Non-Markov random telegraph processes with arbitrary nonlocal memory

We study random dichotomous stationary ergodic processes with nonlocal in-time memory. These processes are characterized by a conditional transition probabilities distribution containing an integral term involving the memory function, which takes into account the past of the process. For the first time, we present a method for solving the integro-differential equation, which connects the arbitrary memory and pair correlation functions in an analytical closed form. Conditions of stationarity for the telegraph process with exponential memory are found. It is shown that the pair correlation function of this process can be of two types: either exponentially decreasing with two different decrements or oscillating with exponentially decreasing amplitude.