[PPT] - Extreme value statistics Density of near-extreme events Sanjib PowerPoint Presentation

SLIDE 1

Extreme value statistics

Density of near-extreme events Sanjib Sabhapandit

Laboratoire de Physique Th´ eorique et Mod` eles Statistiques CNRS UMR 8626 — Universit´ e Paris-Sud 91405 Orsay cedex, France

Collaborator Satya N. Majumdar

Ref. Phys. Rev. Lett. 98, 140201 (2007).

SLIDE 2

Extreme value statistics: The statistics of the maximum or the minimum value

f a set of random observations {X1, X2, . . . , XN}.
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 1 / 26

SLIDE 3

Outline

1

a brief historical introduction to the extreme value statistics

2

extreme value statistics of i.i.d. random variables

◮ the three limiting distributions ◮ their appearence in other problems

3

extreme value statistics in weakly correlated variables

4

near-extreme events

◮ density of states with respect to the extreme value ◮ limiting behavior of the mean density of states ◮ illustration with explicit examples ◮ comparison with Yamal summer temperature data

5

summary

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 2 / 26

SLIDE 4

A brief history of extreme value statistics

Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t.

Ans. t N/(N + 1)

(see [Gumbel (1958)])

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 3 / 26

SLIDE 5

A brief history of extreme value statistics

Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t.

Ans. t N/(N + 1)

(see [Gumbel (1958)]) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range (= max − min) of random samples from the Gaussian distribution.

[von Bortkiewicz (1922)]

—introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks]

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 3 / 26

SLIDE 6

A brief history of extreme value statistics

Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t.

Ans. t N/(N + 1)

(see [Gumbel (1958)]) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range (= max − min) of random samples from the Gaussian distribution.

[von Bortkiewicz (1922)]

—introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value.

[von Mises (1923)]

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 3 / 26

SLIDE 7

A brief history of extreme value statistics

Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t.

Ans. t N/(N + 1)

(see [Gumbel (1958)]) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range (= max − min) of random samples from the Gaussian distribution.

[von Bortkiewicz (1922)]

—introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value.

[von Mises (1923)]

Edward Lewis Dodd (1923): The median.

[Dodd (1923)]

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 3 / 26

SLIDE 8

A brief history of extreme value statistics

Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t.

Ans. t N/(N + 1)

(see [Gumbel (1958)]) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range (= max − min) of random samples from the Gaussian distribution.

[von Bortkiewicz (1922)]

—introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value.

[von Mises (1923)]

Edward Lewis Dodd (1923): The median.

[Dodd (1923)]

Maurice Ren´ e Fr´ echet (1927): Obtained two of the three kinds of extreme-value distribution.

[Fr´ echet (1927)] (not printed until 1928)

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 3 / 26

SLIDE 9

A brief history of extreme value statistics

Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t.

Ans. t N/(N + 1)

(see [Gumbel (1958)]) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range (= max − min) of random samples from the Gaussian distribution.

[von Bortkiewicz (1922)]

—introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value.

[von Mises (1923)]

Edward Lewis Dodd (1923): The median.

[Dodd (1923)]

Maurice Ren´ e Fr´ echet (1927): Obtained two of the three kinds of extreme-value distribution.

[Fr´ echet (1927)] (not printed until 1928)

Sir Ronald Aylmer Fisher & Leonard Henry Caleb Tippett (1928): Obtained three types of distributions and showed that extreme limit distributions can only be one of the three types.

[Fisher & Tippett (1928)]

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 3 / 26

SLIDE 10

A brief history of extreme value statistics

Nicolas Bernouilli (1709): The mean largest distance from the origin given n points lying at random on a straight line of fixed length t.

Ans. t N/(N + 1)

(see [Gumbel (1958)]) Ladislaus von Bortkiewicz (1922): Dealt with the distribution of the range (= max − min) of random samples from the Gaussian distribution.

[von Bortkiewicz (1922)]

—introduced the concept of distribution of largest values. [best known for showing that the Poisson distribution provides a good fit to the number of Prussian army soldiers killed by horse kicks] Richard von Mises (1923): The expected value.

[von Mises (1923)]

Edward Lewis Dodd (1923): The median.

