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Practical drift conditions for subgeometric rates of convergence. (English) Zbl 1082.60062

Let (X,) be a measurable space, P a transition kernel on it, supposed ψ-irreducible (with a maximal ψ) and aperiodic. If ξ is a measure on , a a probability on , then a set C is called ξ a -petite if n0 a n P n (x,B)ξ(B) for all xC, B. Condition D(φ,V,C), where V:X[1,], φ:[1,](0,] is concave, nondecreasing and differentiable, C, means the existence of a constant b with PV+φVV+b1 C . 𝒴 is defined as the set of pairs (Ψ 1 ,Ψ 2 ), Ψ i defined on [1,), ultimately nondecreasing, one of them tending to at , such that Ψ 1 (x)Ψ 2 (y)x+y. For a measure μ on and a f:X[1,), μ f is defined as sup |g|f |gdμ|. The main result, proved after seven lemmas, states: if D(φ,V,C) holds, φ ' 0 at , C is petite, sup C V< and if (Ψ 1 ,Ψ 2 )𝒴, then there exists an invariant probability π and, for x(V<), lim n Ψ 1 (r φ (n))P n (x,·)-π Ψ 2 (φV) =0, where r φ is φK φ , K φ being the inverse of 1 · dx/φ(x). Also that every probability λ on with Vdλ< is (Ψ 2 (φV),Ψ 1 (r φ ))-regular, i.e. E λ ( k=0 τ B Ψ 1 (r φ (k))Ψ 2 (φV(Φ k ))<, for all B, ψ(B)>0, where τ B is the first 1 visit in B of the P-chain Φ k , and there exists a constant c such that, if μ is another such λ,

n0 Ψ 1 (r φ (n))λ(dx)μ(dy)P n (x,·)-P n (y,·) Ψ 2 (φV) cVd(λ+μ)·

As applications (supposing D(φ,V,C)): φ(t)=ct α , α[0,1), c(0,1], Ψ 1 (t)=(qt) q , Ψ 2 (t)=(pt) p , p(0,1), q=1-p (polynomial rates of convergence), φ(t)=c(1+logt) α , α0, c(0,1] and same Ψ’s (logarithmic rates), φ concave, differentiable, φ(t)=ct/log α t for tt 0 , α>0, c>0 (subexponential rates). The above when X=,P(n,i)=0 for i0,n+1 (D(φ,V,C) is shown to hold). Hastings-Metropolis algorithm: P(x,A)= A α(x,x+y)q(y)dμ(y)+c1 A (x), α(x·y)=min(1,π(y)/π(x)), μ being the Lebesgue measure on X= d and π a probability density function (for it and for the following it is proved, as theorems, that D(φ,V,C) is valid, in each case under some conditions, for some φ,V,C). Nonlinear autoregressive model: Φ n+1 =g(Φ n )+ε n+1 ; ε n d -valued, independent, identically distributed, Eε=0, E(e z|ε| γ )<, γ(0,1], g continuous, |g(t)|<|t|(1-r|t| -ϱ ), ϱ[0,2), for |t|R 0 . Stochastic unit root: Φ n+1 =1 (U n+1 g(Φ n )) Φ n +ε n+1 , U n independent, all uniformly distributed on [0,1].


MSC:
60J05Discrete-time Markov processes on general state spaces