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3080 results in Probability theory and stochastic processes

Page 69 of 154

Probability
  • The Classical Limit Theorems
  • Henry McKean
  • Published online:
    05 December 2014
    Print publication:
    27 November 2014
  • Probability theory has been extraordinarily successful at describing a variety of phenomena, from the behaviour of gases to the transmission of messages, and is, besides, a powerful tool with applications throughout mathematics. At its heart are a number of concepts familiar in one guise or another to many: Gauss' bell-shaped curve, the law of averages, and so on, concepts that crop up in so many settings they are in some sense universal. This universality is predicted by probability theory to a remarkable degree. This book explains that theory and investigates its ramifications. Assuming a good working knowledge of basic analysis, real and complex, the author maps out a route from basic probability, via random walks, Brownian motion, the law of large numbers and the central limit theorem, to aspects of ergodic theorems, equilibrium and nonequilibrium statistical mechanics, communication over a noisy channel, and random matrices. Numerous examples and exercises enrich the text.

  • 11 - Statistical Mechanics Out of Equilibrium

  • Henry McKean, New York University
  • Book:
    Probability
    Published online:
    05 December 2014
    Print publication:
    27 November 2014, pp 352-401

  • Summary

    This is much harder. I quote from Feynman [1964] at some length

    With this chapter we begin a new subject which will occupy us for some time. It is the first part of the analysis of the properties of matter from the physical point of view, in which, recognizing that matter is made out of a great many atoms, or elementary parts, which interact electrically and obey the laws of mechanics, we try to understand why various aggregates of atoms behave the way they do.

    It is obvious that this is a difficult subject, and we emphasize at the beginning that it is in fact an extremely difficult subject, and that we have to deal with it differently than we have dealt with the other subjects so far. In the case of mechanics and in the case of light, we were able to begin with a precise statement of some laws, like Newton's laws, or the formula for the field produced by an accelerating charge, from which a whole host of phenomena could be essentially understood, and which would produce a basis for our understanding of mechanics and light from that time on. That is, we may learn more later, but we do not learn different physics, we only learn better methods of mathematical analysis to deal with the situation.

  • 12 - Random Matrices

  • Henry McKean, New York University
  • Book:
    Probability
    Published online:
    05 December 2014
    Print publication:
    27 November 2014, pp 402-446

  • Summary

    This subject is not so classical as what's gone before. It has its roots in practical statistics (1930 or so), quantum mechanics (1958), and, very recently, in an astonishing variety of other questions, touched upon below. §12.7 et seq. are technically more demanding than any thing we've done before, employing for the most part radically new methods, but it's fascinating and worth the effort. Mehta [1967] is the best general reference.

    The Gaussian orthogonal ensemble (GOE)

    Quantum-mechanically speaking, the energy levels E of a large collection of n like atoms, without internal degrees of freedom such as angular momentum or spin, may be found by solving Hψ ≡ −Δψ + Vψ = Eψ, in which Δ is Laplace's operator in ℝ3n = ℝ3 × … × ℝ3, one copy for each atom to record its location, and V is the (classical) total potential energy of such a configuration. But, if n is big, the computation is already way out of reach of the biggest machine. What to do? Well, you can stop there, or you can look for something simpler, something you can compute, to see if that helps. Wigner [1958, 1967] proposed the simplest such caricature in connection with the scattering of neutrinos off heavy nuclei, replacing H by a typical, real symmetric, n × n matrix x = (xij :1 ≤ i, j ≤ n) and looking at its spectrum as n ↑ ∞ far away from Nature perhaps, but never mind.

Page 69 of 154