Tail of stationary probability of Stochastic Dynamical systems - - PowerPoint PPT Presentation

Tail of stationary probability of Stochastic Dynamical systems

Gerold Alsmeyer (University of Muenster) Sara Brofferio (University Paris Sud) Dariusz Buraczewski (University of Wroclaw) June 2016

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 1 / 15

Stationary measure of Stochastic dynamical system

Stochastic dynamical systems

A Stochastic dynamical systems on R

A SDS on R is a stochastic process defined recursively by Xn = Ψn(Xn−1) = Ψn ◦ . . . ◦ Ψ1(X0), where Ψn : R → R are chosen randomly and independently according to a same law µ. Examples : Affine transformation Ψn(x) = Anx + Bn i.i.d Xn = AnXn−1 + Bn Reflected random walks Ψn(x) = |x + Un| Logistic Model Ψn(x) = Rnx(1 − x)

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 2 / 15

Stationary measure of Stochastic dynamical system

Stochastic dynamical systems

A Stochastic dynamical systems on R

A SDS on R is a stochastic process defined recursively by Xn = Ψn(Xn−1) = Ψn ◦ . . . ◦ Ψ1(X0), where Ψn : R → R are chosen randomly and independently according to a same law µ. Examples : Affine transformation Ψn(x) = Anx + Bn i.i.d Xn = AnXn−1 + Bn Reflected random walks Ψn(x) = |x + Un| Logistic Model Ψn(x) = Rnx(1 − x)

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 2 / 15

Stationary measure of Stochastic dynamical system

Stochastic dynamical systems

A Stochastic dynamical systems on R

A SDS on R is a stochastic process defined recursively by Xn = Ψn(Xn−1) = Ψn ◦ . . . ◦ Ψ1(X0), where Ψn : R → R are chosen randomly and independently according to a same law µ. Examples : Affine transformation Ψn(x) = Anx + Bn i.i.d Xn = AnXn−1 + Bn Reflected random walks Ψn(x) = |x + Un| Logistic Model Ψn(x) = Rnx(1 − x)

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 2 / 15

Stationary measure of Stochastic dynamical system

Questions

Xn = Ψn(Xn−1) = Ψn ◦ . . . ◦ Ψ1(X0), Questions : Positive Recurrence and existence of a

Stationary probability ν :

X0 ∼ ν independent of Ψ1 than X0 =d Ψ1(X0) = X1 Behavior at infinity of ν This are asymptotic problems, stable under local perturbation of Ψ

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 3 / 15

Stationary measure of Stochastic dynamical system

Questions

Xn = Ψn(Xn−1) = Ψn ◦ . . . ◦ Ψ1(X0), Questions : Positive Recurrence and existence of a

Stationary probability ν :

X0 ∼ ν independent of Ψ1 than X0 =d Ψ1(X0) = X1 Behavior at infinity of ν This are asymptotic problems, stable under local perturbation of Ψ

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 3 / 15

Stationary measure of Stochastic dynamical system

Questions

Xn = Ψn(Xn−1) = Ψn ◦ . . . ◦ Ψ1(X0), Questions : Positive Recurrence and existence of a

Stationary probability ν :

X0 ∼ ν independent of Ψ1 than X0 =d Ψ1(X0) = X1 Behavior at infinity of ν This are asymptotic problems, stable under local perturbation of Ψ

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 3 / 15

Stationary measure of Stochastic dynamical system

Affine recursions : Recurrence

Xn = AnXn−1 + Bn Contractive case : If E

log A < 0 and E log+ |B| < ∞,

Xn is positive recurrent and has a unique stationary probability.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 4 / 15

Stationary measure of Stochastic dynamical system

Affine recursions : Recurrence

Xn = AnXn−1 + Bn Contractive case : If E

log A < 0 and E log+ |B| < ∞,

Xn is positive recurrent and has a unique stationary probability. Critical : If E

log A = 0 and E || log A| + | log |B|||2+ǫ < ∞

Xn is null recurrent and has a unique infinite invariant measure [BBE97].

