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What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

On the rate of convergence of the 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? Uniform integrability of the Biggins martingale The rate of convergence

What is the branching random walk and the Biggins martingale?

M– a point process on R; L := M(R), possibly infinite. By branching random walk (BRW) on R is meant the sequence of point processes (Mn)n∈N0, where for any Borel set B ⊂ R, M0(B) := ✶{0∈B}, Mn+1(B) :=

• r

Mn,r(B − An,r), n ∈ N0. Here (An,r) are the points of Mn, and (Mn,r) are independent copies of M. Supercriticality : if P{L < ∞} = 1 it is additionally assumed that EL > 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? Uniform integrability of the Biggins martingale The rate of convergence

What is the branching random walk and the Biggins martingale?

M– a point process on R; L := M(R), possibly infinite. By branching random walk (BRW) on R is meant the sequence of point processes (Mn)n∈N0 , where for any Borel set B ⊂ R, M0(B) := ✶{0∈B}, Mn+1(B) :=

• r

Mn,r(B − An,r), n ∈ N0. Here (An,r) are the points of Mn, and (Mn,r) are independent copies of M. Supercriticality : if P{L < ∞} = 1 it is additionally assumed that EL > 1.

Assume that for some γ > 0 m(γ) := E

• R

eγxM(dx) ∈ (0, ∞) and set Wn := Wn(γ) = m(γ)−n

• R

eγxMn(dx), n ∈ N. The sequence (Wn, σ(M1, . . . , Mn))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

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

Uniform integrability of the Biggins martingale:

Sufficient conditions for uniform integrability of the Biggins martingale were obtained by 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

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

Uniform integrability of the Biggins martingale:

Sufficient conditions for uniform integrability of the Biggins martingale were obtained by Biggins (1977) Liu (1997) Lyons (1997) Let (Mk, Qk)k∈N be independent copies of an R2-valued random vector (M, Q) with arbitrary dependence between M and Q. If the series Z := Q1 + M1Q2 + M1M2Q3 + . . . 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

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

Uniform integrability of the Biggins martingale:

Let (Mk, Qk)k∈N be independent copies of an R2-valued random vector (M, Q) with arbitrary dependence between M and Q. If the series Z := Q1 + M1Q2 + M1M2Q3 + . . . converges a.s., the random variable Z is called perpetuity. M– a point process with points (Xi); L := M(R) Standing assumption: there exists γ > 0 such that m(γ) := E L

i=1 eγXi < ∞.

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 P{(M∗, Q∗) ∈ A} = E

L

• i=1

eγXi m(γ) ✶A eγXi m(γ) ,

L

• j=1

eγXj m(γ)

• for any Borel set A in R2.

Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 5/15

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

Uniform integrability of the Biggins martingale:

Theorem (Alsmeyer & I. (2009)) The Biggins martingale (Wn)n∈N is uniformly integrable if, and only if, Z∗ := Q∗

1 + M∗ 1 Q∗ 2 + M∗ 1 M∗ 2 Q∗ 3 + . . . < ∞

a.s., where P{(M∗, Q∗) ∈ A} = E

L

• i=1

eγXi m(γ) ✶A eγXi m(γ) ,

L

• j=1

eγXj m(γ)

• for any Borel set A in R2.

According to Goldie & Maller (2000), Z∗ < ∞ a.s. if, and only if, lim

n→∞M∗ 1 M∗ 2 · . . . · M∗ n = 0

a.s. and EJ(log+ Q∗) = EW1J(log+ W1) < ∞, where J(x) := x x

0 P{− log M∗ > y}dy ,

x > 0. In particular, if E log M∗ ∈ (−∞, 0), then EW1 log+ W1 < ∞ 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

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

CLT for the tail of the Biggins martingale Relevant literature LIL for the tail of the Biggins martingale Other results Exponentially fast a.s. convergence for the tail of the Biggins martingale

The rate of convergence:

CLT for the tail of the Biggins martingale

m(γ) = E

• R eγxM(dx), γ > 0;

Wn(γ) = (m(γ))−n

R eγxMn(dx), n ∈ N;

lim

n→∞Wn(γ) = W∞(γ) a.s.

