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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: ◮ {αi}i∈Z iid random variables with values in [0, 1] ◮ A = σ(αi : i ∈ Z) generated σ-algebra ◮ X0 = 0 and P(Xn+1 = Xn + 1|A, X0, . . . , Xn) = αi on {Xn = i} P(Xn+1 = Xn − 1|A, X0, . . . , Xn) = 1 − αi on {Xn = i} ◮ Xn is not a Markov process ◮ Xn → ∞ a.s., but Xn/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 Tn = min{k : Xk = n} = first hitting time of n. Assume E log 1 − α α < 0, (positive drift) E 1 − α α κ = 1, (Cramér’s condition) E 1 − α α κ log+ 1 − α α < ∞, κ > 0, log 1−α

α

is non-arithmetic (not concentrated on δZ for any δ). 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(Tn − An) D → κ − stable rv. where An ≡ 0 for κ < 1, An = nc1 for κ > 1. For κ > 2 n−1/2(Tn − nc) D → N(0, 1). Moreover, n−κ(Xn − Bn) 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

◮ Un

i = number of steps before Tn from i to i − 1;

−∞ < i ≤ n − 1. ◮ Tn = n + 2

i≤n−1 Un i .

◮ It is enough to handle n

i=1 Un 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

◮ Un

i = number of steps before Tn from i to i − 1;

−∞ < i ≤ n − 1. ◮ Un

j given A, Un j+1, . . . , Un n−1 is the sum of Un j+1 + 1 iid

random variables with joint distribution P(V = k) = αj(1 − αj)k, k = 0, 1, . . . . ◮ U is a GW branching process with random offspring and immigration distribution. Given the environment α both the

• ffspring 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 X0 = 0, and Xn+1 =

Xn

• i=1

A(n+1)

i

+ Bn+1 =: θn+1 ◦ Xn + Bn+1, n ≥ 0, where the offsprings {A(n)

i

: i = 1, 2, . . . , n = 1, 2, . . .} are iid, and independently, {Bn : n = 1, 2, . . .} iid. Subcritical: EA < 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 = EA < 1 and E log B < ∞ then there exists a unique stationary distribution in the form X∞ = B1 + θ1 ◦ B2 + θ1 ◦ θ2 ◦ B3 + . . . =

∞

• i=0

Πi ◦ Bi+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 - tails

Theorem (Basrak & Kulik & Palmowski (2013))

(i) If m = EA < 1, EA2 < ∞ 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 = EA < 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, ̟), (ǫ1, ̟1), (ǫ2, ̟2), . . . iid random elements in ∆2 (the environment) ◮ X0 = 0, and Xn+1 =

Xn

• i=1

A(n+1)

i

+ Bn+1 =: θn+1 ◦ Xn + Bn+1, n ≥ 0, where, conditioned on the environments E, I, the variables {A(n)

i

, Bn : i = 1, 2, . . . , n = 1, 2, . . .} are independent, for n fixed (A(n)

i

)i=1,2,... are iid with distribution ǫn, and Bn has 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, . . .} ◮ Xn+1 = Xn

i=1 A(n+1) i

+ Bn+1 =: θn+1 ◦ Xn + Bn+1 ◮ m(δ) = ∞

i=1 iδ({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∞ = B1 + θ1 ◦ B2 + θ1 ◦ θ2 ◦ B3 + . . . =

∞

• i=0

Πi ◦ Bi+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 Tail asymptotics

Goldie’s setup - the fixed point equation

Xn+1 =

Xn

• i=1

A(n+1)

i

+ Bn+1 =: θn+1 ◦ Xn + Bn+1, n ≥ 0, The stationary distribution satisfies the corresponding fixed point equation X D =

X

• i=1

Ai + B = θ ◦ X + B =: Ψ(X), (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: Em(ǫ)κ = 1 for some κ > 0. ◮ log m(ǫ) is nonarithmetic (not concentrated on δZ) ◮ EAκ∨2 < ∞, (by Jensen: Em(ǫ)κ∨2 < ∞), and EBκ < ∞. For κ > 1 assume further that EAκ+δ < ∞, EBκ+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 C = 1 κEm(ǫ)κ log m(ǫ)E [Ψ(X∞)κ − m(ǫ)κX κ

∞] ≥ 0.

