van Handel Probability in High Dimension ORF 570 Lecture

Ramon van Handel Probability in High Dimension

Probability in High Dimension Report Title Lecture notes from course ORF 570 Probability in High Dimension (educational material made freely available on my website) Ramon van Handel Probability in High Dimension ORF 570 Lecture Notes Princeton University This version: June 30, 2014 Preface These lecture notes were written for the course ORF 570: Probability in High Dimension that I taught at P...

Hidden Markov Models Lecture Notes

The Local Geometry of Finite Mixtures

Abstract. We establish that for q ≥ 1, the class of convex combinations of q translates of a smooth probability density has local doubling dimension proportional to q. The key difficulty in the proof is to control the local geometric structure of mixture classes. Our local geometry theorem yields a bound on the (bracketing) metric entropy of a class of normalized densities, from which a local e...

Convergence in distribution for filtering processes associated to Hidden Markov Models with densities

A Hidden Markov Model generates two basic stochastic processes, a Markov chain, which is hidden, and an observation sequence. The filtering process of a Hidden Markov Model is, roughly speaking, the sequence of conditional distributions of the hidden Markov chain that is obtained as new observations are received. It is well-known, that the filtering process itself, is also a Markov chain. A cla...

Notes on Free Probability Theory

Lecture 1. Free Independence and Free Harmonic Analysis. 2 1.1. Probability spaces. 2 1.2. Non-commutative probability spaces. 3 1.3. Classical independence. 4 1.4. Free products of non-commutative probability spaces. 5 1.5. Free Fock space. 6 1.6. Free Central Limit Theorem. 8 1.7. Free Harmonic Analysis. 10 1.8. Further topics. 16 Lecture 2. Random Matrices and Free Probability. 17 2.1. Rando...