Page 94 - Special Topic Session (STS) - Volume 3

STS517 Zaoli C. et al.
Samorodnitsky (2004), Owada and Samorodnitsky (2015a,b), Owada (2016)
and Lacaux and Samorodnitsky (2016).
We start with d σ-finite, infinite measures on (ℤ , ℬ(ℤ )) defined by
ℕ 0
ℕ 0

()
where for = 1, … , , is the law of an irreducible aperiodic null-recurrent

() () ()
Markov chain ( ) ≥0 on ℤ starting at 0 = ∈ ℤ. Further, ( ) ∈ is its

unique (infinite) invariant measure satisfying 0 () = 1. Given this invariant
()
measure, we can extend the probability measures from measures on ℤ
ℕ 0

to measures on ℤ which, in turn, allows us to extend the measure µ in (2.1)
ℤ

to ℤ as well. We will keep using the same notation as in (2.1).
ℤ
We will work with the product space

of d copies of (ℤ , ℬ(ℤ )), on which we put the product, σ-finite, infinite,
ℤ
ℤ
measure
µ = µ × · × µ .
1

The key assumption is a regular variation assumption on the return times
()
of the Markov chains ( ) ≥0 , = 1, … , . For = (… , , , , , ...) ∈ ℤ
2
0
−1
1

we define the first return time to the origin by () = inf { ≥ 1: = }. We

assume that for = 1, … , we have

()
(2.2) ( > ) ∈ −
0
for some 0 < < 1. This implies that

(2.3) ({: = 0 for some = 0, 1, … , })

.
See Resnick et al. (2000).

On ℤ there is a natural left shift operator
ℤ
.

It is naturally extended to a group action of ℤ on E as follows. Writing an
()
()
()

ℤ

element ∈ = ( (1) , … , ) with = (… , , , , … ∈ ℤ ) for =
−1
1
2
1, … , , we set for = ( , … , , ) ∈ ℤ ,

1

(2.4) x = ( 1 (1) , … . . () ) .

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