What is the branching random walk and the Biggins martingale? Uniform integrability of the On the rate of convergence of the Biggins martingale The rate of convergence Biggins martingale in supercritical branching random walks Alexander Iksanov, Kyiv, Ukraine Conference ‘Probability and Analysis’, May 15-19, 2017, B¸ edlewo, Poland Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 1/15
What is the branching random walk and the Biggins martingale? What is the branching random walk and the Biggins martingale? Uniform integrability of the M – a point process on R ; L := M ( R ) , possibly infinite. Biggins martingale By branching random walk (BRW) on R is meant the sequence of The rate of convergence point processes ( M n ) n ∈ N 0 , where for any Borel set B ⊂ R , M 0 ( B ) := ✶ { 0 ∈ B } , � M n +1 ( B ) := M n,r ( B − A n,r ) , n ∈ N 0 . r Here ( A n,r ) are the points of M n , and ( M n,r ) are independent copies of M . Supercriticality : if P { L < ∞} = 1 it is additionally assumed that E L > 1 . Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 2/15
What is the branching random walk and the Biggins martingale? What is the branching M – a point process on R ; L := M ( R ) , possibly infinite. random walk and the Biggins martingale? By branching random walk (BRW) on R is meant the sequence of point processes ( M n ) n ∈ N 0 , where for any Borel set B ⊂ R , Uniform integrability of the Biggins martingale M 0 ( B ) := ✶ { 0 ∈ B } , The rate of convergence � M n +1 ( B ) := M n,r ( B − A n,r ) , n ∈ N 0 . r Here ( A n,r ) are the points of M n , and ( M n,r ) are independent copies of M . Supercriticality : if P { L < ∞} = 1 it is additionally assumed that E L > 1 . Assume that for some γ > 0 � e γx M (d x ) ∈ (0 , ∞ ) m ( γ ) := E R and set � W n := W n ( γ ) = m ( γ ) − n e γx M n (d x ) , n ∈ N . R The sequence ( W n , σ ( M 1 , . . . , M n )) n ∈ N is a nonnegative martingale (Kingman (1975) and Biggins (1977)) which is called the Biggins martingale. Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 3/15
Uniform integrability of the Biggins martingale: What is the branching random walk and the Biggins martingale? Uniform integrability of the Sufficient conditions for uniform integrability of the Biggins martingale Biggins martingale were obtained by The rate of convergence Biggins (1977) Liu (1997) Lyons (1997) Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 4/15
Uniform integrability of the Biggins martingale: What is the branching random walk and the Biggins martingale? Uniform integrability of the Sufficient conditions for uniform integrability of the Biggins martingale Biggins martingale were obtained by The rate of convergence Biggins (1977) Liu (1997) Lyons (1997) Let ( M k , Q k ) k ∈ N be independent copies of an R 2 -valued random vector ( M, Q ) with arbitrary dependence between M and Q . If the series Z := Q 1 + M 1 Q 2 + M 1 M 2 Q 3 + . . . converges a.s., the random variable Z is called perpetuity. Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 4/15
Uniform integrability of the Biggins martingale: What is the branching Let ( M k , Q k ) k ∈ N be independent copies of an R 2 -valued random vector ( M, Q ) with random walk and the Biggins arbitrary dependence between M and Q . If the series martingale? Uniform integrability of the Z := Q 1 + M 1 Q 2 + M 1 M 2 Q 3 + . . . Biggins martingale converges a.s., the random variable Z is called perpetuity. The rate of convergence M – a point process with points ( X i ) ; L := M ( R ) Standing assumption: there exists γ > 0 such that m ( γ ) := E � L i =1 e γXi < ∞ . Theorem (Alsmeyer & I. (2009)) The Biggins martingale is uniformly integrable if, and only if, Z ∗ := Q ∗ 1 + M ∗ 1 Q ∗ 2 + M ∗ 1 M ∗ 2 Q ∗ 3 + . . . < ∞ a.s. , where � e γX i L L e γX i e γX j � P { ( M ∗ , Q ∗ ) ∈ A } = E � � m ( γ ) ✶ A m ( γ ) , m ( γ ) i =1 j =1 for any Borel set A in R 2 . Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 5/15
