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Motivation On the stationary distribution of a GWIRE On the stationary process Subcritical Galton-Watson branching processes with immigration in random environment Péter Kevei University of Szeged Probability and Analysis 2019, Bedlewo Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Outline Motivation Kesten - Kozlov - Spitzer: RWRE model Galton-Watson processes in deterministic environment On the stationary distribution of a GWIRE Preliminaries Tail asymptotics Proofs On the stationary process Tail process Point process convergence Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process This is ongoing joint work with Bojan Basrak (Zagreb). Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Kesten - Kozlov - Spitzer: RWRE model An RWRE model RWRE model by Kozlov and Solomon: ◮ { α i } i ∈ Z iid random variables with values in [ 0 , 1 ] ◮ A = σ ( α i : i ∈ Z ) generated σ -algebra ◮ X 0 = 0 and P ( X n + 1 = X n + 1 |A , X 0 , . . . , X n ) = α i on { X n = i } P ( X n + 1 = X n − 1 |A , X 0 , . . . , X n ) = 1 − α i on { X n = i } ◮ X n is not a Markov process ◮ X n → ∞ a.s., but X n / n → 0 a.s. Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Kesten - Kozlov - Spitzer: RWRE model KKS result Let T n = min { k : X k = n } = first hitting time of n . Assume E log 1 − α < 0 , (positive drift) α � κ � 1 − α E = 1 , (Cramér’s condition) α � κ � 1 − α 1 − α log + < ∞ , κ > 0 , E α α log 1 − α is non-arithmetic (not concentrated on δ Z for any δ ). α These are the assumption in Goldie’s implicit renewal theorem. Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Kesten - Kozlov - Spitzer: RWRE model KKS result Theorem (Kesten & Kozlov & Spitzer (1975)) Then, for κ ∈ ( 0 , 2 ) , n − 1 /κ ( T n − A n ) D → κ − stable rv. where A n ≡ 0 for κ < 1 , A n = nc 1 for κ > 1 . For κ > 2 n − 1 / 2 ( T n − nc ) D → N ( 0 , 1 ) . Moreover, n − κ ( X n − B n ) also converges. Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Kesten - Kozlov - Spitzer: RWRE model Branching connection ◮ U n i = number of steps before T n from i to i − 1; −∞ < i ≤ n − 1. ◮ T n = n + 2 � i ≤ n − 1 U n i . ◮ It is enough to handle � n i = 1 U n i . Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Kesten - Kozlov - Spitzer: RWRE model Branching connection ◮ U n i = number of steps before T n from i to i − 1; −∞ < i ≤ n − 1. ◮ U n j given A , U n j + 1 , . . . , U n n − 1 is the sum of U n j + 1 + 1 iid random variables with joint distribution P ( V = k ) = α j ( 1 − α j ) k , k = 0 , 1 , . . . . ◮ U is a GW branching process with random offspring and immigration distribution. Given the environment α both the offspring and the immigration distribution is geometric with parameter α . Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Galton-Watson processes in deterministic environment Subcritical GWI Let X 0 = 0, and X n A ( n + 1 ) � X n + 1 = + B n + 1 =: θ n + 1 ◦ X n + B n + 1 , n ≥ 0 , i i = 1 where the offsprings { A ( n ) : i = 1 , 2 , . . . , n = 1 , 2 , . . . } are iid, i and independently, { B n : n = 1 , 2 , . . . } iid. Subcritical: E A < 1. Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Galton-Watson processes in deterministic environment Stationary distribution - existence Theorem (Quine (1970), Foster & Williamson (1971)) If m = E A < 1 and E log B < ∞ then there exists a unique stationary distribution in the form ∞ � X ∞ = B 1 + θ 1 ◦ B 2 + θ 1 ◦ θ 2 ◦ B 3 + . . . = Π i ◦ B i + 1 . i = 0 Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Galton-Watson processes in deterministic environment Stationary distribution - tails Theorem (Basrak & Kulik & Palmowski (2013)) (i) If m = E A < 1 , E A 2 < ∞ and P ( B > x ) is regularly varying with index − α ∈ ( − 2 , 0 ) , then P ( X ∞ > x ) ∼ c P ( B > x ) , c > 0 . (ii) If m = E A < 1 , and P ( A > x ) is regularly varying with index α ∈ ( − 2 , − 1 ) , and P ( B > x ) ∼ c ′ P ( A > x ) , c ′ ≥ 0 then P ( X ∞ > x ) ∼ c P ( A > x ) , c > 0 . Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Preliminaries GWIRE process - notation ◮ ∆ the space of probability measures on N = { 0 , 1 , . . . } ◮ ( ǫ, ̟ ) , ( ǫ 1 , ̟ 1 ) , ( ǫ 2 , ̟ 2 ) , . . . iid random elements in ∆ 2 (the environment) ◮ X 0 = 0, and X n � A ( n + 1 ) X n + 1 = + B n + 1 =: θ n + 1 ◦ X n + B n + 1 , n ≥ 0 , i i = 1 where, conditioned on the environments E , I , the variables { A ( n ) , B n : i = 1 , 2 , . . . , n = 1 , 2 , . . . } are independent, for n i fixed ( A ( n ) ) i = 1 , 2 ,... are iid with distribution ǫ n , and B n has i distribution ̟ n . Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Preliminaries GWIRE process - notation ◮ ∆ the space of probability measures on N = { 0 , 1 , . . . } i = 1 A ( n + 1 ) ◮ X n + 1 = � X n + B n + 1 =: θ n + 1 ◦ X n + B n + 1 i ◮ m ( δ ) = � ∞ i = 1 i δ ( { i } ) for δ ∈ ∆ . ◮ Subcritical branching: E log m ( ǫ ) < 0. Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Preliminaries Existence of the stationary distribution Theorem (Key (1987)) If E log m ( ǫ ) < 0 (offspring) and E log + m ( ̟ ) < ∞ (immigration) then a unique stationary distribution exists: ∞ � X ∞ = B 1 + θ 1 ◦ B 2 + θ 1 ◦ θ 2 ◦ B 3 + . . . = Π i ◦ B i + 1 . i = 0 Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Tail asymptotics Goldie’s setup - the fixed point equation X n A ( n + 1 ) � X n + 1 = + B n + 1 =: θ n + 1 ◦ X n + B n + 1 , n ≥ 0 , i i = 1 The stationary distribution satisfies the corresponding fixed point equation X X D � = A i + B = θ ◦ X + B =: Ψ( X ) , i = 1 ( θ, B ) and X on the right-hand side are independent. Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Tail asymptotics Assumptions We are interested in the tail behavior, so need more assumption: ◮ Cramér’s condition: E m ( ǫ ) κ = 1 for some κ > 0. ◮ log m ( ǫ ) is nonarithmetic (not concentrated on δ Z ) ◮ E A κ ∨ 2 < ∞ , (by Jensen: E m ( ǫ ) κ ∨ 2 < ∞ ), and E B κ < ∞ . For κ > 1 assume further that E A κ + δ < ∞ , E B κ + δ < ∞ for some δ > 0. Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged

Motivation On the stationary distribution of a GWIRE On the stationary process Tail asymptotics Main result Theorem (Basrak & K (2019+)) Then P ( X ∞ > x ) ∼ Cx − κ as x → ∞ , where 1 κ E m ( ǫ ) κ log m ( ǫ ) E [Ψ( X ∞ ) κ − m ( ǫ ) κ X κ C = ∞ ] ≥ 0 . Moreover, C > 0 for κ ≥ 1 . Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged