Given a random word of size $n$ whose letters are drawn independently from an ordered alphabet of size $m$, the fluctuations of the shape of the random RSK Young tableaux are investigated, when $n$ and $m$ converge together to infinity. If $m$ does not grow too fast and if the draws are uniform, then the limiting shape is the same as the limiting spectrum of the GUE. In the non-uniform case, a control of both highest probabilities will ensure the convergence of the first row of the tableau toward the Tracy--Widom distribution.
In this paper, a random graph process ${G(t)}_{tgeq 1}$ is studied and its degree sequence is analyzed. Let $(W_t)_{tgeq 1}$ be an i.i.d. sequence. The graph process is defined so that, at each integer time $t$, a new vertex, with $W_t$ edges attached to it, is added to the graph. The new edges added at time t are then preferentially connected to older vertices, i.e., conditionally on $G(t-1)$, the probability that a given edge is connected to vertex i is proportional to $d_i(t-1)+delta$, where $d_i(t-1)$ is the degree of vertex $i$ at time $t-1$, independently of the other edges. The main result is that the asymptotical degree sequence for this process is a power law with exponent $tau=min{tau_{W}, tau_{P}}$, where $tau_{W}$ is the power-law exponent of the initial degrees $(W_t)_{tgeq 1}$ and $tau_{P}$ the exponent predicted by pure preferential attachment. This result extends previous work by Cooper and Frieze, which is surveyed.
The first aim of the present paper, is to establish strong approximations of the uniform non-overlapping k-spacings process extending the results of Aly et al. (1984). Our methods rely on the invariance principle in Mason and van Zwet (1987). The second goal, is to generalize the Dindar (1997) results for the increments of the spacings quantile process to the uniforme non-overlapping k-spacings quantile process. We apply the last result to characterize the limit laws of functionals of the increments k-spacings quantile process.
In this paper we consider the nonparametric functional estimation of the drift of Gaussian processes using Paley-Wiener and Karhunen-Lo`eve expansions. We construct efficient estimators for the drift of such processes, and prove their minimaxity using Bayes estimators. We also construct superefficient estimators of Stein type for such drifts using the Malliavin integration by parts formula and stochastic analysis on Gaussian space, in which superharmonic functionals of the process paths play a particular role. Our results are illustrated by numerical simulations and extend the construction of James-Stein type estimators for Gaussian processes by Berger and Wolper.
These notes survey some aspects of discrete-time chaotic calculus and its applications, based on the chaos representation property for i.i.d. sequences of random variables. The topics covered include the Clark formula and predictable representation, anticipating calculus, covariance identities and functional inequalities (such as deviation and logarithmic Sobolev inequalities), and an application to option hedging in discrete time.
We propose and study a general method for construction of consistent statistical tests on the basis of possibly indirect, corrupted, or partially available observations. The class of tests devised in the paper contains Neymans smooth tests, data-driven score tests, and some types of multi-sample tests as basic examples. Our tests are data-driven and are additionally incorporated with model selection rules. The method allows to use a wide class of model selection rules that are based on the penalization idea. In particular, many of the optimal penalties, derived in statistical literature, can be used in our tests. We establish the behavior of model selection rules and data-driven tests under both the null hypothesis and the alternative hypothesis, derive an explicit detectability rule for alternative hypotheses, and prove a master consistency theorem for the tests from the class. The paper shows that the tests are applicable to a wide range of problems, including hypothesis testing in statistical inverse problems, multi-sample problems, and nonparametric hypothesis testing.
We provide a new characterization of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure $P^{star}$ which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to $P^{star}$ coincides with the variance-optimal martingale measure relative to the original probability measure $P$.
Let $a$ be a complex random variable with mean zero and bounded variance. Let $N_{n}$ be the random matrix of size $n$ whose entries are iid copies of $a$ and $M$ be a fixed matrix of the same size. The goal of this paper is to give a general estimate for the condition number and least singular value of the matrix $M + N_{n}$, generalizing an earlier result of Spielman and Teng for the case when $a$ is gaussian. Our investigation reveals an interesting fact that the core matrix $M$ does play a role on tail bounds for the least singular value of $M+N_{n} $. This does not occur in Spielman-Teng studies when $a$ is gaussian. Consequently, our general estimate involves the norm $|M|$. In the special case when $|M|$ is relatively small, this estimate is nearly optimal and extends or refines existing results.
In this note we investigate the last passage percolation model in the presence of macroscopic inhomogeneity. We analyze how this affects the scaling limit of the passage time, leading to a variational problem that provides an ODE for the deterministic limiting shape of the maximal path. We obtain a sufficient analytical condition for uniqueness of the solution for the variational problem. Consequences for the totally asymmetric simple exclusion process are discussed.
We derive properties of the rate function in Varadhans (annealed) large deviation principle for multidimensional, ballistic random walk in random environment, in a certain neighborhood of the zero set of the rate function. Our approach relates the LDP to that of regeneration times and distances. The analysis of the latter is possible due to the i.i.d. structure of regenerations.
