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Large deviation for extremes in BRW with regularly varying displacements Ayan Bhattacharya Centrum Wiskunde & Informatica, Amsterdam June 20, 2018 Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 1 / 37
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Page 1: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Large deviation for extremes in BRW withregularly varying displacements

Ayan Bhattacharya

Centrum Wiskunde & Informatica, Amsterdam

June 20, 2018

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 1 / 37

Page 2: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

It starts with a single particle at the origin of the real line. Thisis referred as the 0th generation.

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 2 / 37

Page 3: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

It starts with a single particle at the origin of the real line.

Thisis referred as the 0th generation.

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 2 / 37

Page 4: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

It starts with a single particle at the origin of the real line. Thisis referred as the 0th generation.

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 2 / 37

Page 5: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time the particle at origin produces a random numberof particles according to a distribution (progeny distribution) onN = {1, 2, 3, . . .} (no leaf ) and dies immediately. The newparticles form generation 1.Each new particle comes with a random real-valueddisplacement being independent of others. Displacements areidentically distributed according to the law of X .

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 3 / 37

Page 6: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time the particle at origin produces a random numberof particles according to a distribution (progeny distribution) onN = {1, 2, 3, . . .} (no leaf ) and dies immediately. The newparticles form generation 1.Each new particle comes with a random real-valueddisplacement being independent of others. Displacements areidentically distributed according to the law of X .

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 3 / 37

Page 7: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time the particle at origin produces a random numberof particles according to a distribution (progeny distribution) onN = {1, 2, 3, . . .} and dies immediately. The new particles formgeneration 1.

Each new particle comes with a random real-valueddisplacement being independent of others.

Displacements areidentically distributed according to the law of X .

0 time•

DISP

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 4 / 37

Page 8: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time the particle at origin produces a random numberof particles according to a distribution (progeny distribution) onN = {1, 2, 3, . . .} and dies immediately. The new particles formgeneration 1.

Each new particle comes with a random real-valueddisplacement being independent of others. Displacements areidentically distributed according to the law of X .

0 time•

DISP

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 4 / 37

Page 9: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time, each particle in the first generation produces arandom number of particles according to progeny distribution

being independent of others and whatever happened in the firstgeneration. The new particles form second generation. Eachnew particle comes with a random displacement beingindependent of others.

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 5 / 37

Page 10: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time, each particle in the first generation produces arandom number of particles according to progeny distributionbeing independent of others and

whatever happened in the firstgeneration. The new particles form second generation. Eachnew particle comes with a random displacement beingindependent of others.

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 5 / 37

Page 11: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time, each particle in the first generation produces arandom number of particles according to progeny distributionbeing independent of others and whatever happened in the firstgeneration.

The new particles form second generation. Eachnew particle comes with a random displacement beingindependent of others.

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 5 / 37

Page 12: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time, each particle in the first generation produces arandom number of particles according to progeny distributionbeing independent of others and whatever happened in the firstgeneration. The new particles form second generation.

Eachnew particle comes with a random displacement beingindependent of others.

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 5 / 37

Page 13: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time, each particle in the first generation produces arandom number of particles according to progeny distributionbeing independent of others and whatever happened in the firstgeneration. The new particles form second generation. Eachnew particle comes with a random displacement beingindependent of others.

0 time•

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 5 / 37

Page 14: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time, each particle in the first generation produces arandom number of particles according to progeny distributionbeing independent of others and whatever happened in the firstgeneration. The new particles form second generation. Eachnew particle comes with a random displacement beingindependent of others.

0 time•

DISP

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 6 / 37

Page 15: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

After unit time, each particle in the first generation produces arandom number of particles according to progeny distributionbeing independent of others and whatever happened in the firstgeneration. The new particles form second generation. Eachnew particle comes with a random displacement beingindependent of others.

0 time•

•DISP

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 7 / 37

Page 16: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

This mechanism goes on.

The position of a particle is defined to be its displacementtranslated by position of its parent.

0 time•

•DISP

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 8 / 37

Page 17: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

This mechanism goes on.

The position of a particle is defined to be its displacementtranslated by position of its parent.

0 time•

•DISP

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 8 / 37

Page 18: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Branching random walk on real line

This mechanism goes on.

The position of a particle is defined to be its displacementtranslated by position of its parent.

0 time•

•DISP

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 8 / 37

Page 19: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

The collection of positions in the system is called branchingrandom walk (BRW).

0 time•

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 9 / 37

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In this talk, we shall focus on the position of the topmostparticle in the nth generation.