[Dodd (1923)]

Maurice Ren´ e Fr´ echet (1927): Obtained two of the three kinds of extreme-value distribution.

[Fr´ echet (1927)] (not printed until 1928)

Sir Ronald Aylmer Fisher & Leonard Henry Caleb Tippett (1928): Obtained three types of distributions and showed that extreme limit distributions can only be one of the three types.

[Fisher & Tippett (1928)]

Boris Vladimirovich Gnedenko (1943): Provided rigorous foundation and necessary and sufficient conditions for the weak convergence to the extreme limit distributions.

[Gnedenko (1943)]

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 3 / 26

SLIDE 11

A brief history ... (II): applications

Emil Julius Gumbel

[Gumbel (1958)] ◮ Les intervalles extrˆ

emes entre les ´ emissions radio-actives. (Extreme intervals between radioactive emissions)

J. Phys. Radium. 8, 321-329, (1937).

◮ Les intervalles extrˆ

emes entre les ´ emissions radioactives. II

J. Phys. Radium 8, 446-452 (1937)

◮ The return period of flood flows.

Ann. Math. Statistics 12, 163–190 (1941).

◮ Probability-interpretation of the observed return-periods of floods.

Trans. Amer. Geophys. Union 1941, 836–850 (1941).

◮ On the frequency distribution of extreme values in meteorological data.

Bull. Amer. Meteorol. Soc. 23, 95–105 (1942).

◮ Statistical forecast of droughts.

Bull. Int. Assoc. Sci. Hydrol. 8, 5-23 (1963).
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 4 / 26

SLIDE 12

A brief history ... (II): applications

Emil Julius Gumbel

[Gumbel (1958)] ◮ Les intervalles extrˆ

emes entre les ´ emissions radio-actives. (Extreme intervals between radioactive emissions)

J. Phys. Radium. 8, 321-329, (1937).

◮ Les intervalles extrˆ

emes entre les ´ emissions radioactives. II

J. Phys. Radium 8, 446-452 (1937)

◮ The return period of flood flows.

Ann. Math. Statistics 12, 163–190 (1941).

◮ Probability-interpretation of the observed return-periods of floods.

Trans. Amer. Geophys. Union 1941, 836–850 (1941).

◮ On the frequency distribution of extreme values in meteorological data.

Bull. Amer. Meteorol. Soc. 23, 95–105 (1942).

◮ Statistical forecast of droughts.

Bull. Int. Assoc. Sci. Hydrol. 8, 5-23 (1963).

Wallodi Weibull:

◮ A statistical theory of the strength of material (transl.).

Ingvetensk. Akad. Handl 151, 1-45 (1939).

◮ A statistical distribution function of wide applicability.

J. Appl. Mech. 18, 293-277 (1951).
1. Yield strength of a Bofors steel.
2. Size distribution of fly ash.
3. Fiber strength of Indian cotton.
4. Length of cytoidea (Worm length for ancient sedimentary deposits).
5. Fatigue life of a St-37 steel.
6. Statures for adult males, born in the British Isles.
7. Breadth of beans of Phaseolux Vulgaris.
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 4 / 26

SLIDE 13

Recent applications

Physics

◮ Disordered systems: The low temperature physics of disordered systems is

governed by the statistics of extremely low energy states.

J.-P. Bouchaud & M. M´

ezard, JPA 30, 7997 (1997).

D.S. Dean & S.N. Majumdar, PRE 64, 046121 (2001).

◮ Random matrix theory

C.A. Tracy & H. Widom, Comm. Math. Phys. 159, 151 (1994); 177, 727 (1996).
D.S. Dean & S.N. Majumdar, PRL 97, 160201 (2006).

◮ Fluctuating interfaces

Gy¨
rgyi, Holdsworth, Portelli, & R´

acz, PRE 68, 056116 (2003).

S.N. Majumdar & A. Comtet PRL 92, 225501 (2004).

Computer science

◮ Search tree problems

S.N. Majumdar & P.L. Krapivsky, PRE 65, 036127 (2002).

◮ Optimization problems

S.N. Majumdar & P.L. Krapivsky, PRE 62, 7735 (2000).

