The second one, with covariance $$ (2-h)\biggl(s^h+t^h-\frac{1}{2}[(s+t)^h +|s-t|^h]\biggr), $$ parameter $0 < h\leq 4$, corresponds to sub-fractional Brownian motion for $0 < h < 2 $. The third one, with covariance $$ -\left(s^2\log s + t^2\log t -\frac{1}{2}[(s+t)^2 \log (s+t) +(s-t)^2 \log |s-t|]\right), $$ is related to the second one. These processes come from occupation time fluctuations of certain particle systems for some values of the parameters.

In general, if the large deviation principle holds for a sequence of probability measures and its rate functional admits a unique minimizer, then the measures asymptotically concentrate in its neighborhood so that the law of large numbers follows. This paper discusses the situation that the rate functional has two distinct minimizers, for a simple model described by the pinned Wiener measures with certain densities involving a scaling. We study their asymptotic behavior and determine to which minimizers they converge based on a more precise investigation than the large deviation's level.

We investigate the asymptotic behavior of weighted sums of independent standardized random variables with uniformly bounded third moments. The sequence of weights is given by a family of rectangular matrices with uniformly small entries and approximately orthogonal rows. We prove that the empirical CDF of the resulting partial sums converges to the normal CDF with probability one. This result implies almost sure convergence of empirical periodograms, almost sure convergence of spectral distribution of circulant and reverse circulant matrices, and almost sure convergence of the CDF generated from independent random variables by independent random orthogonal matrices. In the special case of trigonometric weights, the speed of the almost sure convergence is described by a normal approximation as well as a large deviation principle.

Consider the matrix $\Delta_n = ((\ \mathrm{I}(X_i + X_j > 0)\ ))_{i,j = 1,2,...,n}$ where $\{X_i\}$ are i.i.d. and their distribution is continuous and symmetric around $0$. We show that the rank $r_n$ of this matrix is equal in distribution to $2\sum_{i=1}^{n-1}\mathrm{I}(\xi_i =1,\xi_{i+1}=0)+\mathrm{I}(\xi_n=1)$ where $\xi_i \stackrel{i.i.d.}{\sim} \text{Ber} $ is asymptotically normal with mean zero and variance $1/4$. We also show that $n^{-1}r_n$ converges to $1/2$ almost surely.

We derive a unified stochastic picture for the duality of a resampling-selection model with a branching-coalescing particle process (cf. ) and for the self-duality of Feller's branching diffusion with logistic growth (cf. ). The two dual processes are approximated by particle processes which are forward and backward processes in a graphical representation. We identify duality relations between the basic building blocks of the particle processes which lead to the two dualities mentioned above.

Let $\{X, X_{{n}}; n\in\mathbb{N}^{d}\}$ be a field of i.i.d. random variables indexed by $d$-tuples of positive integers and let $S_{{n}}=\sum_{{k}\leq{n}}X_{{k}}$. We prove some strong limit theorems for $S_{{n}}$. Also, when $d\geq 2$ and $h({n})$ satisfies some conditions, we show that there are no LIL type results for $S_{{n}}/\sqrt{|{n}|h({n})}$.

The coordinates $x_i$ of a point $x = (x_1, x_2, \dots, x_n)$ chosen at random according to a uniform distribution on the $\ell_2(n)$-sphere of radius $n^{1/2}$ have approximately a normal distribution when $n$ is large. The coordinates $x_i$ of points uniformly distributed on the $\ell_1(n)$-sphere of radius $n$ have approximately a double exponential distribution. In these and all the $\ell_p(n),1 \le p \le \infty,$ convergence of the distribution of coordinates as the dimension $n$ increases is at the rate $\sqrt{n}$ and is described precisely in terms of weak convergence of a normalized empirical process to a limiting Gaussian process, the sum of a Brownian bridge and a simple normal process.

This note extends Voiculescu's $S$-transform based analytical machinery for free multiplicative convolution to the case where the mean of the probability measures vanishes. We show that with the right interpretation of the $S$-transform in the case of vanishing mean, the usual formula makes perfectly good sense.

We derive a class of ergodic transformation of self-similar Gaussian processes that are Volterra, i.e. of type $X_t = \int^t_0 z_X(t,s)dW_s$, $t \in [0,\infty)$, where $z_X$ is a deterministic kernel and $W$ is a standard Brownian motion.