0 time•

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 10 / 37

Page 21: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Why BRW?

BRW is considered to be very important in the context ofprobability, statistical physics, algorithms etc. It has connectionto Gaussian multplicative chaos, Gaussian free field, randompolymers, percolation etc.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 11 / 37

Page 22: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

An easy to state problem

Suppose that X is positive almost surely.

The displacement of a particle is the lifetime of a bacteria.

The position of the topmost particle in the nth generation canbe interpreted as the last time one can see an nth generationbacteria.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 12 / 37

Page 23: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

An easy to state problem

Suppose that X is positive almost surely.

The displacement of a particle is the lifetime of a bacteria.

The position of the topmost particle in the nth generation canbe interpreted as the last time one can see an nth generationbacteria.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 12 / 37

Page 24: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

An easy to state problem

Suppose that X is positive almost surely.

The displacement of a particle is the lifetime of a bacteria.

The position of the topmost particle in the nth generation canbe interpreted as the last time one can see an nth generationbacteria.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 12 / 37

Page 25: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

An easy to state problem

Suppose that X is positive almost surely.

The displacement of a particle is the lifetime of a bacteria.

The position of the topmost particle in the nth generation canbe interpreted as the last time one can see an nth generationbacteria.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 12 / 37

Page 26: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Challenges

Phase transition in the asymptotic behavior of extremes.

Reason: Non-trivial dependence structure. (Durrett(1979))

0 time•

•DISP

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 13 / 37

Page 27: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Challenges

Phase transition in the asymptotic behavior of extremes.

Reason: Non-trivial dependence structure. (Durrett(1979))

0 time•

•DISP

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 13 / 37

Page 28: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Assumptions on genealogical structure

The genealogy of the particles is given by a Galton-Watsonprocess.

We shall assume that the underlying GW process is supercriticaland satisfies the Kesten-Stigum condition.

Zn denotes the number of particles in the nth generation forevery n ≥ 1.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 14 / 37

Page 29: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Assumptions on genealogical structure

The genealogy of the particles is given by a Galton-Watsonprocess.

We shall assume that the underlying GW process is supercriticaland satisfies the Kesten-Stigum condition.

Zn denotes the number of particles in the nth generation forevery n ≥ 1.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 14 / 37

Page 30: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Assumptions on genealogical structure

The genealogy of the particles is given by a Galton-Watsonprocess.

We shall assume that the underlying GW process is supercriticaland satisfies the Kesten-Stigum condition.

Zn denotes the number of particles in the nth generation forevery n ≥ 1.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 14 / 37

Page 31: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Assumptions on genealogical structure

The genealogy of the particles is given by a Galton-Watsonprocess.

We shall assume that the underlying GW process is supercriticaland satisfies the Kesten-Stigum condition.

Zn denotes the number of particles in the nth generation forevery n ≥ 1.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 14 / 37

Page 32: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

1 < m = E(Z1) <∞.

(m−nZn : n ≥ 1) is a non-negative martingale sequence andhence m−nZn converges to a random variable W almost surelyas n→∞.

Kesten-Stigum condition (E(Z1 log+ Z1) <∞) implies that W is

positive almost surely due to “no leaf” assumption.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 15 / 37

Page 33: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

1 < m = E(Z1) <∞.

(m−nZn : n ≥ 1) is a non-negative martingale sequence andhence m−nZn converges to a random variable W almost surelyas n→∞.

Kesten-Stigum condition (E(Z1 log+ Z1) <∞) implies that W is

positive almost surely due to “no leaf” assumption.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 15 / 37

Page 34: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

1 < m = E(Z1) <∞.

(m−nZn : n ≥ 1) is a non-negative martingale sequence andhence m−nZn converges to a random variable W almost surelyas n→∞.

Kesten-Stigum condition (E(Z1 log+ Z1) <∞) implies that W is

positive almost surely due to “no leaf” assumption.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 15 / 37

Page 35: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Assumptions on the displacements

The displacements are real-valued. For every x > 0,

P(|X | > x) = x−αL(x)

where L is slowly varying function and satisfies tail-balancingconditions

limx→∞

P(X > x)

P(|X | > x)= p and lim

x→∞

P(X < −x)

P(|X | > x)= 1− p

for some p ∈ [0, 1].

