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The Skorokhod embedding problem and its offspring
Skorokhod embedding problem Chacon-Walsh embedding Azéma-Yor embedding Root embedding stopping times
2009/5/18
This is a survey about the Skorokhod embedding problem. It presents all known solutions together with their properties and some applications. Some of the solutions are just described, while others are...
On the Markov chain central limit theorem
Central Limit Theorem Markov Chain Monte Carlo Mixing Condition Drift Condition
2009/5/18
The goal of this expository paper is to describe conditions which guarantee a central limit theorem for functionals of general state space Markov chains. This is done with a view towards Markov chain ...
On the constructions of the skew Brownian motion
skew Brownian motion PDE with singular drift PDE with transmission condition SDE with local time excursions of Brownian motion
2009/5/18
This article summarizes the various ways one may use to construct the Skew Brownian motion, and shows their connections. Recent applications of this process in modelling and numerical simulation motiv...
An essay on the general theory of stochastic processes
General theory of stochastic processes Enlargements of filtrations Random times Submartingales Stopping times Honest times Pseudo-stopping times
2009/5/18
This text is a survey on the general theory of stochastic processes, with a view towards random times and enlargements of filtrations. The first five chapters present standard materials, which where d...
The realization of positive random variables via absolutely continuous transformations of measure on Wiener space
positive random variables continuous transformations Wiener space
2009/5/18
Let $mu$ be a Gaussian measure on some measurable space ${W = {w}, calB (W)}$ and let $nu$ be a measure on the same space which is absolutely continuous with respect to $nu$. The paper surveys results...
Recent advances in invariance principles for stationary sequences
Brownian motion weakly dependent sequences martingale differences linear process central limit theorem invariance principle
2009/5/18
In this paper we survey some recent results on the central limit theorem and its weak invariance principle for stationary sequences. We also describe several maximal inequalities that are the main too...
Proofs of the martingale FCLT
functional central limit theorems martingales diffusion approximations invariance principles
2009/5/18
This is an expository review paper elaborating on the proof of the martingale functional central limit theorem (FCLT). This paper also reviews tightness and stochastic boundedness, highlighting one-di...
A survey of random processes with reinforcement
urn model urn scheme Pólya’s urn stochastic approximation dynamical system exchangeability Lyapunov function reinforced random walk ERRW VRRW learning agent-based model evolutionary game theory self-avoiding walk
2009/5/18
The models surveyed include generalized Polya urns, reinforced random walks, interacting urn models, and continuous reinforced processes. Emphasis is on methods and results, with sketches provided of ...
Stochastic analysis of Bernoulli processes
Malliavin calculus Bernoulli processes discrete time chaotic calculus functional inequalities option hedging
2009/5/18
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 ...
This survey treats the problem of ruin in a risk model when assets earn investment income. In addition to a general presentation of the problem, topics covered are a presentation of the relevant integ...
Existence and spatial limit theorems for lattice and continuum particle systems
Interacting particle system functional central limit theorem point process
2009/5/18
We give a general existence result for interacting particle systems with local interactions and bounded jump rates but noncompact state space at each site. We allow for jump events at a site that affe...
Excursions Into a New Duality Relation for Diffusion Processes
Excursions Diffusion Processes
2009/5/12
This work was motivated by the recent work by H. Dette, J. Pitman and W.J. Studden on a new duality relation for random walks. In this note we consider the diffusion process limit of their result, and...
We slightly improve the upper bounds of disconnection exponents for planar Brownian motion that we derived in an earlier paper. We also give a proof of the lower bound $1/(2pi)$ for the disconnection ...
Moderate Deviations for Martingales with Bounded Jumps
Moderate deviations martingales bounded martingale difference
2009/5/11
We prove that the Moderate Deviation Principle (MDP) holds for the trajectory of a locally square integrable martingale with bounded jumps as soon as its quadratic covariation, properly scaled, conver...
On the Non-Convexity of the Time Constant in First-Passage Percolation
First-passage percolation timeconstant convexity
2009/5/11
We give a counterexample to a conjecture of Hammersley and Welsh (1965) about the convexity of the time constant in first-passage percolation, as a functional on the space of distribution functions. T...