On smoothed analysis in dense graphs and formulas

We study a model of random graphs, where a random instance is obtained by adding random edges to a large graph of a given density. The research on this model has been started by Bohman and colleagues (Random Struct Algor 3 ; Random Struct Algor 4 ). Here we obtain a sharp threshold for the appearance of a fixed subgraph and for certain Ramsey properties. We also consider a related model of random k‐SAT formulas, where an instance is obtained by adding random k‐clauses to a fixed formula with a given number of clauses, and derive tight bounds for the non‐satisfiability of the thus‐obtained random formula. © 2006 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

MOMENT GENERATING FUNCTIONS OF RANDOM VARIABLES AND ASYMPTOTIC BEHAVIOUR FOR GENERALIZED FELLER OPERATORS

1. Introduction and NotationsThe generalized Feller operators which include many famous operators, such ajsBernstein, Szasz-Mirakjan, BaskakoV, Meyer--K5nig and Zeller operators, can be constructed by making use of the probabilistic method. In the paper [1][2], Xu JihuaPr(--lvjdetl a general scheme f(,r its construction, and Zhao Jillghui showed that theFeller type operators are of good approximations f'or unbounded functions.Our purpose is to present representation of moment generating …     

M/M/∞ queues in semi-Markovian random environment

In this paper we investigate an M/M/∞ queue whose parameters depend on an external random environment that we assume to be a semi-Markovian process with finite state space. For this model we show a recursive formula that allows to compute all the factorial moments for the number of customers in the system in steady state. The used technique is based on the calculation of the raw moments of the measure of a bidimensional random set. Finally the case when the random environment has only two states is deeper analyzed. We obtain an explicit formula to compute the above mentioned factorial moments when at least one of the two states has sojourn time exponentially distributed. Part of this research took place while the author was still post-doc at EURANDOM, Eindhoven, The Netherlands. The work was supported by the Spanish Ministry of Education and Science by the Grant MTM2007-63140.     

随机针偶与凸体K相交的几何概率问题

本文研究了随机针偶与凸体K相交的几何概率,利用有向直线偶的运动不变密度公式,获得了针偶的运动不变密度公式,从而进一步得到随机针偶与凸体K相交且针偶的交点属于K的几何概率.     

Random walks and the effective resistance of networks

In this article we present an interpretation ofeffective resistance in electrical networks in terms of random walks on underlying graphs. Using this characterization we provide simple and elegant proofs for some known results in random walks and electrical networks. We also interpret the Reciprocity theorem of electrical networks in terms of traversals in random walks. The byproducts are (a) precise version of thetriangle inequality for effective resistances, and (b) an exact formula for the expectedone-way transit time between vertices.     

Bentley’s conjecture on popularity toplist turnover under random copying

Bentley et al. studied the turnover rate in popularity toplists in a ‘random copying’ model of cultural evolution. Based on simulations of a model with population size N, list length ? and invention rate μ, they conjectured a remarkably simple formula for the turnover rate: $\ell \sqrt{\mu}$ . Here we study an overlapping generations version of the random copying model, which can be interpreted as a random walk on the integer partitions of the population size. In this model we show that the conjectured formula, after a slight correction, holds asymptotically.     

Intersection probabilities and kinematic formulas for polyhedral cones

For polyhedral convex cones in \({\mathbb{R}^d}\), we give a proof for the conic kinematic formula for conic curvature measures, which avoids the use of characterization theorems. For the random cones defined as typical cones of an isotropic random central hyperplane arrangement, we find probabilities for non-trivial intersection, either with a fixed cone, or for two independent random cones of this type.     

An expectation formula with applications

In this paper, we derive an expectation formula of a random variable having distribution W(x;q). As applications of the expectation formula, we give a transformation formula and an expansion of Sears? transformation formula.     