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 4 / 15

Stationary measure of Stochastic dynamical system

Affine recursions : Recurrence

Xn = AnXn−1 + Bn Contractive case : If E

log A < 0 and E log+ |B| < ∞,

Xn is positive recurrent and has a unique stationary probability. Critical : If E

log A = 0 and E || log A| + | log |B|||2+ǫ < ∞

Xn is null recurrent and has a unique infinite invariant measure [BBE97]. Divergent case : If E

log A > 0 and E log+ |B| < ∞

Xn is transient.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 4 / 15

Stationary measure of Stochastic dynamical system

Affine recursion : tail of the stationary probability

Xn = AnXn−1 + Bn Contractive case E log A < 0

If there exists κ > 0 such that E [|A|κ] = 1, E

|A|κ log+ |A| < ∞ and E [|B|κ] < ∞ and log A non-arithmetic.

Then ν(z, +∞) ∼ C+z−κ, i.e. ν(dx) ∼ C+dx

xκ+1

Kesten 1973 - affine recursions A > 0 Goldie 1991 - A generic and other recursions. Guivarch and Le Page 2014 -multidimensional affine recursion

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 5 / 15

Stationary measure of Stochastic dynamical system

Affine recursion : tail of the stationary probability

Xn = AnXn−1 + Bn Contractive case E log A < 0

If there exists κ > 0 such that E [|A|κ] = 1, E

|A|κ log+ |A| < ∞ and E [|B|κ] < ∞ and log A non-arithmetic.

Then ν(z, +∞) ∼ C+z−κ, i.e. ν(dx) ∼ C+dx

xκ+1

Kesten 1973 - affine recursions A > 0 Goldie 1991 - A generic and other recursions. Guivarch and Le Page 2014 -multidimensional affine recursion

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 5 / 15

Stationary measure of Stochastic dynamical system

Affine recursion : tail of the stationary probability

Xn = AnXn−1 + Bn Contractive case E log A < 0

If there exists κ > 0 such that E [|A|κ] = 1, E

|A|κ log+ |A| < ∞ and E [|B|κ] < ∞ and log A non-arithmetic.

Then ν(z, +∞) ∼ C+z−κ, i.e. ν(dx) ∼ C+dx

xκ+1

Kesten 1973 - affine recursions A > 0 Goldie 1991 - A generic and other recursions. Guivarch and Le Page 2014 -multidimensional affine recursion

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 5 / 15

Stationary measure of Stochastic dynamical system

Affine recursion : tail of the stationary probability

Xn = AnXn−1 + Bn Contractive case E log A < 0

If there exists κ > 0 such that E [|A|κ] = 1, E

|A|κ log+ |A| < ∞ and E [|B|κ] < ∞ and log A non-arithmetic.

Then ν(z, +∞) ∼ C+z−κ, i.e. ν(dx) ∼ C+dx

xκ+1

Kesten 1973 - affine recursions A > 0 Goldie 1991 - A generic and other recursions. Guivarch and Le Page 2014 -multidimensional affine recursion

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 5 / 15

Stationary measure of Stochastic dynamical system

Tail for other SDS

Goal : generalize this result to a wide class of SDS Xn = Ψn(Xn−1) Previous results : Ψ at bounded distance of an affine recursion |Ψ(x) − A(Ψ)x| ≤ B(Ψ)

◮ Goldie 1991 - specific recursions Ψ(x) = max{Ax, B}... ◮ Mirek 2011 - AL type of hypothesis (higher dimension), ◮ Reflected random walks

Other models, such as (see Alsmeyer ’15) :

◮ Logistic Model Ψn(x) = Rnx(1 − x) - Athreya ... ◮ AR(1)-model with ARCH(1) errors Ψn(x) = αnx + εn(βn + λnx2)1/2

Partial result- Borkovec and Klüppelberg,

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 6 / 15

Stationary measure of Stochastic dynamical system

Tail for other SDS

Goal : generalize this result to a wide class of SDS Xn = Ψn(Xn−1) Previous results : Ψ at bounded distance of an affine recursion |Ψ(x) − A(Ψ)x| ≤ B(Ψ)

