Time operator of Markov chains and mixing times. Applications to financial data

Entropy, Age and Time Operator

“…is a resolution of the identity in H, i.e.,: E 0 = O, E ∞ = I and E t 1 ≤ E t 2 , t 1 < t 2 . We have omitted the mathematical and technical details, as they are presented elsewhere [19,20], and they are not necessary for the scope of this work. Definition 1.…”

“…The time operator is a self-adjoint operator on the space of fluctuations, with eigenvalues the clock times and corresponding eigenspaces the successive innovations. The time operator of Markov chains has been presented in [20] as an extension of the time operator of Bernoulli processes [21]. From the time operator, we can compute the internal age Age(X t ) of some random variable at each clock time t, as the expectation of the time operator (Section 2).…”

“…After extending the time operator theory to more general processes, like Markov chains, we found non-canonical age formulas, where the correction to the canonical formula is proportional to a decaying exponential term [20]:…”

“…The internal age also determines the mixing time of Markov chains [20]. Mixing time estimations are useful in non-equilibrium statistical physics, in web analysis and network dynamics, computer science and statistics.…”

“…Moreover, the non-canonical age formula indicates a link of aging with the associated Lyapunov functional, in this case defined by the total variation distance [20]. As entropies were historically the first Lyapunov functionals, we investigate in this work which entropies fit better the age formula and are compatible with the monotonic approach to equilibrium, as stated by the second law of thermodynamics (Section 4).…”