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 X n = Ψ n ( X n − 1 ) = Ψ n ◦ . . . ◦ Ψ 1 ( X 0 ) , where Ψ n : R → R are chosen randomly and independently according to a same law µ . Examples : Affine transformation Ψ n ( x ) = A n x + B n i.i.d X n = A n X n − 1 + B n Reflected random walks Ψ n ( x ) = | x + U n | Logistic Model Ψ n ( x ) = R n x (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 X n = Ψ n ( X n − 1 ) = Ψ n ◦ . . . ◦ Ψ 1 ( X 0 ) , where Ψ n : R → R are chosen randomly and independently according to a same law µ . Examples : Affine transformation Ψ n ( x ) = A n x + B n i.i.d X n = A n X n − 1 + B n Reflected random walks Ψ n ( x ) = | x + U n | Logistic Model Ψ n ( x ) = R n x (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 X n = Ψ n ( X n − 1 ) = Ψ n ◦ . . . ◦ Ψ 1 ( X 0 ) , where Ψ n : R → R are chosen randomly and independently according to a same law µ . Examples : Affine transformation Ψ n ( x ) = A n x + B n i.i.d X n = A n X n − 1 + B n Reflected random walks Ψ n ( x ) = | x + U n | Logistic Model Ψ n ( x ) = R n x (1 − x ) Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 2 / 15

Stationary measure of Stochastic dynamical system Questions X n = Ψ n ( X n − 1 ) = Ψ n ◦ . . . ◦ Ψ 1 ( X 0 ) , Questions : Positive Recurrence and existence of a Stationary probability ν : X 0 ∼ ν independent of Ψ 1 than X 0 = d Ψ 1 ( X 0 ) = X 1 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 X n = Ψ n ( X n − 1 ) = Ψ n ◦ . . . ◦ Ψ 1 ( X 0 ) , Questions : Positive Recurrence and existence of a Stationary probability ν : X 0 ∼ ν independent of Ψ 1 than X 0 = d Ψ 1 ( X 0 ) = X 1 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 X n = Ψ n ( X n − 1 ) = Ψ n ◦ . . . ◦ Ψ 1 ( X 0 ) , Questions : Positive Recurrence and existence of a Stationary probability ν : X 0 ∼ ν independent of Ψ 1 than X 0 = d Ψ 1 ( X 0 ) = X 1 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 X n = A n X n − 1 + B n � log A � log + | B | � < 0 and E � < ∞ , Contractive case : If E X n 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 X n = A n X n − 1 + B n � log A � log + | B | � < 0 and E � < ∞ , Contractive case : If E X n is positive recurrent and has a unique stationary probability . � log A � = 0 and E � || log A | + | log | B ||| 2+ ǫ � < ∞ Critical : If E X n 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 X n = A n X n − 1 + B n � log A � log + | B | � < 0 and E � < ∞ , Contractive case : If E X n is positive recurrent and has a unique stationary probability . � log A � = 0 and E � || log A | + | log | B ||| 2+ ǫ � < ∞ Critical : If E X n is null recurrent and has a unique infinite invariant measure [BBE97]. � log A � > 0 and E � log + | B | � < ∞ Divergent case : If E X n 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 X n = A n X n − 1 + B n Contractive case E log A < 0 If there exists κ > 0 such that E [ | A | κ ] = 1, � < ∞ and E [ | B | κ ] < ∞ and log A non-arithmetic. � | A | κ log + | A | E i.e. ν ( dx ) ∼ C + dx Then ν ( z , + ∞ ) ∼ C + z − κ , x κ +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 X n = A n X n − 1 + B n Contractive case E log A < 0 If there exists κ > 0 such that E [ | A | κ ] = 1, � < ∞ and E [ | B | κ ] < ∞ and log A non-arithmetic. � | A | κ log + | A | E i.e. ν ( dx ) ∼ C + dx Then ν ( z , + ∞ ) ∼ C + z − κ , x κ +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 X n = A n X n − 1 + B n Contractive case E log A < 0 If there exists κ > 0 such that E [ | A | κ ] = 1, � < ∞ and E [ | B | κ ] < ∞ and log A non-arithmetic. � | A | κ log + | A | E i.e. ν ( dx ) ∼ C + dx Then ν ( z , + ∞ ) ∼ C + z − κ , x κ +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 X n = A n X n − 1 + B n Contractive case E log A < 0 If there exists κ > 0 such that E [ | A | κ ] = 1, � < ∞ and E [ | B | κ ] < ∞ and log A non-arithmetic. � | A | κ log + | A | E i.e. ν ( dx ) ∼ C + dx Then ν ( z , + ∞ ) ∼ C + z − κ , x κ +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 X n = Ψ n ( X n − 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 Ψ n ( x ) = R n x (1 − x ) - Athreya ... ◮ AR(1)-model with ARCH(1) errors Ψ n ( x ) = α n x + ε n ( β n + λ n x 2 ) 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 X n = Ψ n ( X n − 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 Ψ n ( x ) = R n x (1 − x ) - Athreya ... ◮ AR(1)-model with ARCH(1) errors Ψ n ( x ) = α n x + ε n ( β n + λ n x 2 ) 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 , + ∞ ) Ψ n is a.s. asymptotically linear if | Ψ n ( x ) − Λ n ( x ) | ≤ B n ∀ x If | Λ | := Lip (Λ) = max { − A , + A } : n � | Ψ n · · · Ψ 1 ( x ) − Λ n · · · Λ 1 ( x ) | ≤ Y n := | Λ n · · · Λ k − 1 | B k k =1 Alsmeyer, Brofferio, Buraczewski Stationary probability of SDS June 2016 7 / 15