Theorem (I. & Kabluchko (2016)) Suppose that m(1) = 1, σ2 := Var W1(1) ∈ (0, ∞) and m(2) < 1. Then, as n → ∞, W∞(1) − Wn+r(1) (m(2))(n+r)/2

• r∈N0

f.d.

⇒

• v2W∞(2) Ur
• r∈N0

, where v2 := Var W∞(1) = σ2(1 − m(2))−1, and (Ur)r∈N0 is a stationary zero-mean Gaussian sequence which is independent of W∞(2) and has the covariance Cov (Ur, Us) = (m(2))|r−s|/2, r, s ∈ N0.

Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 7/15

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

CLT for the tail of the Biggins martingale Relevant literature LIL for the tail of the Biggins martingale Other results Exponentially fast a.s. convergence for the tail of the Biggins martingale

The rate of convergence:

CLT for the tail of the Biggins martingale

m(γ) = E

• R eγxM(dx), γ > 0;

Wn(γ) = (m(γ))−n

R eγxMn(dx), n ∈ N;

lim

n→∞Wn(γ) = W∞(γ) a.s.

Theorem (I. & Kabluchko (2016)) Suppose that m(1) = 1, σ2 := Var W1(1) ∈ (0, ∞) and m(2) < 1. Then, as n → ∞,

• W∞(1) − Wn+r(1)

(m(2))(n+r)/2

• r∈N0

f.d.

⇒

• v2W∞(2) Ur
• r∈N0

, where v2 := Var W∞(1) = σ2(1 − m(2))−1, and (Ur)r∈N0 is a stationary zero-mean Gaussian sequence which is independent of W∞(2) and has the covariance Cov (Ur, Us) = (m(2))|r−s|/2, r, s ∈ N0.

Corollary (I. & Kabluchko (2016)) Suppose that m(1) = 1, Var W1(1) ∈ (0, ∞) and m(2) < 1. Then, as n → ∞, W∞(1) − Wn(1) (m(2))n/2

d

→ normal (0, v2W∞(2)).

Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 8/15

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

CLT for the tail of the Biggins martingale Relevant literature LIL for the tail of the Biggins martingale Other results Exponentially fast a.s. convergence for the tail of the Biggins martingale

The rate of convergence:

relevant literature

Here are several related results, with pointers to the literature. a CLT for the tail martingale of a Galton-Watson process – Athreya (1968) and Heyde (1970) a functional CLT for the tail martingale of a Galton-Watson process – Heyde & Brown (1971) CLT’s for multitype branching processes – Kesten & Stigum (1966), Athreya (1968) and Asmussen & Keiding (1978) a CLT for the tail martingale of a weighted branching processes – R¨

• sler, 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

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

CLT for the tail of the Biggins martingale Relevant literature LIL for the tail of the Biggins martingale Other results Exponentially fast a.s. convergence for the tail of the Biggins martingale

The rate of convergence:

LIL for the tail of the Biggins martingale

m(γ) = E

• R eγxM(dx), γ > 0;

Wn(γ) = (m(γ))−n

R eγxMn(dx), n ∈ N;

lim

n→∞Wn(γ) = W∞(γ) a.s.

Theorem (I. & Kabluchko (2016)) Assume that m(1) = 1, σ2 = Var W1(1) ∈ (0, ∞), EW1(2) log+ W1(2) < ∞, and that the function r → (m(r))1/r is finite and decreasing on [1, 2] with − log m(2) 2 < − m′(2) m(2) , where m′ denotes the left derivative. Then W∞(1) and W∞(2) are positive almost surely on the survival set, and lim sup

n→∞

W∞(1) − Wn(1)

• (m(2))n log n

=

• 2v2W∞(2),

lim inf

n→∞

W∞(1) − Wn(1)

• (m(2))n log n

= −

• 2v2W∞(2)

almost surely, where v2 = Var W∞(1) = σ2(1 − m(2))−1 < ∞.

Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 10/15

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

CLT for the tail of the Biggins martingale Relevant literature LIL for the tail of the Biggins martingale Other results Exponentially fast a.s. convergence for the tail of the Biggins martingale

The rate of convergence:

LIL for the tail of the Biggins martingale

m(γ) = E

• R eγxM(dx), γ > 0;

Wn(γ) = (m(γ))−n

R eγxMn(dx), n ∈ N;

lim

n→∞Wn(γ) = W∞(γ) a.s.