Consider a sequence of constants (bn : n ≥ 1) such thatmnP(b−1

n X ∈ ·) v→ να(·) in the space [−∞,∞] \ {0} and

να(dx) = α(px−α−1

1(x > 0) + (1− p)(−x)−α1(x < 0)).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 16 / 37

Page 36: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Assumptions on the displacements

The displacements are real-valued. For every x > 0,

P(|X | > x) = x−αL(x)

where L is slowly varying function and satisfies tail-balancingconditions

limx→∞

P(X > x)

P(|X | > x)= p and lim

x→∞

P(X < −x)

P(|X | > x)= 1− p

for some p ∈ [0, 1].

Consider a sequence of constants (bn : n ≥ 1) such thatmnP(b−1

n X ∈ ·) v→ να(·) in the space [−∞,∞] \ {0} and

να(dx) = α(px−α−1

1(x > 0) + (1− p)(−x)−α1(x < 0)).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 16 / 37

Page 37: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Assumptions on the displacements

The displacements are real-valued. For every x > 0,

P(|X | > x) = x−αL(x)

where L is slowly varying function and satisfies tail-balancingconditions

limx→∞

P(X > x)

P(|X | > x)= p and lim

x→∞

P(X < −x)

P(|X | > x)= 1− p

for some p ∈ [0, 1].

Consider a sequence of constants (bn : n ≥ 1) such thatmnP(b−1

n X ∈ ·) v→ να(·) in the space [−∞,∞] \ {0} and

να(dx) = α(px−α−1

1(x > 0) + (1− p)(−x)−α1(x < 0)).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 16 / 37

Page 38: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Literature

Pioneering work on extremes of BRW has been done byHammerseley-Kingman-Biggins.

Weak convergence of extremes and extremal processes forlight-tailed displacements are known. See Bachman (2000),Eidekon (2011), Maillard (2015), Madaule (2017), Mallein(2016).

Large deviation is derived for topmost particle in branchingBrownian motion (BBM). See Chauvin and Rouault (1988).

Large deviation for topmost position in different variants of themodel BBM: Derrida and Shi(2017).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 17 / 37

Page 39: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Literature

Pioneering work on extremes of BRW has been done byHammerseley-Kingman-Biggins.

Weak convergence of extremes and extremal processes forlight-tailed displacements are known. See Bachman (2000),Eidekon (2011), Maillard (2015), Madaule (2017), Mallein(2016).

Large deviation is derived for topmost particle in branchingBrownian motion (BBM). See Chauvin and Rouault (1988).

Large deviation for topmost position in different variants of themodel BBM: Derrida and Shi(2017).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 17 / 37

Page 40: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Literature

Pioneering work on extremes of BRW has been done byHammerseley-Kingman-Biggins.

Weak convergence of extremes and extremal processes forlight-tailed displacements are known. See Bachman (2000),Eidekon (2011), Maillard (2015), Madaule (2017), Mallein(2016).

Large deviation is derived for topmost particle in branchingBrownian motion (BBM). See Chauvin and Rouault (1988).

Large deviation for topmost position in different variants of themodel BBM: Derrida and Shi(2017).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 17 / 37

Page 41: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Literature

Pioneering work on extremes of BRW has been done byHammerseley-Kingman-Biggins.

Weak convergence of extremes and extremal processes forlight-tailed displacements are known. See Bachman (2000),Eidekon (2011), Maillard (2015), Madaule (2017), Mallein(2016).

Large deviation is derived for topmost particle in branchingBrownian motion (BBM). See Chauvin and Rouault (1988).

Large deviation for topmost position in different variants of themodel BBM: Derrida and Shi(2017).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 17 / 37

Page 42: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Literature

Pioneering work on extremes of BRW has been done byHammerseley-Kingman-Biggins.

Weak convergence of extremes and extremal processes forlight-tailed displacements are known. See Bachman (2000),Eidekon (2011), Maillard (2015), Madaule (2017), Mallein(2016).

Large deviation is derived for topmost particle in branchingBrownian motion (BBM). See Chauvin and Rouault (1988).

Large deviation for topmost position in different variants of themodel BBM: Derrida and Shi(2017).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 17 / 37

Page 43: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Regularly varying displacements

Let Mn be the position of the topmost particle in the nthgeneration.

b−1n Mn ⇒ M where M is a W -mixture of Frechet distributions.

(Durrett(1983))

Let v denote the generic vertex, |v| denote generation of thevertex v and S(v) denote the position. Consider

Pn =∑|v|=n

δb−1n S(v)

Let M = { space of all measures on [−∞,∞] \ {0}}

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 18 / 37

Page 44: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Regularly varying displacements

Let Mn be the position of the topmost particle in the nthgeneration.

b−1n Mn ⇒ M where M is a W -mixture of Frechet distributions.

