Nonasymptotic upper bounds on the probability of the /spl epsi/-atypical set for Markov chains

摘要

For a stationary, irreducible and aperiodic Markov chain with finite alphabet A, starting symbol X/sub 0/=/spl sigma/, transition probability matrix P, stationary distribution /spl pi/, support S(/spl pi/,P)={(j,k):/spl pi//sub j/P/sub k|j/<0} and for a function f such that M=/spl Delta/E/sub /spl pi/P/f(X/sub 1/,X/sub 2/)>+/spl infin/ it is known that limsup/sub l/spl rarr//spl infin// 1/l log/sub 2/ /spl mu/(|M-1/l /spl Sigma//sub i=1//sup l/f(X/sub i-1/,X/sub i/)|/spl ges//spl epsiv/)/spl les/-inf/sub (q,Q)/spl isin//spl Gamma/(e)/ D(Q/spl par/P), where /spl Gamma//sub /spl epsiv//={(q,Q):|E/sub /spl pi/,P/f(X/sub 1/,X/sub 2/)-E/sub q,Q/f(X/sub 1/,X/sub 2/)|/spl ges//spl epsiv/ Qq=q,S(q,Q)/spl sub/S(/spl pi/,P)}. In this paper we demonstrate that inf/sub (q,Q)/spl isin//spl Gamma/(e)/ D(Q/spl par/P)/spl ges/ 1/(2ln2)[sup/sub n/spl ges/1:Kn<0/{K/sub n//(1+K/sub n/)}/spl epsiv//(max/sub j,k:P(k|j)/<0|f(j,k)|]/sup 2/ where K/sub n/=(1-|A|max/sub j,k/|P/sub k|j//sup n/-/spl pi//sub k/)). Under the conditions stated, the set over which the sup is taken is nonempty and therefore the sup exists and is positive; it is also shown that the sup is attained at a finite value of n. A nonasymptotic version of this result is also given based on the method of Markov types.