Simple random walk and Brownian motion are two strongly interconnected mathematical concepts. They are widely involved in not only pure math, but also in many other scientific fields. In this paper I will first introduce and define some basic concepts of discrete-time random walk. Then I will construct Brownian Motion with some basic properties, and use a method called the strong approximation ...

The l1 regularized Gaussian maximum likelihood estimator has been shown to have strong statistical guarantees in recovering a sparse inverse covariance matrix, or alternatively the underlying graph structure of a Gaussian Markov Random Field, from very limited samples. We propose a novel algorithm for solving the resulting optimization problem which is a regularized log-determinant program. In ...

We derive new and improved non-asymptotic deviation inequalities for the sample average approximation (SAA) of an optimization problem. Our results give strong error probability bounds that are “sub-Gaussian” even when randomness problem is fairly heavy tailed. Additionally, we obtain good (often optimal) dependence on size geometrical parameters Finally, allow random constraints SAA unbounded ...

A piecewise linear (PWL) function approximation scheme is described by a lattice algebra of modified operators that allows for the interpolation of PWL function vertexes. A new recursive method called Centred Recursive Interpolation (CRI) based on such modified operators is analysed for successive function smoothing and more accurate approximation. This approximation method, simple but accurate...

The Poisson model is frequently employed to describe count data, but in a Bayesian context it leads to an analytically intractable posterior probability distribution. In this work, we analyze a variational Gaussian approximation to the posterior distribution arising from the Poisson model with a Gaussian prior. This is achieved by seeking an optimal Gaussian distribution minimizing the Kullback...