Branching random walk with stretched exponential tails Piotr - - PowerPoint PPT Presentation

Branching random walk with stretched exponential tails

Piotr Dyszewski (TUM & UWr) June 5, 2020

X X ′ Mn

◮ the particles reproduce according to a Galton-Watson process with

reproduction mean m > 1

◮ the displacements are iid copies of X such that P[X > t] ∼ e−tr ,

r ∈ (0, 1)

◮ Mn = the position of the rightmost particle

Theorem (P .D, N. Gantert, T. Höfelsauer)

Take X such that EX = 0 and EX 2 = 1. Suppose that (. . . ) and that X has a stretched exponential tail, i.e. P[X > t] ∼ e−tr , r ∈ (0, 1). Then there are constants γ, σ, α such that: for r ∈

• 0, 2

3

• ,

Mn − αn

1 r

σn

1 r −1

→d F(x) = E

• exp

−γWe−x and for r ∈ 2

3, 1

• Mn − αn

1 r

n2− 1

r

→ 1

2σ, where W is a martingale limit associated with the underlying Galton-Watson process.

Let (Zn) be a Galton-Watson process with reproduction mean m > 1. Assume for simplicity P[Z1 = 0] = 0. Let T ⊆

n≥0 Nn be the corresponding Galton-Watson tree.

|x| = 1 |x| = 2 |x| = 3 ∅

1 00 12 11 10 002 001 000 003

|x| = 1 |x| = 2 |x| = 3 |x| = n ∅

X0 1 X1 00 X00 12 X12 11 X11 10 X10 002

X002

001

X001

000

X000

003

X003

Xy, y ∈ T iid For x ∈ T, V(x) = ∑

y≤x

Xy Mn = max

|x|=n

V(x)

P[X > t] ∼ e−tr , r ∈ (0, 1) P [X1 + X2 > t] ∼ P [max{X1, X2} > t] Sn = ∑1≤k≤n Xk, X ∗

n = max1≤k≤n Xk

P [Sn > tn] ∼? P

• Sn > cn

1 r

• ≈ sup

s P

• X ∗

n > cn

1 r − s

• P[Sn−1 > s]

≈ sup

s

n exp

• −
• cn

1 r − s

r − s2 2n

• ≈ P
• X ∗

n > cn

1 r − rcr−1n2− 1 r

• P
• Sn−1 > rcr−1n2− 1

r

• P
• Sn > cn

1 r

• ∼ n exp
• −crn + O
• n3− 2

r

If |x| = n, then V(x)

d

= Sn =

n

∑

k=1

Xk. P

• Mn > cn

1 r

• = P
• ∃ |x| = n, V(x) > cn

1 r

• = P

  ∑

|x|=n

✶

V(x)>cn

1 r

≥ 1

  ≤ E   ∑

|x|=n

✶

V(x)>cn

1 r

• 

 = E[Zn]P

• Sn > cn

1 r

• = mnP
• Sn > cn

1 r

• ∼ mn exp {−crn(1 + o(1))}

Mn n

1 r

→ α = log(m)

1 r

The first term in the asymptotic expansion of Mn is related to the biggest displacement, i.e. Nn = max

|y|≤n

Xy.

Proposition

Nn − αn

1 r

σn

1 r −1

→d H

d

= E[exp{−γ′We−x}]

Proof.

xn → ∞ P[Nn ≤ xn] = E [ P[Nn ≤ xn | (Zn) ] ] = E

• P[X ≤ xn]Yn
• ∼ E [exp{−YnP[X > xn]}]

Yn = #{|x| ≤ n} = ∑n

k=1 Zk.

Proof continued .

m−nZn is a positive martingale lim

n→∞

Zn mn = W > 0 Yn ∼ m m − 1 W mn xn = αn

1 r + σn 1 r −1x

YnP[X > xn] ∼ m m − 1 W mn exp

• −
• αn

1 r + σn 1 r −1x

r ∼ m m − 1 W mn exp

• −αrn − αr−1rσx
• ∼ γ′ W exp {−x}

σ = α1−r r γ′ = m m − 1 P

• Nn − αn

1 r

σn

1 r −1

≤ x

• = P[Nn ≤ xn] → E[exp{−γ′We−x}]

Mn n

1 r

→ α

Nn n

1 r

→ α

max

|x|=n

• V(x) = ∑

v≤x

Xv : Xv ≪ αn

1 r

• = o(. . .)

#{|v| ≤ n} = Yn ≈ mn #

• |v| ≤ n : Xv ≈ αn

1 r

• ≈ mo(n)
• |v| ≤ n : Xv ≈ αn

1 r

• is a small subset of {|v| ≤ n}

≈ αn

1 r

V(x) = R1(x) + V0(x) + N(x) + R2(x)

= V0(x) + N(x)

Mn ≈ max {V0(x) + N(x)} N(x) ≈ αn

1 r

V0(x)

d

= Sn−o(n)

• Xk ≪ αn

1 r

The contribution of V0(x) is at most V0(x) = O

• n2− 1

r

• n the other hand

Nn = max

|v|≤n

Xv

d

= αn

1 r + σn 1 r −1H(1 + o(1))

If r < 2

3, n2− 1

r = o

• n

1 r −1

and so Mn ≈ Nn

d

= αn

1 r + σn 1 r −1H(1 + o(1))

Mn − αn

1 r

σn

1 r −1

∼ Nn − αn

1 r

σn

1 r −1

→d H

d

= E[exp{−γ′We−x}]

Mn − Nn n

1 r −1

→ 0

If r > 2

3, n

1 r −1 = o

• n2− 1

r

• ≈ αn

1 r

Mn ≈ max {V0(x) + N(x)} Mn − αn

1 r

n2− 1

r

→ 1

2σ

If r = 2

3, n2− 1

r = n 1 r −1 = √

n Mn ≈ max{N(x) + V0(x)}

Nx

d

= Φ(·)

d

= N (0, 1).

Mn − αn

1 r

σ√ n ≈ max

• N(x) − αn

1 r

σ√ n + 1 σNx

• P
• Mn − αn

1 r

σ√ n ≤ t

• ≈ E

 P

• N(x) − αn

1 r

σ√ n + 1 σN > t Yn  ≈ E

• exp
• γ′W
• e−y(1 − Φ(σ(t − y))dy
• = E
• exp

−γWe−t

Theorem (P .D, N. Gantert, T. Höfelsauer)

Take X such that EX = 0 and EX 2 = 1. Suppose that (. . . ) and that X has a stretched exponential tail, i.e. P[X > t] ∼ e−tr , r ∈ (0, 1). Then there are constants γ, σ, α such that: for r ∈

• 0, 2

3

• ,

Mn − αn

1 r

σn

1 r −1

→d F(x) = E

• exp

−γWe−x , and for r ∈ 2

3, 1

• Mn − αn

1 r

n2− 1

r

→ 1

2σ where W is a martingale limit associated with the underlying Galton-Watson process.

References I

Piotr Dyszewski, Nina Gantert, and Thomas Höfelsauer, The maximum of a branching random walk with stretched exponantial tails, https://arxiv.org/abs/2004.03871 Nina Gantert, The maximum of a branching random walk with semiexponential increments, Annals of Probability 28 (2000), no. 3, 1219–1229.

• A. V. Nagaev, Integral limit theorems taking large deviations into

account when Cramér’s condition does not hold. I, Theory of Probability & Its Applications 14 (1969), no. 1, 51–64.