Finance e.g. see P. Embrechts, C. Kl¨ uppelberg, & T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer, Berlin, 1997).

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 5 / 26

SLIDE 14

Extreme value statistics of i.i.d. random variables

{X1, X2, . . . , XN

each drawn from p(X) ←

− parent distribution

}

set of i.i.d. random variables

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 6 / 26

SLIDE 15

Extreme value statistics of i.i.d. random variables

{X1, X2, . . . , XN

each drawn from p(X) ←

− parent distribution

}

set of i.i.d. random variables Xmax := max(X1, X2, . . . , XN) random variable

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 6 / 26

SLIDE 16

Extreme value statistics of i.i.d. random variables

{X1, X2, . . . , XN

each drawn from p(X) ←

− parent distribution

}

set of i.i.d. random variables Xmax := max(X1, X2, . . . , XN) random variable QN(x) := Prob[Xmax ≤ x] = Prob[X1 ≤ x, X2 ≤ x, . . . , XN ≤ x]

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 6 / 26

SLIDE 17

Extreme value statistics of i.i.d. random variables

{X1, X2, . . . , XN

each drawn from p(X) ←

− parent distribution

}

set of i.i.d. random variables Xmax := max(X1, X2, . . . , XN) random variable QN(x) := Prob[Xmax ≤ x] = Prob[X1 ≤ x, X2 ≤ x, . . . , XN ≤ x] Independence ⇒ QN(x) =

x

−∞

p(x′) dx′

N

=

1 −

∞

x

p(x′) dx′

N

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 6 / 26

SLIDE 18

Extreme value statistics of i.i.d. random variables

{X1, X2, . . . , XN

each drawn from p(X) ←

− parent distribution

}

set of i.i.d. random variables Xmax := max(X1, X2, . . . , XN) random variable QN(x) := Prob[Xmax ≤ x] = Prob[X1 ≤ x, X2 ≤ x, . . . , XN ≤ x] Independence ⇒ QN(x) =

x

−∞

p(x′) dx′

N

=

1 −

∞

x

p(x′) dx′

N

Scaling limit: X large, N large QN(x)

x→∞, N→∞

− − − − − − − − − − →

(x−aN)/bN fixed

F

x − aN

bN

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 6 / 26

SLIDE 19

Extreme value statistics of i.i.d. random variables

{X1, X2, . . . , XN

each drawn from p(X) ←

− parent distribution

}

set of i.i.d. random variables Xmax := max(X1, X2, . . . , XN) random variable QN(x) := Prob[Xmax ≤ x] = Prob[X1 ≤ x, X2 ≤ x, . . . , XN ≤ x] Independence ⇒ QN(x) =

x

−∞

p(x′) dx′

N

=

1 −

∞

x

p(x′) dx′

N

Scaling limit: X large, N large QN(x)

x→∞, N→∞

− − − − − − − − − − →

(x−aN)/bN fixed

F

x − aN

bN

r

lim

N→∞ QN(aN + bNz) = F(z)

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 6 / 26

SLIDE 20

Extreme value statistics of i.i.d. random variables

{X1, X2, . . . , XN

each drawn from p(X) ←

− parent distribution

}

set of i.i.d. random variables Xmax := max(X1, X2, . . . , XN) random variable QN(x) := Prob[Xmax ≤ x] = Prob[X1 ≤ x, X2 ≤ x, . . . , XN ≤ x] Independence ⇒ QN(x) =

x

−∞

p(x′) dx′

N

=

1 −

∞

x

p(x′) dx′

N

Scaling limit: X large, N large QN(x)

x→∞, N→∞

− − − − − − − − − − →

(x−aN)/bN fixed

F

x − aN

bN

r

lim

N→∞ QN(aN + bNz) = F(z)

aN, bN non-universal scale factors dependent on p(X)