Uniform integrability of the Biggins martingale: What is the branching Theorem (Alsmeyer & I. (2009)) random walk and the Biggins The Biggins martingale ( W n ) n ∈ N is uniformly integrable if, and only if, martingale? Z ∗ := Q ∗ 1 + M ∗ 1 Q ∗ 2 + M ∗ 1 M ∗ 2 Q ∗ 3 + . . . < ∞ a.s. , Uniform integrability of the Biggins martingale where � e γXi e γXj L e γXi L The rate of convergence � P { ( M ∗ , Q ∗ ) ∈ A } = E � � , ✶ A m ( γ ) m ( γ ) m ( γ ) i =1 j =1 for any Borel set A in R 2 . According to Goldie & Maller (2000), Z ∗ < ∞ a.s. if, and only if, n →∞ M ∗ 1 M ∗ 2 · . . . · M ∗ lim n = 0 a.s. and E J (log + Q ∗ ) = E W 1 J (log + W 1 ) < ∞ , where x J ( x ) := x > 0 . 0 P {− log M ∗ > y } d y , � x In particular, if E log M ∗ ∈ ( −∞ , 0) , then E W 1 log + W 1 < ∞ is a necessary and sufficient condition for uniform integrability of the Biggins martingale. Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 6/15
The rate of convergence: CLT for the tail of the Biggins martingale What is the branching random walk and the Biggins martingale? � R e γx M (d x ) , γ > 0 ; m ( γ ) = E W n ( γ ) = ( m ( γ )) − n � R e γx M n (d x ) , n ∈ N ; Uniform integrability of the Biggins martingale n →∞ W n ( γ ) = W ∞ ( γ ) a.s. lim The rate of convergence CLT for the tail of the Biggins Theorem (I. & Kabluchko (2016)) martingale Suppose that m (1) = 1 , σ 2 := Var W 1 (1) ∈ (0 , ∞ ) and m (2) < 1 . Relevant literature LIL for the tail of the Biggins Then, as n → ∞ , martingale Other results � W ∞ (1) − W n + r (1) Exponentially fast a.s. convergence � �� � f . d . for the tail of the Biggins ⇒ v 2 W ∞ (2) U r , martingale ( m (2)) ( n + r ) / 2 r ∈ N 0 r ∈ N 0 where v 2 := Var W ∞ (1) = σ 2 (1 − m (2)) − 1 , and ( U r ) r ∈ N 0 is a stationary zero-mean Gaussian sequence which is independent of W ∞ (2) and has the covariance Cov ( U r , U s ) = ( m (2)) | r − s | / 2 , r, s ∈ N 0 . Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 7/15
The rate of convergence: CLT for the tail of the Biggins martingale R e γx M (d x ) , γ > 0 ; What is the branching � m ( γ ) = E random walk and the Biggins W n ( γ ) = ( m ( γ )) − n � R e γx M n (d x ) , n ∈ N ; martingale? n →∞ W n ( γ ) = W ∞ ( γ ) a.s. lim Uniform integrability of the Biggins martingale Theorem (I. & Kabluchko (2016)) Suppose that m (1) = 1 , σ 2 := Var W 1 (1) ∈ (0 , ∞ ) and m (2) < 1 . Then, as The rate of convergence n → ∞ , CLT for the tail of the Biggins martingale � � W ∞ (1) − W n + r (1) �� � Relevant literature f . d . ⇒ v 2 W ∞ (2) U r , LIL for the tail of the Biggins ( m (2)) ( n + r ) / 2 r ∈ N 0 r ∈ N 0 martingale Other results Exponentially fast a.s. convergence where v 2 := Var W ∞ (1) = σ 2 (1 − m (2)) − 1 , and ( U r ) r ∈ N 0 is a stationary zero-mean for the tail of the Biggins martingale Gaussian sequence which is independent of W ∞ (2) and has the covariance Cov ( U r , U s ) = ( m (2)) | r − s | / 2 , r, s ∈ N 0 . Corollary (I. & Kabluchko (2016)) Suppose that m (1) = 1 , Var W 1 (1) ∈ (0 , ∞ ) and m (2) < 1 . Then, as n → ∞ , W ∞ (1) − W n (1) d → normal (0 , v 2 W ∞ (2)) . ( m (2)) n/ 2 Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 8/15
The rate of convergence: relevant literature What is the branching random walk and the Biggins Here are several related results, with pointers to the literature. martingale? Uniform integrability of the a CLT for the tail martingale of a Galton-Watson Biggins martingale process – Athreya (1968) and Heyde (1970) The rate of convergence a functional CLT for the tail martingale of a Galton-Watson CLT for the tail of the Biggins martingale process – Heyde & Brown (1971) Relevant literature LIL for the tail of the Biggins CLT’s for multitype branching processes – Kesten & Stigum martingale (1966), Athreya (1968) and Asmussen & Keiding (1978) Other results Exponentially fast a.s. convergence for the tail of the Biggins a CLT for the tail martingale of a weighted branching martingale processes – R¨ osler, Topchii & Vatutin (2002) a CLT for the tail martingale of a complex-valued branching Brownian motion – Hartung & Klimovsky (2017+) CLT’s for tail martingales associated with random trees – Neininger (2015), Gr¨ ubel & Kabluchko (2016) and Sulzbach (2017) CLT’s for branching diffusions and superprocesses – Adamczak & Mi� lo´ s (2015) and Ren, Song & Zhang (2015) Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 9/15
Recommend
More recommend
