Sample Paths and Limit Theorems

  • Sean P. Meyn
  • Richard L. Tweedie
Part of the Communications and Control Engineering Series book series (CCE)


Most of this chapter is devoted to the analysis of the series Sn(g), where we define for any function g on X,
$${S_n}(g): = \sum\limits_{k = 1}^n {g(\Phi k)}$$
We are concerned primarily with four types of limit theorems for positive recurrent chains possessing an invariant probability π:
  1. (i)

    those which are based upon the existence of martingales associated with the chain;

  2. (ii)

    the Strong Law of Large Numbers (LLN), which states that n-1Sn(g) converges to π (g) = E π[g0)], the steady state expectation of g0) ;

  3. (iii)

    the Central Limit Theorem (CLT) , which states that the sum Sn(g-π(g)), when properly normalized, is asymptotically normally distributed;

  4. (iv)

    the Law of the Iterated Logarithm (LIL) which gives precise upper and lower bounds on the limit supremum of the sequence Sn(g-π(g)) again when properly normalized.



Markov Chain Harmonic Function Central Limit Theorem Poisson Equation Sample Path 
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Copyright information

© Springer-Verlag London Limited 1993

Authors and Affiliations

  • Sean P. Meyn
    • 1
  • Richard L. Tweedie
    • 2
  1. 1.Coordinated Science Laboratory and the Department of Electrical and Computer EngineeringUniversity of IllinoisUrbanaUSA
  2. 2.Department of StatisticsColorado State UniversityFort CollinsUSA

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