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 6 / 26

SLIDE 21

Extreme value statistics of i.i.d. random variables

{X1, X2, . . . , XN

each drawn from p(X) ←

− parent distribution

}

set of i.i.d. random variables Xmax := max(X1, X2, . . . , XN) random variable QN(x) := Prob[Xmax ≤ x] = Prob[X1 ≤ x, X2 ≤ x, . . . , XN ≤ x] Independence ⇒ QN(x) =

x

−∞

p(x′) dx′

N

=

1 −

∞

x

p(x′) dx′

N

Scaling limit: X large, N large QN(x)

x→∞, N→∞

− − − − − − − − − − →

(x−aN)/bN fixed

F

x − aN

bN

r

lim

N→∞ QN(aN + bNz) = F(z)

aN, bN non-universal scale factors dependent on p(X) F1(z) or F2(z) or F3(z) — depending on the tails of p(X) F(z) universal scaling function: only of three possible types

[Fr´ echet (1927), Fisher & Tippett (1928), Gnedenko (1943), Gumbel (1958)]

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 6 / 26

SLIDE 22

The three limiting extreme value distributions

1

Fr´ echet: If p(X) has power-law tail — p(X) ∼ X −(1+α). F1(z) =

exp
−z−α

for z ≥ 0, for z ≤ 0.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 7 / 26

SLIDE 23

The three limiting extreme value distributions

1

Fr´ echet: If p(X) has power-law tail — p(X) ∼ X −(1+α). F1(z) =

exp
−z−α

for z ≥ 0, for z ≤ 0. pdf: f1(z) = α exp

−z−α

z1+α

, α = 3/2

z ∈ [0, ∞).

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 7 / 26

4 2 2 4 6 0.2 0.4 0.6 0.8 Z

SLIDE 24

The three limiting extreme value distributions

1

Fr´ echet: If p(X) has power-law tail — p(X) ∼ X −(1+α). F1(z) =

exp
−z−α

for z ≥ 0, for z ≤ 0. pdf: f1(z) = α exp

−z−α

z1+α

, α = 3/2

z ∈ [0, ∞).

2

Gumbel: If p(X) has faster than power-law but unbounded tail — p(X) ∼ exp(−X δ). F2(z) = exp

−e−z

.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 7 / 26

4 2 2 4 6 0.2 0.4 0.6 0.8 Z

SLIDE 25

The three limiting extreme value distributions

1

Fr´ echet: If p(X) has power-law tail — p(X) ∼ X −(1+α). F1(z) =

exp
−z−α

for z ≥ 0, for z ≤ 0. pdf: f1(z) = α exp

−z−α

z1+α

, α = 3/2

z ∈ [0, ∞).

2

Gumbel: If p(X) has faster than power-law but unbounded tail — p(X) ∼ exp(−X δ). F2(z) = exp

−e−z

.

pdf: f2(z) = exp

−z − e−z

,

z ∈ (−∞, ∞).

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 7 / 26

4 2 2 4 6 0.2 0.4 0.6 0.8 4 2 2 4 6 0.2 0.4 0.6 0.8 Z

SLIDE 26

The three limiting extreme value distributions

1

Fr´ echet: If p(X) has power-law tail — p(X) ∼ X −(1+α). F1(z) =

exp
−z−α

for z ≥ 0, for z ≤ 0. pdf: f1(z) = α exp

−z−α

z1+α

, α = 3/2

z ∈ [0, ∞).

2

Gumbel: If p(X) has faster than power-law but unbounded tail — p(X) ∼ exp(−X δ). F2(z) = exp

−e−z

.

pdf: f2(z) = exp

−z − e−z

,

z ∈ (−∞, ∞).

3

Weibull: If p(X) is bounded — p(X) ∼ (a − X)β−1 as X → a−. F3(z) =

exp
−(−z)β

for z ≤ 0, 1 for z ≥ 0.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 7 / 26

4 2 2 4 6 0.2 0.4 0.6 0.8 4 2 2 4 6 0.2 0.4 0.6 0.8 Z

SLIDE 27

The three limiting extreme value distributions

1

Fr´ echet: If p(X) has power-law tail — p(X) ∼ X −(1+α). F1(z) =

exp
−z−α

for z ≥ 0, for z ≤ 0. pdf: f1(z) = α exp

−z−α

z1+α

, α = 3/2

z ∈ [0, ∞).

2

Gumbel: If p(X) has faster than power-law but unbounded tail — p(X) ∼ exp(−X δ). F2(z) = exp

−e−z

.

pdf: f2(z) = exp

−z − e−z

,

z ∈ (−∞, ∞).

