Information entropy and temperature of binary Markov chains.

We propose two different approaches for introducing the information temperature of binary Nth-order Markov chains. The first approach is based on a comparison of Markov sequences with equilibrium Ising chains at given temperatures. The second approach uses probabilities of finite-length subsequences of symbols occurring, which determine their entropies.

Continuous stochastic processes with nonlocal memory.

We study the non-Markovian random continuous processes described by the Mori-Zwanzig equation. As a starting point, we use the Markovian Gaussian Ornstein-Uhlenbeck process and introduce an integral memory term depending on the past of the process into an expression for the higher-order transition probability function and the stochastic differential equation. We show that the proposed processes can be considered as continuous-time interpolations of discrete-time higher-order autoregressive sequences.