◮ Goldie 1991 - specific recursions Ψ(x) = max{Ax, B}... ◮ Mirek 2011 - AL type of hypothesis (higher dimension), ◮ Reflected random walks

Other models, such as (see Alsmeyer ’15) :

◮ Logistic Model Ψn(x) = Rnx(1 − x) - Athreya ... ◮ AR(1)-model with ARCH(1) errors Ψn(x) = αnx + εn(βn + λnx2)1/2

Partial result- Borkovec and Klüppelberg,

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 6 / 15

Asymptotically linear Stochastic dynamical Systems

Asymptotically linear SDS

Λ(x) =

−

Ax for x ∈ (−∞, 0] and Λ(x) =

+

Ax for x ∈ [0, +∞)

Ψn is a.s. asymptotically linear if

|Ψn(x) − Λn(x)| ≤ Bn ∀x If |Λ| := Lip(Λ) = max{ − A, + A} : |Ψn · · · Ψ1(x) − Λn · · · Λ1(x)| ≤ Yn :=

n

• k=1

|Λn · · · Λk−1|Bk

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 7 / 15

Asymptotically linear Stochastic dynamical Systems

Asymptotically linear SDS

Λ(x) =

−

Ax for x ∈ (−∞, 0] and Λ(x) =

+

Ax for x ∈ [0, +∞)

Ψn is a.s. asymptotically linear if

|Ψn(x) − Λn(x)| ≤ Bn ∀x If |Λ| := Lip(Λ) = max{ − A, + A} : |Ψn · · · Ψ1(x) − Λn · · · Λ1(x)| ≤ Yn :=

n

• k=1

|Λn · · · Λk−1|Bk

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 7 / 15

Asymptotically linear Stochastic dynamical Systems

Types Asymptotically linear Ψ

Ψ at bounded distance of the line

+

Ax near +∞ and of

−

Ax near −∞

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 8 / 15

Asymptotically linear Stochastic dynamical Systems

Types Asymptotically linear Ψ

Ψ at bounded distance of the line

+

Ax near +∞ and of

−

Ax near −∞

+

A > 0 and

−

A > 0 :

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 8 / 15

Asymptotically linear Stochastic dynamical Systems

Types Asymptotically linear Ψ

Ψ at bounded distance of the line

+

Ax near +∞ and of

−

Ax near −∞

+

A > 0 and

−

A > 0 :

+

A < 0 and

−

A < 0

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 8 / 15

Asymptotically linear Stochastic dynamical Systems

Types Asymptotically linear Ψ

Ψ at bounded distance of the line

+

Ax near +∞ and of

−

Ax near −∞

+

A > 0 and

−

A > 0 :

+

A < 0 and

−

A < 0

+

A > 0 and

−

A < 0

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 8 / 15

Asymptotically linear Stochastic dynamical Systems

Types Asymptotically linear Ψ

Ψ at bounded distance of the line

+

Ax near +∞ and of

−

Ax near −∞

+

A > 0 and

−

A > 0 :

+

A < 0 and

−

A < 0

+

A > 0 and

−

A < 0

+

A < 0 and

−

A > 0

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 8 / 15

Asymptotically linear Stochastic dynamical Systems

Markov chain of sign of Λ

The Λn induce a Markov chain on S = {−, +} : ξn = sign(Λn(ξn−1)) = sign(Λn · · · Λ1(ξ0)) S = {−, +} irreducible {−} is attracting ⇒ there exists a unique invariant probability π on S = {−, +} {+} and {−} are attractor is conjugated to an "almost" affine recursion

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 9 / 15

Asymptotically linear Stochastic dynamical Systems

Markov chain of sign of Λ

The Λn induce a Markov chain on S = {−, +} : ξn = sign(Λn(ξn−1)) = sign(Λn · · · Λ1(ξ0)) S = {−, +} irreducible {−} is attracting ⇒ there exists a unique invariant probability π on S = {−, +} {+} and {−} are attractor is conjugated to an "almost" affine recursion