3

Weibull: If p(X) is bounded — p(X) ∼ (a − X)β−1 as X → a−. F3(z) =

exp
−(−z)β

for z ≤ 0, 1 for z ≥ 0. pdf: f3(z) = β(−z)β−1 exp

−(−z)β

, β = 3/2

z ∈ (−∞, 0].

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 7 / 26

4 2 2 4 6 0.2 0.4 0.6 0.8 4 2 2 4 6 0.2 0.4 0.6 0.8 4 2 2 4 6 0.2 0.4 0.6 0.8 Z

SLIDE 28

...in Bose gas or integer partition problem

[A. Comtet, P. Leboeuf & S.N. Majumdar (2007)]

Q(N|E) := Prob[Nex ≤ N, given energy E] — the cumulative distribution of the no. of excited particles Nex particles

f an ideal gas of bosons with fixed total energy E and with single particle

density of states ρ(ǫ) = νǫν−1.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 8 / 26

SLIDE 29

...in Bose gas or integer partition problem

[A. Comtet, P. Leboeuf & S.N. Majumdar (2007)]

Q(N|E) := Prob[Nex ≤ N, given energy E] — the cumulative distribution of the no. of excited particles Nex particles

f an ideal gas of bosons with fixed total energy E and with single particle

density of states ρ(ǫ) = νǫν−1. := C(N|E)

# partitions of E into ≤ N parts

Ω(E)

# partitions of E

E =

i

nii1/ν — the cumulative distribution of the no. of parts in partitions of E.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 8 / 26

SLIDE 30

...in Bose gas or integer partition problem

[A. Comtet, P. Leboeuf & S.N. Majumdar (2007)]

Q(N|E) := Prob[Nex ≤ N, given energy E] — the cumulative distribution of the no. of excited particles Nex particles

f an ideal gas of bosons with fixed total energy E and with single particle

density of states ρ(ǫ) = νǫν−1. := C(N|E)

# partitions of E into ≤ N parts

Ω(E)

# partitions of E

E =

i

nii1/ν — the cumulative distribution of the no. of parts in partitions of E. In terms of a suitable rescaled variable z ≡ z(N, E, ν): Q(N|E)

scaling limit

− − − − − − − →

1≪N≪E

        

F1(z) if 0 < ν < 1 (Fr´ echet) F2(z) if ν = 1 (Gumbel) F3(z) if ν > 1 (Weibull) Is there any connection to the extreme value statistics?

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 8 / 26

SLIDE 31

Single-file diffusion front

Particles diffusing in a narrow channel —so they cannot pass one another.

ρ := density

D := diffusion constant y(t) P(x, t) := Prob

y(t) ≤ x

ρ

√

Dt≫1

− − − − − → F2

x − a(t)

b(t)

a(t) =

√

4Dt

ln

ρ √

Dt 2√π

1/2

+ · · · b(t) =

√

Dt

ln

ρ √

Dt 2√π

−1/2

+ · · · F2(z) = exp

− exp(−z)
[SS, J. Stat. Mech. (2007) L05002]
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 9 / 26

SLIDE 32

Maximum of weakly correlated random variables

ξ ξ ξ ξ ξ ξ M1 M2M3 M4 M5 M6 Xi

Xmax = max(X1, X2, . . . , XN

correlated

) = max(M1, M2, . . . , Mn

“uncorrelated”

), n = N

ξ ≫ 1.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 10 / 26

SLIDE 33

Maximum of weakly correlated random variables

ξ ξ ξ ξ ξ ξ M1 M2M3 M4 M5 M6 Xi

“limit laws of i.i.d. random variables” Xmax = max(X1, X2, . . . , XN

correlated

) = max(M1, M2, . . . , Mn

“uncorrelated”

), n = N

ξ ≫ 1.

Extreme values in samples from m-dependent stationary stochastic processes

[Watson (1954)]

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 10 / 26

SLIDE 34

For a stationary Gaussian sequence

p(X1, X2, . . . , XN) = 1

(2π)Ndet(Σ)

exp

− 1

2 XTΣ−1X

X =

   

X1 X2 . . . XN

    ,

Σ =

   

C0 C1

. . .