Theorem (I. & Kabluchko (2016)) Assume that m(1) = 1, σ2 = Var W1(1) ∈ (0, ∞), EW1(2) log+ W1(2) < ∞, and that the function r → (m(r))1/r is finite and decreasing on [1, 2] with − log m(2) 2 < − m′(2) m(2) , where m′ denotes the left derivative. Then W∞(1) and W∞(2) are positive almost surely

• n the survival set, and

lim sup

n→∞

W∞(1) − Wn(1)

• (m(2))n log n

=

• 2v2W∞(2),

lim inf

n→∞

W∞(1) − Wn(1)

• (m(2))n log n

= −

• 2v2W∞(2)

almost surely, where v2 = Var W∞(1) = σ2(1 − m(2))−1 < ∞.

Why is it a law of the iterated logarithm? In view of Var [W∞(1) − Wn(1)] = v2(m(2))n

• ne may replace log n by the asymptotically equivalent expression

log log(v2(m(2))n), whence the term.

Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 11/15

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

CLT for the tail of the Biggins martingale Relevant literature LIL for the tail of the Biggins martingale Other results Exponentially fast a.s. convergence for the tail of the Biggins martingale

The rate of convergence:

• ther results

(Wn)n∈N0 – uniformly integrable Biggins martingale W – its a.s. limit For a > 0 and p > 1, the Lp-convergence of

n≥0 ean(W − Wn) – Alsmeyer, I., Polotskiy

& R¨

• sler (2009)

The a.s. convergence of

n≥0 ean(W − Wn) – I.& Meiners (2010)

For a function b : R+ → R+ regularly varying at ∞ of index α, α > −1, the a.s. convergence of

n≥0 b(n)(W − Wn) – I. (2006)

Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 12/15

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

CLT for the tail of the Biggins martingale Relevant literature LIL for the tail of the Biggins martingale Other results Exponentially fast a.s. convergence for the tail of the Biggins martingale

The rate of convergence:

exponentially fast a.s. convergence for the tail of the Biggins martingale

m(γ) = E

• R eγxM(dx), γ > 0;

Wn(γ) = (m(γ))−n

R eγxMn(dx), n ∈ N;

lim

n→∞Wn(γ) = W∞(γ) a.s.

Theorem (I. & Meiners (2010)) Let a > 0 be given. Assume that m(1) = 1, ea m1/r(r) ≤ 1 for some r ∈ (1, 2) and define θ to be the minimal r > 1 such that earm(r) = 1. Assume further that EW1(1)θ < ∞, and in case when a = − log infr≥1 m1/r(r) (which implies that θ = θ0 satisfies m1/θ0(θ0) = inf1≤θ≤2 m1/θ(θ)) assume that − log m(θ0) θ0 < − m′(θ0) m(θ0) . Then the Biggins martingale (Wn(1))n∈N0 is uniformly integrable and

• n≥0 ean(W∞(1) − Wn(1)) converges almost surely.

Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 13/15

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

CLT for the tail of the Biggins martingale Relevant literature LIL for the tail of the Biggins martingale Other results Exponentially fast a.s. convergence for the tail of the Biggins martingale

The rate of convergence:

exponentially fast a.s. convergence for the tail of the Biggins martingale

A typical situation in which the previous theorem applies. The bottom point of the graph of m1/x(x) is marked by a filled red circle. The vertical dashed blue line connects this point to the x-axis indicating the point θ0. The red horizontal line and the black horizontal line at 1 indicate the open interval of possible values of e−a such that a > 0 and eam1/θ0 m(θ0) < 1. For those a’s,

n≥0 ean(W∞(1) − Wn(1)) converges a.s.

1 2 θ0 x 1

m(x)1/x m(x)

Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 14/15

What is the branching random walk and the Biggins martingale? Uniform integrability of the Biggins martingale The rate of convergence

CLT for the tail of the Biggins martingale Relevant literature LIL for the tail of the Biggins martingale Other results Exponentially fast a.s. convergence for the tail of the Biggins martingale

THANK YOU FOR YOUR ATTENTION!

Alexander Iksanov On the rate of convergence of the Biggins martingale in supercritical branching random walks May 16, 2017 15/15