2025/10/30 by Fredes, Luis, Marckert, Jean-François
#FOS: Mathematics #Probability (math.PR)
paper · doi:10.48550/arxiv.2510.26207
Let M be an irreducible transition matrix on a finite state space V. For a Markov chain C=(Ck,k≥ 0) with transition matrix M, let τ≥ 1u denote the first positive hitting time of u by C, and ρ the unique invariant measure of M. Kemeny proved that if X is sampled according to ρ independently of C, the expected value of the first positive hitting time of X by C does not depend on the starting state of the chain: all the values (E(τ≥ 1X~|~C0=u), u ∈ V) are equal. \par In this paper, we show that this property follows from a more general result: the generating function ∑v∈ VE(x^τv≥ 1~|~C0=u)det(Id-xM(v)) is independent of the starting state u, where M(v) is obtained from M by deleting the row and column corresponding to the state v. The factors appearing in this generating function are: first, the probability generating function of τ≥ 1v, and second, the sequence of determinants (det(Id-xM(v)),v∈ V), which, for x=1, is known to be proportional to the invariant measure (ρu,u∈ V). From this property, we deduce several further results, including relations involving higher moments of τX≥ 1, which are of independent interest.