Simple Monte Carlo integration with respect to Bernoulli convolutions

Monte Carlo integration with subtraction

This paper investigates a class of algorithms for numerical integration of a function in d dimensions over a compact domain by Monte Carlo methods. We construct a histogram approximation to the function using a partition of the integration domain into a set of bins specified by some parameters. We then consider two adaptations; the first is to subtract the histogram approximation, whose integra...

Absolute Continuity of Bernoulli Convolutions, a Simple Proof

A bstract . The distribution νλ of the random series ∑ ±λn has been studied by many authors since the two seminal papers by Erdős in 1939 and 1940. Works of Alexander and Yorke, Przytycki and Urbański, and Ledrappier showed the importance of these distributions in several problems in dynamical systems and Hausdorff dimension estimation. Recently the second author proved a conjecture made by Gar...

Spectra of Bernoulli Convolutions

Monte Carlo Integration With Acceptance-Rejection

This article considers Monte Carlo integration under rejection sampling or Metropolis-Hastings sampling. Each algorithm involves accepting or rejecting observations from proposal distributions other than a target distribution. While taking a likelihood approach, we basically treat the sampling scheme as a random design, and define a stratified estimator of the baseline measure. We establish tha...

Monte Carlo integration with Markov chain

There are two conceptually distinct tasks in Markov chain Monte Carlo (MCMC): a sampler is designed for simulating a Markov chain and then an estimator is constructed on the Markov chain for computing integrals and expectations. In this article, we aim to address the second task by extending the likelihood approach of Kong et al. for Monte Carlo integration. We consider a general Markov chain s...