Let be a measurable space, a transition kernel on it, supposed -irreducible (with a maximal ) and aperiodic. If is a measure on , a probability on , then a set is called -petite if for all , . Condition , where , is concave, nondecreasing and differentiable, , means the existence of a constant with . is defined as the set of pairs , defined on , ultimately nondecreasing, one of them tending to at , such that . For a measure on and a , is defined as . The main result, proved after seven lemmas, states: if holds, at , is petite, and if , then there exists an invariant probability and, for , , where is , being the inverse of . Also that every probability on with is -regular, i.e. , for all , , where is the first visit in of the -chain , and there exists a constant such that, if is another such ,
As applications (supposing ): , , , , , , (polynomial rates of convergence), , , and same ’s (logarithmic rates), concave, differentiable, for , , (subexponential rates). The above when for ( is shown to hold). Hastings-Metropolis algorithm: , , being the Lebesgue measure on and a probability density function (for it and for the following it is proved, as theorems, that is valid, in each case under some conditions, for some ). Nonlinear autoregressive model: ; -valued, independent, identically distributed, , , , continuous, , , for . Stochastic unit root: , independent, all uniformly distributed on .