(Durrett(1983))

Let v denote the generic vertex, |v| denote generation of thevertex v and S(v) denote the position. Consider

Pn =∑|v|=n

δb−1n S(v)

Let M = { space of all measures on [−∞,∞] \ {0}}

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 18 / 37

Page 45: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Regularly varying displacements

Let Mn be the position of the topmost particle in the nthgeneration.

b−1n Mn ⇒ M where M is a W -mixture of Frechet distributions.

(Durrett(1983))

Let v denote the generic vertex, |v| denote generation of thevertex v and S(v) denote the position. Consider

Pn =∑|v|=n

δb−1n S(v)

Let M = { space of all measures on [−∞,∞] \ {0}}

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 18 / 37

Page 46: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Regularly varying displacements

Let Mn be the position of the topmost particle in the nthgeneration.

b−1n Mn ⇒ M where M is a W -mixture of Frechet distributions.

(Durrett(1983))

Let v denote the generic vertex, |v| denote generation of thevertex v and S(v) denote the position. Consider

Pn =∑|v|=n

δb−1n S(v)

Let M = { space of all measures on [−∞,∞] \ {0}}

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 18 / 37

Page 47: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Regularly varying displacements

Let Mn be the position of the topmost particle in the nthgeneration.

b−1n Mn ⇒ M where M is a W -mixture of Frechet distributions.

(Durrett(1983))

Let v denote the generic vertex, |v| denote generation of thevertex v and S(v) denote the position. Consider

Pn =∑|v|=n

δb−1n S(v)

Let M = { space of all measures on [−∞,∞] \ {0}}

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 18 / 37

Page 48: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Weak convergence of Pn

Theorem (B. Hazra and Roy (2016))There exists a Cox cluster process P such that Pn ⇒P as n→∞in the space M where

Pd=∞∑l=1

ZGl δW 1/αjl

with (jl : l ≥ 1) be the atoms of the PRM(να) on R.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 19 / 37

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Aim

Consider an increasing sequence (cn : n ≥ 1) such that

limn→∞

c−1n bn = 0.

c−1n Mn converges to 0 in probability.

QuestionWhat is the rate of convergence for P(Mn > cnx) ?

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 20 / 37

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Aim

Consider an increasing sequence (cn : n ≥ 1) such that

limn→∞

c−1n bn = 0.

c−1n Mn converges to 0 in probability.

QuestionWhat is the rate of convergence for P(Mn > cnx) ?

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 20 / 37

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Aim

Consider an increasing sequence (cn : n ≥ 1) such that

limn→∞

c−1n bn = 0.

c−1n Mn converges to 0 in probability.

QuestionWhat is the rate of convergence for P(Mn > cnx) ?

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 20 / 37

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Aim

Consider an increasing sequence (cn : n ≥ 1) such that

limn→∞

c−1n bn = 0.

c−1n Mn converges to 0 in probability.

QuestionWhat is the rate of convergence for P(Mn > cnx) ?

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 20 / 37

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Generalization

Same questions can be asked for second, third, . . . topmostpositions in the nth generation.

Joint distribution of the first k largest positions and gapstatistics.Consider the sequence of point processes

Nn =∑|v|=n

δc−1n S(v)

QuestionHow does Nn behave asymptotically?

Hult and Samorodnitsky (2010). Large deviation of extremalprocesses.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 21 / 37

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Generalization

Same questions can be asked for second, third, . . . topmostpositions in the nth generation.Joint distribution of the first k largest positions and gapstatistics.

Consider the sequence of point processes

Nn =∑|v|=n

δc−1n S(v)

QuestionHow does Nn behave asymptotically?

Hult and Samorodnitsky (2010). Large deviation of extremalprocesses.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 21 / 37

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Generalization

Same questions can be asked for second, third, . . . topmostpositions in the nth generation.Joint distribution of the first k largest positions and gapstatistics.Consider the sequence of point processes

Nn =∑|v|=n

δc−1n S(v)

QuestionHow does Nn behave asymptotically?

Hult and Samorodnitsky (2010). Large deviation of extremalprocesses.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 21 / 37

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Aim

Recall M = {space of all point measures on [−∞,∞] \ {0}}.

Vague convergence on the space M is metrizable and Mequipped with vague topology is complete and separable.

Nn converges to null measure (∅) in the space M almost surely.

Consider A ⊂M such that ∅ /∈ A. Then it is clear thatP(Nn ∈ A)→ 0.

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Aim

Recall M = {space of all point measures on [−∞,∞] \ {0}}.

Vague convergence on the space M is metrizable and Mequipped with vague topology is complete and separable.