We prove that the expected Lebesgue measure of the range of an additive Levy process is positive if and only if the product $\Re([1+\Psi_1(\xi)]^{-1})...\Re([1+\Psi_N(\xi)]^{-1})$ is integrable. This was previously proved by Khoshnevisan, Xiao and Zhong [1] under a sector condition.

In a 2-parameter scale free model of random graphs it is shown that the asymptotic degree distribution is the same in the neighbourhood of every vertex. This degree distribution is still a power law with characteristic exponent 2, but this exponent is different from the one observed in the whole graph.

We show that a slight modification of a theorem of Ruzmaikina and Aizenman on competing particle systems on the real line leads to a characterization of Poisson-Dirichlet distributions $PD(\alpha,0)$. Precisely, let $\xi$ be a proper random mass-partition i.e. a random sequence $(\xi_i,i\in N)$ such that $\xi_1 \geq \xi_2 \geq \dots \geq 0$ and $\sum_i \xi_i =1$ a.s. Consider $\{W_i\}_{i\in N}$, an iid sequence of random positive numbers whose distribution is absolutely continuous with respect to the Lebesgue measure and $E[W^\lambda]<\infty$ for all $\lambda \in R$. It is shown that, if the law of $\xi$ is invariant under the random reshuffling $$( \xi_i , i \in N) \to \left(\frac{\xi_i W_i}{\sum_j \xi_jW_j } , i \in N \right)$$ where the weights are reordered after evolution, then it must be a mixture of Poisson-Dirichlet distributions $PD(\alpha,0), \alpha\in$.

Let $T$ be a random field weakly invariant under the action of a compact group $G$. We give conditions ensuring that independence of the random Fourier coefficients is equivalent to Gaussianity. As a consequence, in general it is not possible to simulate a non-Gaussian invariant random field through its Fourier expansion using independent coefficients

We prove that a law of large numbers and a central limit theorem hold for the excited random walk model in every dimension $d\geq 2$.

Suppose that $T_n$ is a Toeplitz matrix whose entries come from a sequence of independent but not necessarily identically distributed random variables with mean zero. Under some additional tail conditions, we show that the spectral norm of $T_n$ is of the order $\sqrt{n \log n}$. The same result holds for random Hankel matrices as well as other variants of random Toeplitz matrices which have been studied in the literature.

In this note we present a simplified proof of the zero-one law by Merkl and Zerner (2001) for directional transience of random walks in i.i.d. random environments (RWRE) on the square lattice. Also, we indicate how to construct a two-dimensional counterexample in a non-uniformly elliptic and stationary environment which has better ergodic properties than the example given by Merkl and Zerner.

We study a class of adaptive Markov Chain Monte Carlo (MCMC) processes which aim at behaving as an ``optimal'' target process via a learning procedure. We show, under appropriate conditions, that the adaptive MCMC chain and the ``optimal'' (nonadaptive) MCMC process share many asymptotic properties. The special case of adaptive MCMC algorithms governed by stochastic approximation is considered in details and we apply our results to the adaptive Metropolis algorithm of [Haario, Saksman, Tamminen].

Fix $k\geq 1$, and let $I(l), l \geq 1$, be a sequence of $k$-dimensional vectors of multiple Wiener-Itô integrals with respect to a general Gaussian process. We establish necessary and sufficient conditions to have that, as $l \to\infty$, the law of $I(l)$ is asymptotically close (for example, in the sense of Prokhorov's distance) to the law of a $k$-dimensional Gaussian vector having the same covariance matrix as $I(l)$. The main feature of our results is that they require minimal assumptions (basically, boundedness of variances) on the asymptotic behaviour of the variances and covariances of the elements of $I(l)$. In particular, we will not assume that the covariance matrix of $I(l)$ is convergent. This generalizes the results proved in Nualart and Peccati (2005), Peccati and Tudor (2005) and Nualart and Ortiz-Latorre (2007). As shown in Marinucci and Peccati (2007b), the criteria established in this paper are crucial in the study of the high-frequency behaviour of stationary fields defined on homogeneous spaces.

Let $U(N)$ denote the maximal length of arithmetic progressions in a random uniform subset of $\{0,1\}^N$. By an application of the Chen-Stein method, we show that $U(N)- 2 \log(N)/\log(2)$ converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length $W(N)$ of arithmetic progressions (mod $N$). When considered in the natural way on a common probability space, we observe that $U(N)/\log(N)$ converges almost surely to $2/\log(2)$, while $W(N)/\log(N)$ does not converge almost surely (and in particular, $\limsup W(N)/\log(N)$ is at least $3/\log(2)$).

