Page 440 - Special Topic Session (STS) - Volume 3

STS555 Patrice Bertail et al.
In short, Harris recurrence property ensures that X visits set A infinitely often a.s..
It is well-known that in Harris recurrent case it is also possible to recover the
regeneration properties via splitting technique introduced in [5].

Definition 3. We say that a set S ∈ E is small if there exists a parameter δ > 0,

a positive probability measure Φ supported by S and an integer m ∈ N such
∗
that

where Π denotes the m-th iterate of the transition probability Π.
m
The inequality (1) from above definition is called minorization condition
M(m, S, δ, ψ) and gives a uniform bound from below on the transition
probabilities. Note that the parameter δ controls how fast our chain X forgets
its past.
It is assumed throughout the rest of this paper that the minorization
condition M is satisfied with m = 1. The family of the conditional distributions
{Π(x, dy)}x∈E and the initial distribution ν are dominated by a σ-finite measure λ of
reference, so that ν(dy) = f (y)λ(dy) and Π(x, dy) = p(x, y)λ(dy), for all x ∈ E and
that p(x, y) ≥ δφ(y), λ(dy) a.s. for any x ∈ S, with Φ(dy) = φ(y)dy. We then split the
series as in the same way as it was originally done in [1].

Algorithm (Approximate regeneration blocks construction)
1. Construct an estimator (may be on part of the data) pn(x, y) of the
transition density using sample Xn+1. An estimator pn must satisfy the
following conditions

pn(x, y) ≥ δγ(y), λ(dy) a.s. and pn(Xi, Xi+1) > 0, 1 ≤ i ≤ n.

ˆ
2. Conditioned on Xn+1 draw Y ’s only at those time points when Xi ∈ S. That
is because only then the split chain can regenerate. At such time point i,
ˆ
draw Yi from the Bernoulli distribution with parameter δγ(Xi+1)\pn(Xi, Xi+1).
ˆ
ˆ
3. Count the number of visits ln = ∑ "{Xi ∈ S, Yi = 1) to the atom S1 = S
=1
× {1} up to time n.
ˆ
4. Divide the trajectory Xn+1 into ln + 1 approximate regeneration blocks
ˆ
according to the consecutive visits of (X, Y ) to S1. Approximated blocks
are of the form

where

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