Asymptotic entropy of transformed random walks

Abstract

We consider general transformations of random walks on groups determined by Markov stopping times and prove that the asymptotic entropy (respectively, rate of escape) of the transformed random walks is equal to the asymptotic entropy (respectively, rate of escape) of the original random walk multiplied by the expectation of the corresponding stopping time. This is an analogue of the well-known Abramov formula from ergodic theory; its particular cases were established earlier by Kaimanovich [Differential entropy of the boundary of a random walk on a group. Uspekhi Mat. Nauk38(5(233)) (1983), 187–188] and Hartman et al [An Abramov formula for stationary spaces of discrete groups. Ergod. Th. & Dynam. Sys.34(3) (2014), 837–853].

Asymptotic entropy of random walks on Fuchsian buildings and Kac–Moody groups

“…Remark 3.17. The results of Forghani [10] are somehow in the same spirit as Theorem 3.16. It is shown there that the asymptotic entropy of transformed random walks (by stopping times) equals the product of the original entropy and the expectation of the corresponding stopping times.…”

“…1 n E[log π n (X n )] if the limit exists, where π n is the distribution of X n . Intuitively, the entropy can be seen as the "asymptotic uncertainty" of the random walk and it plays an important role in the description of the asymptotic behavior of the random walk, see Derriennic [9,10], Guivarc?h [15], Kaimanovich [17], Kaimanovich and Vershik [18] and Vershik [24], amongst others. It is well-known that the limit defining h necessarily exists for random walks on groups whenever E[− log π 1 (X 1 )] < ∞ (this is an application of Kingman's subadditive ergodic theorem [17]).…”

Shannon's theorem for locally compact groups

“…Remark 6.6. Note that when G is a discrete group and µ has finite entropy, then H(µ τ ) ≤ E(τ )H 1 , see [For17]. However, in the case of locally compact groups, the same statement does not hold for differential entropies.…”

“…L , and µ τ = θ = n=0 β * n * α. The probability measure θ was introduced by Willis [Wil90]; transformations of random walks via stopping times are due to the first author and Kaimanovich, who show these transformations preserve the Poisson boundaries for random walks on discrete groups (for more details see [For15] and [For17]).…”

Random walks of infinite moment on free semigroups

“…We also need the following Abramov-type formula, which generalizes Lemma 2.5 of [For17]. Proposition 6.4.…”