On the length of the longest head-run for the Markov chain with two states

In the article the asymptotic formula for the lenght of the longest head-run in the trajectory of a Markov chain with two states is obtained. The main theorem is the amplification and generalization (on the Markov chain) of the results proved by P. Erdös and P. Révész for the sequence of independent identically distributed random variables $\xi_i$ with
$$ \mathbf P\{\xi_i=1\}=\mathbf P\{\xi_i=0\}=1/2. $$