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2 results

  • 15 - Information Projection and Large Deviations

  • from Part III - Hypothesis Testing and Large Deviations
  • Yury Polyanskiy, Massachusetts Institute of Technology, Yihong Wu, Yale University, Connecticut
  • Book:
    Information Theory
    Published online:
    09 January 2025
    Print publication:
    02 January 2025, pp 296-316

  • Summary

    In this chapter we introduce the problem of analyzing low-probability events, known as large deviation theory. It is usually solved by computing moment-generating functions and Fenchel-Legendre conjugation. It turns out, however, that these steps can be interpreted information-theoretically in terms of information projection. We show how to solve information projection in a special case of linear constraints, connecting the solution to exponential families.

  • The existence of moments for stationary Markov chains

  • R. L. Tweedie
  • Journal:
    Journal of Applied Probability / Volume 20 / Issue 1 / March 1983
    Published online by Cambridge University Press:
    14 July 2016, pp. 191-196
    Print publication:
    March 1983

  • We give conditions under which the stationary distribution π of a Markov chain admits moments of the general form ∫ f(x)π(dx), where f is a general function; specific examples include f(x) = xr and f(x) = esx. In general the time-dependent moments of the chain then converge to the stationary moments. We show that in special cases this convergence of moments occurs at a geometric rate. The results are applied to random walk on [0, ∞).