An Erratum is available in ECP volume 17 paper number 18.

We show how the evolving set methodology of Morris and Peres can be used to show Cheeger inequalities for bounding the spectral gap of a finite Markov kernel. This leads to sharp versions of several previous Cheeger inequalities, including ones involving edge-expansion, vertex-expansion, and mixtures of both. A bound on the smallest eigenvalue also follows.

We study the dynamical properties of the Brownian diffusions having $\sigma\,{\rm Id}$ as diffusion coefficient matrix and $b=\nabla U$ as drift vector. We characterize this class through the equality $D^2_+=D^2_-$, where $D_{+}$ (resp. $D_-$) denotes the forward (resp. backward) stochastic derivative of Nelson's type. Our proof is based on a remarkable identity for $D_+^2-D_-^2$ and on the use of the martingale problem.

Talagrand's positivity principle states that one can slightly perturb a Hamiltonian in the Sherrington-Kirkpatrick model in such a way that the overlap of two configurations under the perturbed Gibbs' measure will become typically nonnegative. In this note we observe that abstracting from the setting of the SK model only improves the result and does not require any modifications in Talagrand's argument. In this version, for example, positivity principle immediately applies to the setting of replica symmetry breaking interpolation. Also, abstracting from the SK model improves the conditions in the Ghirlanda-Guerra identities and as a consequence results in a perturbation of smaller order necessary to ensure positivity of the overlap.

We consider invasion percolation on the square lattice. van den Berg, Peres, Sidoravicius and Vares have proved that the probability that the radius of a so-called pond is larger than $n$, differs at most a factor of order log $n$ from the probability that in critical Bernoulli percolation the radius of an open cluster is larger than $n$. We show that these two probabilities are, in fact, of the same order. Moreover, we prove an analogous result for the volume of a pond.

Let $X$ be a Banach space. Suppose that for all $p\in $ a constant $C_{p,X}$ depending only on $X$ and $p$ exists such that for any two $X$-valued martingales $f$ and $g$ with tangent martingale difference sequences one has $$\mathbb{E}\|f\|^p \leq C_{p,X} \mathbb{E}\|g\|^p \qquad (*).$$ This property is equivalent to the UMD condition. In fact, it is still equivalent to the UMD condition if in addition one demands that either $f$ or $g$ satisfy the so-called (CI) condition. However, for some applications it suffices to assume that $(*)$ holds whenever $g$ satisfies the (CI) condition. We show that the class of Banach spaces for which $(*)$ holds whenever only $g$ satisfies the (CI) condition is more general than the class of UMD spaces, in particular it includes the space $L^1$. We state several problems related to $(*)$ and other decoupling inequalities.

In this paper, we provide a necessary and sufficient condition for general $d$-dimensional arrays of random variables to satisfy strong law of large numbers. Then, we apply the result to obtain some strong laws of large numbers for $d$-dimensional arrays of blockwise independent and blockwise orthogonal random variables.

A one dimensional stochastic differential equation of the form \[dX=A X dt+\tfrac12 (-A)^{-\alpha}\partial_\xi[((-A)^{-\alpha}X)^2]dt+\partial_\xi dW(t),\qquad X(0)=x\] is considered, where $A=\tfrac12 \partial^2_\xi$. The equation is equipped with periodic boundary conditions. When $\alpha=0$ this equation arises in the Kardar-Parisi-Zhang model. For $\alpha\ne 0$, this equation conserves two important properties of the Kardar-Parisi-Zhang model: it contains a quadratic nonlinear term and has an explicit invariant measure which is gaussian. However, it is not as singular and using renormalization and a fixed point result we prove existence and uniqueness of a strong solution provided $\alpha>\frac18$.

Central limit theorems for Markov chains are considered, and in particular the relationships between various expressions for asymptotic variance known from the literature. These turn out to be equal under fairly general conditions, although not always. We also investigate the existence of CLTs, and pose some open problems.

We are going to study the limiting spectral measure of fixed dimensional Hermitian block-matrices with large dimensional Wigner blocks. We are going also to identify the limiting spectral measure when the Hermitian block-structure is Circulant. Using the limiting spectral measure of a Hermitian Circulant block-matrix we will show that the spectral measure of a Wigner matrix with $k$-weakly dependent entries need not to be the semicircle law in the limit.