重尾情形下NA随机变量的随机加权和

In 2003, Tang Qihe et al. obtained a simple asymptotic formula for independent identically distributed （i.i.d.） random variables with heavy tails. In this paper, under certain moment conditions, we establish a formula as the same as Tang’s, when random variables are negatively associated （NA）.     

n维有向de Bruijn图和无向de Bruijn图上的随机游动

本文对有向和无向de Bruijn图上的随机游动进行了研究，得出了有向de Bruijn图上简单随机游动任意两点之间平均击中时间的显式表达式，并证明了有向和无向de Bruijn图上随机游动的快速收敛性。     

随机环境中r维分支链的协方差计算

定义了各种条件多元概率母函数，并利用条件多元概率母函数这一强有力工具研究随机环境中r-维分支链的性质，并给出了其协方差阵的精确计算公式．     

Gauge and conditional gauge on negatively curved graphs

In this paper we study certain edge-weighted random walks on infinite graphs with bounded vertex degree for which the second smallest eigenvalue of the Laplacian is negative.We find analogues of the Feynman-Kac functional and give several conditions equivalent to the boundeness of the corresponding gauge function.From this result we derive a new formula for the second smallest eigenvalue of the Laplacian and we apply this formula to the theory of graph coverings.An analogue of the conditional gauge theorem is shown to hold for certain Schrödinger operators.     

Asymptotics of matrix integrals and tensor invariants of compact Lie groups

In this paper we give an asymptotic formula for a matrix integral which plays a crucial role in the approach of Diaconis et al. to random matrix eigenvalues. The choice of parameter for the asymptotic analysis is motivated by an invariant-theoretic interpretation of this type of integral. For arbitrary regular irreducible representations of arbitrary connected semisimple compact Lie groups, we obtain an asymptotic formula for the trace of permutation operators on the space of tensor invariants, thus extending a result of Biane on the dimension of these spaces.

     

Integral representation of renormalized self-intersection local times

In this paper we apply Clark-Ocone formula to deduce an explicit integral representation for the renormalized self-intersection local time of the d-dimensional fractional Brownian motion with Hurst parameter H∈(0,1). As a consequence, we derive the existence of some exponential moments for this random variable.     

On non-Markovian forward–backward SDEs and backward stochastic PDEs

In this paper, we establish an equivalence relationship between the wellposedness of forward–backward SDEs (FBSDEs) with random coefficients and that of backward stochastic PDEs (BSPDEs). Using the notion of the “decoupling random field”, originally observed in the well-known Four Step Scheme (Ma et al., 1994 [13]) and recently elaborated by Ma et al. (2010) [14], we show that, under certain conditions, the FBSDE is wellposed if and only if this random field is a Sobolev solution to a degenerate quasilinear BSPDE, extending the existing non-linear Feynman–Kac formula to the random coefficient case. Some further properties of the BSPDEs, such as comparison theorem and stability, will also be discussed.     

A new approach to stochastic evolution equations with adapted drift

In this paper we develop a new approach to stochastic evolution equations with an unbounded drift A which is dependent on time and the underlying probability space in an adapted way. It is well-known that the semigroup approach to equations with random drift leads to adaptedness problems for the stochastic convolution term. In this paper we give a new representation formula for the stochastic convolution which avoids integration of non-adapted processes. Here we mainly consider the parabolic setting. We establish connections with other solution concepts such as weak solutions. The usual parabolic regularity properties are derived and we show that the new approach can be applied in the study of semilinear problems with random drift. At the end of the paper the results are illustrated with two examples of stochastic heat equations with random drift.     

Entropy formula of Pesin type for noninvertible random dynamical systems

In this paper we consider random dynamical systems (abbreviated henceforth as RDS's) generated by compositions of random endomorphisms (maybe noninvertible and with singularities) of class of a compact manifold. Entropy formula of Pesin type is proved for such RDS's under some absolute continuity conditions on the associated invariant measures. Received October 17, 1997; in final form January 5, 1998     

Quantization dimension of random self-similar measures

In this paper, we study the quantization dimension of a random self-similar measure μ supported on the random self-similar set K(ω). We establish a relationship between the quantization dimension of μ and its distribution. At last we give a simple example to show that how to use the formula of the quantization dimension.     

A Generalized Itô-Ventzell Formula to Derive Forward Utility Models in a Jump Market

Our main topic in this article is the forward utility field, which is a quite a new concept introduced by Musiela and Zariphopoulou. Different from most article in this field discussing forward utility in a continuous market, we extend this concept to jump market case. We first provide a generalized Itô-Ventzell formula, which can be applied in a general jump semimartingale driven by Brownian motion and Poisson random measure. Then three special forward utility models are discussed by exploiting this generalized Itô-Ventzell formula.