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 9 / 15

Asymptotically linear Stochastic dynamical Systems

Markov chain of sign of Λ

The Λn induce a Markov chain on S = {−, +} : ξn = sign(Λn(ξn−1)) = sign(Λn · · · Λ1(ξ0)) S = {−, +} irreducible {−} is attracting ⇒ there exists a unique invariant probability π on S = {−, +} {+} and {−} are attractor is conjugated to an "almost" affine recursion

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 9 / 15

Asymptotically linear Stochastic dynamical Systems

Modulus of Λ as Markov Random walk

Λ1(±1) = ξ1| ξ0 A1| with ξ0 = ± Sn := log|Λn · · · Λ1(ξ0)| = log|

ξn−1

An|+ · · · + log|

ξ0

A1| Sn is a Markov random walk driven by the Markov chain ξn on {+, −}. Then under moment condition Sn n → π+E log

+

A + π−E log

−

A

Corollary

Suppose E[| log ± A|] < ∞ and E[log+ B] < ∞ . If π+E log + A + π−E log − A < 0 then X x

n = Ψn(Xn−1) is positive recurrent

and admit at least one stationary measure.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 10 / 15

Asymptotically linear Stochastic dynamical Systems

Modulus of Λ as Markov Random walk

Λ1(±1) = ξ1| ξ0 A1| with ξ0 = ± Sn := log|Λn · · · Λ1(ξ0)| = log|

ξn−1

An|+ · · · + log|

ξ0

A1| Sn is a Markov random walk driven by the Markov chain ξn on {+, −}. Then under moment condition Sn n → π+E log

+

A + π−E log

−

A

Corollary

Suppose E[| log ± A|] < ∞ and E[log+ B] < ∞ . If π+E log + A + π−E log − A < 0 then X x

n = Ψn(Xn−1) is positive recurrent

and admit at least one stationary measure.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 10 / 15

Asymptotically linear Stochastic dynamical Systems

Modulus of Λ as Markov Random walk

Λ1(±1) = ξ1| ξ0 A1| with ξ0 = ± Sn := log|Λn · · · Λ1(ξ0)| = log|

ξn−1

An|+ · · · + log|

ξ0

A1| Sn is a Markov random walk driven by the Markov chain ξn on {+, −}. Then under moment condition Sn n → π+E log

+

A + π−E log

−

A

Corollary

Suppose E[| log ± A|] < ∞ and E[log+ B] < ∞ . If π+E log + A + π−E log − A < 0 then X x

n = Ψn(Xn−1) is positive recurrent

and admit at least one stationary measure.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 10 / 15

Asymptotically linear Stochastic dynamical Systems

Modulus of Λ as Markov Random walk

Λ1(±1) = ξ1| ξ0 A1| with ξ0 = ± Sn := log|Λn · · · Λ1(ξ0)| = log|

ξn−1

An|+ · · · + log|

ξ0

A1| Sn is a Markov random walk driven by the Markov chain ξn on {+, −}. Then under moment condition Sn n → π+E log

+

A + π−E log

−

A

Corollary

Suppose E[| log ± A|] < ∞ and E[log+ B] < ∞ . If π+E log + A + π−E log − A < 0 then X x

n = Ψn(Xn−1) is positive recurrent

and admit at least one stationary measure.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 10 / 15

Asymptotically linear Stochastic dynamical Systems

Modulus of Λ as Markov Random walk

Λ1(±1) = ξ1| ξ0 A1| with ξ0 = ± Sn := log|Λn · · · Λ1(ξ0)| = log|

ξn−1

An|+ · · · + log|

ξ0

A1| Sn is a Markov random walk driven by the Markov chain ξn on {+, −}. Then under moment condition Sn n → π+E log

+

A + π−E log

−

A

Corollary

Suppose E[| log ± A|] < ∞ and E[log+ B] < ∞ . If π+E log + A + π−E log − A < 0 then X x

n = Ψn(Xn−1) is positive recurrent

and admit at least one stationary measure.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 10 / 15

