Type II error exponent for identity test of Markov processes

Goodness-of-fit (or identity) test proposed by Ryabko and Astola (RA-test) is considered. The purpose of this test is to determine whether the sample was generated by the process with the particular distribution pi (main hypothesis) or by a stationary and ergodic source which differs from the source under the main hypothesis. RA-test is based on the ideas of data compression and makes use of an arbitrary universal code. In this paper the rate at which probability of Type II error of RA-test tends to zero when the size of a sample tends to infinity is studied. It is shown that for certain codes this probability goes to zero at exponential rate of D(pi||tau ), where D(ldr||ldr) denotes Kullback-Leibler divergence and tau is the real distribution of the source.