CN−1 C1 C0

. . .

CN−2 . . . . . . ... . . . CN−1 CN−2

. . .

C0

    ,

Cn ≡ XiXi+n Xmax := max(X1, X2, . . . , XN) If either lim

n→∞ Cn ln n = 0

r

∞

n=1

C2

n < ∞

      

then lim

N→∞ Prob[Xmax ≤ aN + bNz] = exp(−e−z)

Gumbel

[Berman (1964)]

aN =

√

2 ln N − ln ln N + ln 4π 2

√

2 ln N bN = 1

√

2 ln N

      

same as in the case of independent Gaussian random variables.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 11 / 26

SLIDE 35

Near-extreme events

While EVS is very important, an equally important issue concerns the near-extreme events: How many events occur with their values near the extreme? Whether the global maximum (or minimum) value is very far from

thers,
r

whether there are many other events whose values are close to the maximum (minimum) value.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 12 / 26

SLIDE 36

Near-extreme events

While EVS is very important, an equally important issue concerns the near-extreme events: How many events occur with their values near the extreme? Whether the global maximum (or minimum) value is very far from

thers,
r

whether there are many other events whose values are close to the maximum (minimum) value. The issue of the crowding of near-extreme events arises in many problems:

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 12 / 26

SLIDE 37

Near-extreme events

While EVS is very important, an equally important issue concerns the near-extreme events: How many events occur with their values near the extreme? Whether the global maximum (or minimum) value is very far from

thers,
r

whether there are many other events whose values are close to the maximum (minimum) value. The issue of the crowding of near-extreme events arises in many problems: Disordered systems at low temperature: the spectral density function of the excited states near the ground state.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 12 / 26

SLIDE 38

Near-extreme events

While EVS is very important, an equally important issue concerns the near-extreme events: How many events occur with their values near the extreme? Whether the global maximum (or minimum) value is very far from

thers,
r

whether there are many other events whose values are close to the maximum (minimum) value. The issue of the crowding of near-extreme events arises in many problems: Disordered systems at low temperature: the spectral density function of the excited states near the ground state. Weather and climate extremes: how often do extreme temperature events such as heat waves and cold waves occur?

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 12 / 26

SLIDE 39

Near-extreme events

While EVS is very important, an equally important issue concerns the near-extreme events: How many events occur with their values near the extreme? Whether the global maximum (or minimum) value is very far from

thers,
r

whether there are many other events whose values are close to the maximum (minimum) value. The issue of the crowding of near-extreme events arises in many problems: Disordered systems at low temperature: the spectral density function of the excited states near the ground state. Weather and climate extremes: how often do extreme temperature events such as heat waves and cold waves occur? Insurance: unexpectedly high number of excessively large claims.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 12 / 26

SLIDE 40

Near-extreme events

While EVS is very important, an equally important issue concerns the near-extreme events: How many events occur with their values near the extreme? Whether the global maximum (or minimum) value is very far from

thers,
r

whether there are many other events whose values are close to the maximum (minimum) value. The issue of the crowding of near-extreme events arises in many problems: Disordered systems at low temperature: the spectral density function of the excited states near the ground state. Weather and climate extremes: how often do extreme temperature events such as heat waves and cold waves occur? Insurance: unexpectedly high number of excessively large claims. Optimization problems: number of near-optimal solutions.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 12 / 26

SLIDE 41

A measure of the crowding of near-extreme events

The density of states with respect to the maximum:

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

Normalization:

∞ ρ(r, N) dr = 1 − 1

N Note: even though the random variables are independent, the different terms become correlated through their common maximum Xmax.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 13 / 26

SLIDE 42

A measure of the crowding of near-extreme events

The density of states with respect to the maximum:

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

Normalization:

∞ ρ(r, N) dr = 1 − 1

N Note: even though the random variables are independent, the different terms become correlated through their common maximum Xmax.

ρ(r, N) fluctuates from one realization of {X1, X2, . . . , XN} to another:

— the statistical properties of ρ(r, N) ? Does it show any general limiting behavior? (In the same sense, as one finds for the extreme value statistics).