Nn converges to null measure (∅) in the space M almost surely.

Consider A ⊂M such that ∅ /∈ A. Then it is clear thatP(Nn ∈ A)→ 0.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 22 / 37

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Aim

Recall M = {space of all point measures on [−∞,∞] \ {0}}.

Vague convergence on the space M is metrizable and Mequipped with vague topology is complete and separable.

Nn converges to null measure (∅) in the space M almost surely.

Consider A ⊂M such that ∅ /∈ A. Then it is clear thatP(Nn ∈ A)→ 0.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 22 / 37

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Aim

Recall M = {space of all point measures on [−∞,∞] \ {0}}.

Vague convergence on the space M is metrizable and Mequipped with vague topology is complete and separable.

Nn converges to null measure (∅) in the space M almost surely.

Consider A ⊂M such that ∅ /∈ A.

Then it is clear thatP(Nn ∈ A)→ 0.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 22 / 37

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Aim

Recall M = {space of all point measures on [−∞,∞] \ {0}}.

Vague convergence on the space M is metrizable and Mequipped with vague topology is complete and separable.

Nn converges to null measure (∅) in the space M almost surely.

Consider A ⊂M such that ∅ /∈ A. Then it is clear thatP(Nn ∈ A)→ 0.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 22 / 37

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QuestionDoes there exist (rn : n ≥ 1) and a non-trivial measure λ on M suchthat rnP(Nn ∈ A) converges to λ(A) for every nice measurable setA ⊂M ?

“nice measurable set” A means

λ(∂A) = 0. (∂A means boundary of A)

“bounded away” means ∅ /∈ A (∅ is the null measure in M )

“non-trivial measure” λ means the measure λ such that0 < λ(A) <∞ for a “nice” set A.

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QuestionDoes there exist (rn : n ≥ 1) and a non-trivial measure λ on M suchthat rnP(Nn ∈ A) converges to λ(A) for every nice measurable setA ⊂M ?

“nice measurable set” A means

λ(∂A) = 0. (∂A means boundary of A)

“bounded away” means ∅ /∈ A (∅ is the null measure in M )

“non-trivial measure” λ means the measure λ such that0 < λ(A) <∞ for a “nice” set A.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 23 / 37

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QuestionDoes there exist (rn : n ≥ 1) and a non-trivial measure λ on M suchthat rnP(Nn ∈ A) converges to λ(A) for every nice measurable setA ⊂M ?

“nice measurable set” A means

λ(∂A) = 0. (∂A means boundary of A)

“bounded away” means ∅ /∈ A (∅ is the null measure in M )

“non-trivial measure” λ means the measure λ such that0 < λ(A) <∞ for a “nice” set A.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 23 / 37

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QuestionDoes there exist (rn : n ≥ 1) and a non-trivial measure λ on M suchthat rnP(Nn ∈ A) converges to λ(A) for every nice measurable setA ⊂M ?

“nice measurable set” A means

λ(∂A) = 0. (∂A means boundary of A)

“bounded away” means ∅ /∈ A (∅ is the null measure in M )

“non-trivial measure” λ means the measure λ such that0 < λ(A) <∞ for a “nice” set A.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 23 / 37

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Consider the space M = { space of all measures on M }.(rnP(Nn ∈ ·) : n ≥ 1) is a sequence of elements in M.M0 = {ξ ∈M : ξ(A) <∞ for all measurable subsets A ⊂M \ {∅}}.

Definition (Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014))Consider a complete separable metric space S and an element s0 ∈ S.Let M0 be the space of all locally finite measures on the spaceS \ {s0}.A sequence of measures (ξn : n ≥ 1) is said to converge inM0 to a measure ξ ∈M0 if

∫fdξn →

∫fdξ for every bounded,

continuous positive function f : S→ [0,∞) such that f vanishes in aneighbourhood of s0.

We can use M0 convergence with S = M and s0 = ∅.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 24 / 37

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Consider the space M = { space of all measures on M }.

(rnP(Nn ∈ ·) : n ≥ 1) is a sequence of elements in M.M0 = {ξ ∈M : ξ(A) <∞ for all measurable subsets A ⊂M \ {∅}}.

Definition (Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014))Consider a complete separable metric space S and an element s0 ∈ S.Let M0 be the space of all locally finite measures on the spaceS \ {s0}.A sequence of measures (ξn : n ≥ 1) is said to converge inM0 to a measure ξ ∈M0 if

∫fdξn →

∫fdξ for every bounded,

continuous positive function f : S→ [0,∞) such that f vanishes in aneighbourhood of s0.