Asymptotically linear Stochastic dynamical Systems

Modulus of Λ as Markov Random walk

Λ1(±1) = ξ1| ξ0 A1| with ξ0 = ± Sn := log|Λn · · · Λ1(ξ0)| = log|

ξn−1

An|+ · · · + log|

ξ0

A1| Sn is a Markov random walk driven by the Markov chain ξn on {+, −}. Then under moment condition Sn n → π+E log

+

A + π−E log

−

A

Corollary

Suppose E[| log ± A|] < ∞ and E[log+ B] < ∞ . If π+E log + A + π−E log − A < 0 then X x

n = Ψn(Xn−1) is positive recurrent

and admit at least one stationary measure.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 10 / 15

Asymptotically linear Stochastic dynamical Systems

Λ as linear map

⇋

• | −

A| | + A|

• ⇋
• | −

A| | + A|

• Λ(x) =

−

Ax1[x<0] + + Ax1[x>0] ⇋

• | −

A|1[ −

A≥0]

| + A|1[ +

A<0]

| − A|1[ −

A<0]

| + A|1[ +

A≥0]

• cf. Xn = AnXn−1 + Bn ∈ Rd- Kesten 73 and Guivarch-Le Page 14.

Ps :=

• E[| −

A|s1[ −

A≥0]]

E[| + A|s1[ +

A<0]]

E[| − A|s1[ −

A<0]]

E[| + A|s1[ +

A≥0]]

• ρ(s) = spectral radius of Ps

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 11 / 15

Asymptotically linear Stochastic dynamical Systems

Λ as linear map

⇋

• | −

A| | + A|

• ⇋
• | −

A| | + A|

• Λ(x) =

−

Ax1[x<0] + + Ax1[x>0] ⇋

• | −

A|1[ −

A≥0]

| + A|1[ +

A<0]

| − A|1[ −

A<0]

| + A|1[ +

A≥0]

• cf. Xn = AnXn−1 + Bn ∈ Rd- Kesten 73 and Guivarch-Le Page 14.

Ps :=

• E[| −

A|s1[ −

A≥0]]

E[| + A|s1[ +

A<0]]

E[| − A|s1[ −

A<0]]

E[| + A|s1[ +

A≥0]]

• ρ(s) = spectral radius of Ps

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 11 / 15

Asymptotically linear Stochastic dynamical Systems

Λ as linear map

⇋

• | −

A| | + A|

• ⇋
• | −

A| | + A|

• Λ(x) =

−

Ax1[x<0] + + Ax1[x>0] ⇋

• | −

A|1[ −

A≥0]

| + A|1[ +

A<0]

| − A|1[ −

A<0]

| + A|1[ +

A≥0]

• cf. Xn = AnXn−1 + Bn ∈ Rd- Kesten 73 and Guivarch-Le Page 14.

Ps :=

• E[| −

A|s1[ −

A≥0]]

E[| + A|s1[ +

A<0]]

E[| − A|s1[ −

A<0]]

E[| + A|s1[ +

A≥0]]

• ρ(s) = spectral radius of Ps

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 11 / 15

Asymptotically linear Stochastic dynamical Systems

Λ as linear map

⇋

• | −

A| | + A|

• ⇋
• | −

A| | + A|

• Λ(x) =

−

Ax1[x<0] + + Ax1[x>0] ⇋

• | −

A|1[ −

A≥0]

| + A|1[ +

A<0]

| − A|1[ −

A<0]

| + A|1[ +

A≥0]

• cf. Xn = AnXn−1 + Bn ∈ Rd- Kesten 73 and Guivarch-Le Page 14.

Ps :=

• E[| −

A|s1[ −

A≥0]]

E[| + A|s1[ +

A<0]]

E[| − A|s1[ −

A<0]]

E[| + A|s1[ +

A≥0]]

• ρ(s) = spectral radius of Ps

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 11 / 15

Asymptotically linear Stochastic dynamical Systems

Λ as linear map

⇋

• | −

A| | + A|

• ⇋
• | −

A| | + A|

• Λ(x) =

−

Ax1[x<0] + + Ax1[x>0] ⇋

• | −

A|1[ −

A≥0]

| + A|1[ +

A<0]

| − A|1[ −

A<0]

| + A|1[ +

A≥0]

• cf. Xn = AnXn−1 + Bn ∈ Rd- Kesten 73 and Guivarch-Le Page 14.