Moreover, C > 0 for κ ≥ 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 Proofs

Moments of the stationary distribution - deterministic

Lemma

Let Xn be a subcritical GWI Xn+1 = Xn

i=1 A(n+1) i

+ Bn+1 =: θn+1 ◦ Xn + Bn, X∞ = B1 + θ1 ◦ B2 + θ1 ◦ θ2 ◦ B3 + . . . =

∞

• i=0

Πi ◦ Bi+1. EX α

∞ < ∞ whenever EBα < ∞ and EAα < ∞, α ≥ 1.

◮ Quine (1970): α = 2; ◮ Barczy, Nedényi, Pap (2018): α = 2, 3. ◮ Sz˝ ucs (2014): under ergodicity conditions, general α ◮ K - Wiandt (2019+): multitype case

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 Proofs

Proof

◮ Enough to show that Mα(n) := E(θ1 ◦ θ2 ◦ . . . θn ◦ Bn+1)α decreases exponentially. ◮ ∀λ ∈ (µα, 1) ∃n0 ∈ N such that for all n ≥ n0 µαE n

i=1 Ai

nµ α < λ. ◮ C0 = E n0

i=1 Ai

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 Proofs

Proof

◮ µαE n

i=1 Ai

nµ

α < λ for n ≥ n0. ◮ C0 = E n0

i=1 Ai

α ◮ Y independent of A1, A2, . . ., E Y

• i=1

Ai α =

∞

• n=1

P(Y = n)E n

• i=1

Ai α ≤ P(Y ≥ 1)C0 +

∞

• n=n0

P(Y = n)nαλ ≤ C0EY + λEY 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 Proofs

Proof

Mα(n) := E(θ1 ◦ θ2 ◦ . . . ◦ θn ◦ Bn+1)α E Y

i=1 Ai

α ≤ C0EY + λEY α implies the recursion Mα(n) ≤ C0M1(n − 1) + λMα(n − 1). Iterating the inequality, we obtain Mα(n) ≤ C0

n−1

• i=0

λiµn−i + λnEBα.

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 Proofs

Moments of the stationary distribution - random

Lemma

Assume that Em(ǫ)α < 1, EAα < ∞, and EBα < ∞. Then EX α

∞ < ∞.

By the conditional Jensen inequality Em(ǫ)t ≤ EAt for t ≥ 1, while for t ≤ 1 Em(ǫ)t ≥ EAt.

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 Proofs

Goldie’s condition

Ψ(n) = n

i=1 Ai + B. The fixed point equation

X D = Ψ(X),

Lemma

Assume Em(ǫ)κ = 1, EAκ∨2 < ∞, (Em(ǫ)κ∨2 < ∞), and EBκ < ∞. For κ > 1 assume further that EAκ+δ < ∞, EBκ+δ < ∞ for some δ > 0. Then E |Ψ(X)κ − (m(ǫ)X)κ| < ∞.

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 process

The setup

◮ Xn+1 = Xn

i=1 A(n+1) i

+ Bn+1 =: θn+1 ◦ Xn + Bn+1, n ∈ Z, strictly stationary. ◮ Assume P(X0 > x) ∼ cx−κ, c > 0. ◮ Let an such that nP(X0 > an) ∼ 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 Tail process

Tail process

Theorem

For any integers k, ℓ ≥ 0 L X0 x , X−k X0 , . . . , X0 X0 , . . . , Xℓ X0

• X0 > x
• d

− → (Y0, Θ−k, . . . , Θ0, . . . , Θℓ) . where Θn is a multiplicative random walk, independent of Y0, and P(Y0 > u) = u−κ, u ≥ 1. Consequence of a general results by Janssen & Segers (2014).

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 Point process convergence

Anticlustering and A′(an) conditions

Anticlustering: If rn = o(n), then for any u > 0 lim

k→∞ lim sup n→∞ P

• max

k≤|t|≤rn

Xt > anu|X0 > anu

• = 0.

Ergodicity implies (A′(an)): E exp

• −

n

• i=1

f i n, Xi an

• −

kn

• k=1

E exp

• −

rn

• i=1

f krn n , Xi an

• → 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 Point process convergence

Point process convergence

Anticlustering and A′(an) implies for each u > 0

n

• i=1

δ(i/n,Xi/an)

• [0,1]×(R\[−u,u])

D

− → N(u).

Subcritical Galton-Watson branching processes with immigration in random environment University of Szeged