We can use M0 convergence with S = M and s0 = ∅.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 24 / 37

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Consider the space M = { space of all measures on M }.(rnP(Nn ∈ ·) : n ≥ 1) is a sequence of elements in M.

M0 = {ξ ∈M : ξ(A) <∞ for all measurable subsets A ⊂M \ {∅}}.

Definition (Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014))Consider a complete separable metric space S and an element s0 ∈ S.Let M0 be the space of all locally finite measures on the spaceS \ {s0}.A sequence of measures (ξn : n ≥ 1) is said to converge inM0 to a measure ξ ∈M0 if

∫fdξn →

∫fdξ for every bounded,

continuous positive function f : S→ [0,∞) such that f vanishes in aneighbourhood of s0.

We can use M0 convergence with S = M and s0 = ∅.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 24 / 37

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Consider the space M = { space of all measures on M }.(rnP(Nn ∈ ·) : n ≥ 1) is a sequence of elements in M.M0 = {ξ ∈M : ξ(A) <∞ for all measurable subsets A ⊂M \ {∅}}.

Definition (Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014))Consider a complete separable metric space S and an element s0 ∈ S.Let M0 be the space of all locally finite measures on the spaceS \ {s0}.A sequence of measures (ξn : n ≥ 1) is said to converge inM0 to a measure ξ ∈M0 if

∫fdξn →

∫fdξ for every bounded,

continuous positive function f : S→ [0,∞) such that f vanishes in aneighbourhood of s0.

We can use M0 convergence with S = M and s0 = ∅.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 24 / 37

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Consider the space M = { space of all measures on M }.(rnP(Nn ∈ ·) : n ≥ 1) is a sequence of elements in M.M0 = {ξ ∈M : ξ(A) <∞ for all measurable subsets A ⊂M \ {∅}}.

Definition (Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014))Consider a complete separable metric space S and an element s0 ∈ S.

Let M0 be the space of all locally finite measures on the spaceS \ {s0}.A sequence of measures (ξn : n ≥ 1) is said to converge inM0 to a measure ξ ∈M0 if

∫fdξn →

∫fdξ for every bounded,

continuous positive function f : S→ [0,∞) such that f vanishes in aneighbourhood of s0.

We can use M0 convergence with S = M and s0 = ∅.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 24 / 37

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Consider the space M = { space of all measures on M }.(rnP(Nn ∈ ·) : n ≥ 1) is a sequence of elements in M.M0 = {ξ ∈M : ξ(A) <∞ for all measurable subsets A ⊂M \ {∅}}.

Definition (Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014))Consider a complete separable metric space S and an element s0 ∈ S.Let M0 be the space of all locally finite measures on the spaceS \ {s0}.

A sequence of measures (ξn : n ≥ 1) is said to converge inM0 to a measure ξ ∈M0 if

∫fdξn →

∫fdξ for every bounded,

continuous positive function f : S→ [0,∞) such that f vanishes in aneighbourhood of s0.

We can use M0 convergence with S = M and s0 = ∅.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 24 / 37

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Consider the space M = { space of all measures on M }.(rnP(Nn ∈ ·) : n ≥ 1) is a sequence of elements in M.M0 = {ξ ∈M : ξ(A) <∞ for all measurable subsets A ⊂M \ {∅}}.

Definition (Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014))Consider a complete separable metric space S and an element s0 ∈ S.Let M0 be the space of all locally finite measures on the spaceS \ {s0}.A sequence of measures (ξn : n ≥ 1) is said to converge inM0 to a measure ξ ∈M0 if

∫fdξn →

∫fdξ for every bounded,

continuous positive function f : S→ [0,∞) such that f vanishes in aneighbourhood of s0.

We can use M0 convergence with S = M and s0 = ∅.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 24 / 37

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Consider the space M = { space of all measures on M }.(rnP(Nn ∈ ·) : n ≥ 1) is a sequence of elements in M.M0 = {ξ ∈M : ξ(A) <∞ for all measurable subsets A ⊂M \ {∅}}.

Definition (Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014))Consider a complete separable metric space S and an element s0 ∈ S.Let M0 be the space of all locally finite measures on the spaceS \ {s0}.A sequence of measures (ξn : n ≥ 1) is said to converge inM0 to a measure ξ ∈M0 if

∫fdξn →

∫fdξ for every bounded,

continuous positive function f : S→ [0,∞) such that f vanishes in aneighbourhood of s0.

We can use M0 convergence with S = M and s0 = ∅.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 24 / 37

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More questions

Can we write down rn in terms of cn?