Ps :=

• E[| −

A|s1[ −

A≥0]]

E[| + A|s1[ +

A<0]]

E[| − A|s1[ −

A<0]]

E[| + A|s1[ +

A≥0]]

• ρ(s) = spectral radius of Ps

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 11 / 15

Stationary probability for AL-SDS

Tail of Stationary probability for AL-SDS. Case 1

Case

Theorem

Let κ be such that ρ(κ) = 1 and E(|

±

A|κ log+ |

±

A|) < ∞ E(Bα) < ∞ and the law of log ± A is "non-arithmetic". Then ν(t, +∞) ∼ C+t−κ and ν(−∞, −t) ∼ C−t−κ with πα(−)C+ = πα(+)C− and C±(> 0).

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 12 / 15

Stationary probability for AL-SDS

Tail of Stationary probability for AL-SDS. Case 2

Case

Theorem

Let κ+ be such that E(| + A|κ+1 +

A>0) = 1 + moment+non-arithmetic.

Then ν(t, +∞) ∼ C+t−κ+. Let κ− be such that E(| − A|κ−1 −

A>0) = 1 and E(| +

A|κ−1 +

A>0) < 1

+ moment+non-arithmetic. Then ν(t, +∞) ∼ C+t−κ−.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 13 / 15

Stationary probability for AL-SDS

Tail of Stationary probability for AL-SDS. Case 2

Case

Theorem

Let κ+ be such that E(| + A|κ+1 +

A>0) = 1 + moment+non-arithmetic.

Then ν(t, +∞) ∼ C+t−κ+. Let κ− be such that E(| − A|κ−1 −

A>0) = 1 and E(| +

A|κ−1 +

A>0) < 1

+ moment+non-arithmetic. Then ν(t, +∞) ∼ C+t−κ−.

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 13 / 15

Stationary probability for AL-SDS

Some ideas of the proof

Lemma

If Λn · · · Λ1(x) → 0 in probability for all x ∈ R and 0 ∈ supp(φ) then ν(φ) =

∞

• n=0

E [gφ(Λ1 · · · Λn)] where gφ(Λ) = E(φ(Λ(Ψ1(X0))) − φ(Λ(Λ1(X0)) when X0 ∼ ν

1 Renewal theorem for Λ1 · · · Λn 2 If φ is Lipschitz gφ(t) = E(φ(tΨ1(X0))) − φ(tΛ1(X0)) is directly

Riemann integrable

3 C± > 0 Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 14 / 15

Stationary probability for AL-SDS

Some ideas of the proof

Lemma

If Λn · · · Λ1(x) → 0 in probability for all x ∈ R and 0 ∈ supp(φ) then ν(φ) =

∞

• n=0

E [gφ(Λ1 · · · Λn)] where gφ(Λ) = E(φ(Λ(Ψ1(X0))) − φ(Λ(Λ1(X0)) when X0 ∼ ν

1 Renewal theorem for Λ1 · · · Λn 2 If φ is Lipschitz gφ(t) = E(φ(tΨ1(X0))) − φ(tΛ1(X0)) is directly

Riemann integrable

3 C± > 0 Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 14 / 15

Stationary probability for AL-SDS

On going work and open questions

Critical case π+E log + A + π−E log − A = 0 Multidimensional situation Uniqueness of the invariant probability

Thank you

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 15 / 15

Stationary probability for AL-SDS

On going work and open questions

Critical case π+E log + A + π−E log − A = 0 Multidimensional situation Uniqueness of the invariant probability

Thank you

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 15 / 15

Stationary probability for AL-SDS

On going work and open questions

Critical case π+E log + A + π−E log − A = 0 Multidimensional situation Uniqueness of the invariant probability

Thank you

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 15 / 15

Stationary probability for AL-SDS

On going work and open questions

Critical case π+E log + A + π−E log − A = 0 Multidimensional situation Uniqueness of the invariant probability

Thank you

Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 15 / 15