Characterization of potential well spectrum
in renewal processes
Miguel Abadi
NUMEC - IME-USP. Joint work with Liliam Cardeño - Universidad de Antioquia
and Sandro Gallo - UFRJ
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Definitions and notation
(Yn )n∈N House of cards Markov chain over N
Q(i, i + 1) = 1 − qi ,
Q(i, 0) = qi
with 0 < qi < 1, i ≥ 0.
Positive recurrent if
X m−1
Y
(1 − qi ) < +∞.
m≥1 i=0
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Definitions and notation
(Yn )n∈N House of cards Markov chain over N
Q(i, i + 1) = 1 − qi ,
Q(i, 0) = qi
with 0 < qi < 1, i ≥ 0.
Positive recurrent if
X m−1
Y
(1 − qi ) < +∞.
m≥1 i=0
(Xn )n∈N renewal process
Xn = 1{Yn = 0} ,
Miguel Abadi
for any n ≥ 0.
Characterization of potential well spectrum in renewal processes
Definitions
Fix a0n−1 ∈ {0, 1}n
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Definitions
Fix a0n−1 ∈ {0, 1}n
Classical hitting time (of the realization x0∞ ) TO a0n−1 :
τan−1 (x0∞ ) = inf{k ≥ 1 | xkk +n−1 = a0n−1 }
0
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Definitions
Fix a0n−1 ∈ {0, 1}n
Classical hitting time (of the realization x0∞ ) TO a0n−1 :
τan−1 (x0∞ ) = inf{k ≥ 1 | xkk +n−1 = a0n−1 }
0
First possible return OF a0n−1
Tn (a0n−1 ) =
inf
x0∞ :x0n−1 =a0n−1
τan−1 (x0∞ )
0
a0n−1 is set as initial condition
Tn (a0n−1 ) minimum of the returns to a0n−1
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Behaviour of Tn
Saussol et al.(02), Afraimovich et al. (03): ergodic, positive
entropy, finite alphabet
Tn
=1
n→∞ n
lim
Miguel Abadi
a.e.
Characterization of potential well spectrum in renewal processes
Behaviour of Tn
Saussol et al.(02), Afraimovich et al. (03): ergodic, positive
entropy, finite alphabet
Tn
=1
n→∞ n
lim
a.e.
Abadi and Lambert (13) for i.i.d. (Abadi and Rada, work in
progress for β-mixing)
n − Tn =⇒ L(µ, σ 2 )
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Potential well
Definition: Exit probability
ρ(a0n−1 ) = µ(τan−1 > Tn (a0n−1 ) | X0n−1 = a0n−1 )
0
Figure : 1 − ρ(a0n−1 ) = µ(τan−1 = Tn (a0n−1 ) | X0n−1 = a0n−1 ) Depth of
0
the well
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Main results
Theorem
lim sup µ ρ(a0n−1 )µ(a0n−1 )τan−1 > k = e−k
n→∞ k ∈N
lim sup
n→∞ k ∈N
0
µ ρ(a0n−1 )µ(a0n−1 )τan−1 > k | X0n−1 = a0n−1
0
ρ(a0n−1 )
= e−k
∀a0∞
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Main results
Theorem (a)
For any a0∞ 6= 0∞
0 ,
1
2
3
ρ(a0n−1 ) converges to 1 if, and only if, a0∞ is an aperiodic
sequence,
∃C1 ∈ (0, 1) s.t. for any periodic sequence,
∃ limn∈∞ ρ(a0n−1 ) =: ρ(a0∞ ) ∈ [C1 , 1).
{ρ(a0∞ ) : a0∞ 6= 0∞
0 } has a unique accumulation point
which is 1.
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Main results
Theorem (b)
Let
R(n) :=
i+n
XY
(1 − qj ),
n≥1.
i≥0 j=n
The sequence ρ(0n−1
0 ), n ∈ N may
1
converge to a limiting value belonging to the interval (0, 1)
if, and only if, R(n) converges,
2
oscillate between two constants 0 < c1 < c2 < 1 if, and
only if, R(n) oscillates,
3
converge to 0 if, and only if, R(n) diverges.
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Example
Regime (1) is when qi → q∞ > 0.
Regime (2) is when qi = q0 ∈ (0, 1) for i even and
qi = q1 6= q0 for i odd.
Regime (3) is when qi & 0.
Miguel Abadi
Characterization of potential well spectrum in renewal processes
Example
Assume that for any i ≥ 0,
(r )
qi
=1−
i +1
i +2
r
,
n→+∞
for some real number r > 1. Then ρ(00n−1 ) −→ 0 and
1
If r > 2 then limn→∞
2
If r = 2 then limn→∞
3
If 1 < r < 2 then lim
ρ(0n−1
)
0
µ(0n−1
)
0
ρ(0n−1
)
0
µ(0n−1
)
0
ρ(0n−1
)
0
µ(0n−1
)
0
Miguel Abadi
= ∞.
= c ∈ (0, ∞).
= 0.
Characterization of potential well spectrum in renewal processes