Can we identify the limit measure λ?

Consequence: rnP(Mn > cnx) converges to some non-null functionf of x . The function f can also be identified.

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More questions

Can we write down rn in terms of cn?

Can we identify the limit measure λ?

Consequence: rnP(Mn > cnx) converges to some non-null functionf of x . The function f can also be identified.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 25 / 37

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More questions

Can we write down rn in terms of cn?

Can we identify the limit measure λ?

Consequence: rnP(Mn > cnx) converges to some non-null functionf of x . The function f can also be identified.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 25 / 37

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More questions

Can we write down rn in terms of cn?

Can we identify the limit measure λ?

Consequence: rnP(Mn > cnx) converges to some non-null functionf of x . The function f can also be identified.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 25 / 37

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Literature on large deviation for extremes

Large deviation results for maxima in BRW with light-taileddisplacement (exponentially decaying tail) have been derived byGantert and Höfelsauer (2018).

Large deviation for extremal process Hult and Samorodnitsky(2010) and Fasen and Roy (2016). (Regularly varying case).

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Main result

Theorem (B. 2018(arXiv:1802.05938v1))There exists rn such that for every “nice set” A ⊂M ,

rnP(Nn ∈ A)M0−→ λ(A)

where

λ(A) =∞∑l=1

m−lE[να(x ∈ R : Zlδx ∈ A)

].

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Main result

Theorem (B. 2018(arXiv:1802.05938v1))

There exists rn(=(mnP(|X | > cn)

)−1) such that for every “nice set”

A ⊂M ,rnP(Nn ∈ A)

M0−→ λ(A)

where

λ(A) =∞∑l=1

m−lE[να(x ∈ R : Zlδx ∈ A)

].

W (martingale limit) does not appear in the limit measure ν.

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Main result

Theorem (B. 2018(arXiv:1802.05938v1))

There exists rn(=(mnP(|X | > cn)

)−1) such that for every “nice set”

A ⊂M ,rnP(Nn ∈ A)

M0−→ λ(A)

where

λ(A) =∞∑l=1

m−lE[να(x ∈ R : Zlδx ∈ A)

].

W (martingale limit) does not appear in the limit measure ν.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 28 / 37

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Large deviation for the topmost position

CorollaryRecall that Mn denotes the position of the topmost particle in thenth generation.

Then

limn→∞

rnP(Mn > cnx) = p1

m − 1x−α for all x > 0.

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Large deviation for the topmost position

CorollaryRecall that Mn denotes the position of the topmost particle in thenth generation. Then

limn→∞

rnP(Mn > cnx) = p1

m − 1x−α for all x > 0.

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Proof of consequence: large deviation for maxima

Fix x > 0.rnP(Mn > cnx

)

= rnP(Nn(x ,∞) ≥ 1

)= rnP

(Nn ∈

{ξ ∈M : ξ(x ,∞) ≥ 1

})n→∞−→ λ

({ξ : ξ(x ,∞) ≥ 1

})= p 1

m−1x−α

This can be done for the joint distribution of topmost andbottommost position, first k-order statistics.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 30 / 37

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Proof of consequence: large deviation for maxima

Fix x > 0.rnP(Mn > cnx

)= rnP

(Nn(x ,∞) ≥ 1

)

= rnP(Nn ∈

{ξ ∈M : ξ(x ,∞) ≥ 1

})n→∞−→ λ

({ξ : ξ(x ,∞) ≥ 1

})= p 1

m−1x−α

This can be done for the joint distribution of topmost andbottommost position, first k-order statistics.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 30 / 37

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Proof of consequence: large deviation for maxima

Fix x > 0.rnP(Mn > cnx

)= rnP

(Nn(x ,∞) ≥ 1

)= rnP

(Nn ∈

{ξ ∈M : ξ(x ,∞) ≥ 1

})

n→∞−→ λ({ξ : ξ(x ,∞) ≥ 1

})= p 1

m−1x−α

This can be done for the joint distribution of topmost andbottommost position, first k-order statistics.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 30 / 37

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Proof of consequence: large deviation for maxima

Fix x > 0.rnP(Mn > cnx

)= rnP

(Nn(x ,∞) ≥ 1

)= rnP

(Nn ∈

{ξ ∈M : ξ(x ,∞) ≥ 1

})n→∞−→ λ

({ξ : ξ(x ,∞) ≥ 1

})

= p 1m−1x

−α

This can be done for the joint distribution of topmost andbottommost position, first k-order statistics.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 30 / 37

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Proof of consequence: large deviation for maxima

Fix x > 0.rnP(Mn > cnx

)= rnP

(Nn(x ,∞) ≥ 1

)= rnP

(Nn ∈

{ξ ∈M : ξ(x ,∞) ≥ 1

})n→∞−→ λ

({ξ : ξ(x ,∞) ≥ 1

})= p 1

m−1x−α

This can be done for the joint distribution of topmost andbottommost position, first k-order statistics.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 30 / 37

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Proof of consequence: large deviation for maxima

Fix x > 0.rnP(Mn > cnx

)= rnP

(Nn(x ,∞) ≥ 1

)= rnP

(Nn ∈

{ξ ∈M : ξ(x ,∞) ≥ 1

})n→∞−→ λ

({ξ : ξ(x ,∞) ≥ 1

})= p 1

m−1x−α

This can be done for the joint distribution of topmost andbottommost position, first k-order statistics.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 30 / 37

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Proof strategy: Principle of single large disp.

Step 1 - One large displacement. It is enough to study anotherpoint process of the displacements upto nth generation due to atmost one large jump in every path.

0 time•

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 31 / 37

Page 90: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Proof strategy: Principle of single large disp.

Step 1 - One large displacement. It is enough to study anotherpoint process of the displacements upto nth generation due to atmost one large jump in every path.

0 time•

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 31 / 37

Page 91: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Proof strategy: contd.......

Step 2 - Cutting the tree (locate the large displacement). Cutthe tree at the (n − K )th generation and forget whateverhappened in the first (n−K ) generations. With high probability,one large displacement is contained in the last K generations.

0 time

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 32 / 37

Page 92: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Proof strategy: continued ......

Advantages of cutting: Get Zn−K independent copies of theindependently and identically point processes.Each of the subtrees have equal probability to contain the largejump.

0 time

••

••

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 33 / 37

Page 93: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Proof strategy: contd.......

Compute the contribution of the large jump at the K th generation ofthe subtrees.

Step 3 - Pruning

Step 4 - Regularization

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 34 / 37

Page 94: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Proof strategy: contd.......

Compute the contribution of the large jump at the K th generation ofthe subtrees.

Step 3 - Pruning

Step 4 - Regularization

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 34 / 37

Page 95: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Proof strategy: contd.......

Compute the contribution of the large jump at the K th generation ofthe subtrees.

Step 3 - Pruning

Step 4 - Regularization

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 34 / 37

Page 96: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Proof strategy: contd.......

Compute the contribution of the large jump at the K th generation ofthe subtrees.

Step 3 - Pruning

Step 4 - Regularization

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 34 / 37

Page 97: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Weakening assumptions

No leaf assumption is not necessary.

Large deviation for P(Nn ∈ A| survival of tree).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 35 / 37

Page 98: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Weakening assumptions

No leaf assumption is not necessary.

Large deviation for P(Nn ∈ A| survival of tree).

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 35 / 37

Page 99: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

The displacements associated to the children from same parentcan be dependent.

If the number of children of a particle is bounded almost surely,then it is easy to use multivariate regular variation.

In general, it is not customary to have bounded number ofchildren of a particle. Remedy: regular variation on the spaceRN developed in Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014).

The limit measure λ changes.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 36 / 37

Page 100: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

The displacements associated to the children from same parentcan be dependent.

If the number of children of a particle is bounded almost surely,then it is easy to use multivariate regular variation.

In general, it is not customary to have bounded number ofchildren of a particle. Remedy: regular variation on the spaceRN developed in Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014).

The limit measure λ changes.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 36 / 37

Page 101: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

The displacements associated to the children from same parentcan be dependent.

If the number of children of a particle is bounded almost surely,then it is easy to use multivariate regular variation.

In general, it is not customary to have bounded number ofchildren of a particle. Remedy: regular variation on the spaceRN developed in Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014).

The limit measure λ changes.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 36 / 37

Page 102: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

The displacements associated to the children from same parentcan be dependent.

If the number of children of a particle is bounded almost surely,then it is easy to use multivariate regular variation.

In general, it is not customary to have bounded number ofchildren of a particle. Remedy: regular variation on the spaceRN developed in Hult and Lindskog (2006), Lindskog, Resnickand Roy (2014).

The limit measure λ changes.

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 36 / 37

Page 103: Large deviation for extremes in BRW with regularly varying ...€¦ · Large deviation for extremes in BRW with regularly varying displacements AyanBhattacharya Centrum Wiskunde &

Thank you

Ayan Bhattacharya (C.W.I.) LDP for BRW June 20, 2018 37 / 37


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