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 13 / 26

SLIDE 43

A measure of the crowding of near-extreme events

The density of states with respect to the maximum:

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

Normalization:

∞ ρ(r, N) dr = 1 − 1

N Note: even though the random variables are independent, the different terms become correlated through their common maximum Xmax.

ρ(r, N) fluctuates from one realization of {X1, X2, . . . , XN} to another:

— the statistical properties of ρ(r, N) ? Does it show any general limiting behavior? (In the same sense, as one finds for the extreme value statistics). We find that ρ(r, N) displays general limiting behavior as N → ∞.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 13 / 26

SLIDE 44

Limiting behavior of mean density of states

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

ρ(r, N) converges to three different limiting forms depending on whether:

1

p(X) has slower than exp(−x) tail

2

p(X) has faster than exp(−x) tail

3

p(X) has exp(−x) tail

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 14 / 26

SLIDE 45

Limiting behavior of mean density of states

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

ρ(r, N) converges to three different limiting forms depending on whether:

1

p(X) has slower than exp(−x) tail:

ρ(r, N)

N→∞

− − − →

1 bN f

r − aN

bN

where

f (z) ≡ f1(z) or f2(z)

2

p(X) has faster than exp(−x) tail

3

p(X) has exp(−x) tail

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 14 / 26

SLIDE 46

Limiting behavior of mean density of states

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

ρ(r, N) converges to three different limiting forms depending on whether:

1

p(X) has slower than exp(−x) tail:

ρ(r, N)

N→∞

− − − →

1 bN f

r − aN

bN

where

f (z) ≡ f1(z) or f2(z)

2

p(X) has faster than exp(−x) tail:

ρ(r, N)

N→∞

− − − → p(aN − r)

3

p(X) has exp(−x) tail

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 14 / 26

SLIDE 47

Limiting behavior of mean density of states

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

ρ(r, N) converges to three different limiting forms depending on whether:

1

p(X) has slower than exp(−x) tail:

ρ(r, N)

N→∞

− − − →

1 bN f

r − aN

bN

where

f (z) ≡ f1(z) or f2(z)

2

p(X) has faster than exp(−x) tail:

ρ(r, N)

N→∞

− − − → p(aN − r)

3

p(X) has exp(−x) tail:

ρ(r, N) = g(r − aN),

where g(z) = ez 1 −

1 + e−z

e−e−z

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 14 / 26

SLIDE 48

Derivation of the mean density of states

1

First consider

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

for a given value of the maximum at Xmax = y.
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 15 / 26

SLIDE 49

Derivation of the mean density of states

1

First consider

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

for a given value of the maximum at Xmax = y.

2

Given Xmax = y, the rest of the (N − 1) variables are distributed independently according to the common conditional pdf pcond(X|y) =

y

−∞

p(w) dw

−1

p(X) θ(y − X).

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 15 / 26

SLIDE 50

Derivation of the mean density of states

1

First consider

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

for a given value of the maximum at Xmax = y.

2

Given Xmax = y, the rest of the (N − 1) variables are distributed independently according to the common conditional pdf pcond(X|y) =

y

−∞

p(w) dw

−1

p(X) θ(y − X).

3

The conditional mean is

ρcond(r, N|y) =

N − 1

N

pcond(y − r|y).
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 15 / 26

SLIDE 51

Derivation of the mean density of states

1

First consider

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

for a given value of the maximum at Xmax = y.

2

Given Xmax = y, the rest of the (N − 1) variables are distributed independently according to the common conditional pdf pcond(X|y) =

y

−∞

p(w) dw

−1

p(X) θ(y − X).

3

The conditional mean is

ρcond(r, N|y) =

N − 1

N

pcond(y − r|y).

4

In terms of the conditional mean

ρ(r, N) = ∞

−∞

ρcond(r, N|y) pmax(y, N)

= Np(y)

y

−∞

p(w) dw

N−1

dy.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 15 / 26

SLIDE 52

Derivation of the mean density of states

1

First consider

ρ(r, N) = 1

N

N−1

{Xi=Xmax}

δ

r − ({

= max(X1, X2, . . . , XN

i.i.d. from p(X)

) Xmax − Xi)

for a given value of the maximum at Xmax = y.

2

Given Xmax = y, the rest of the (N − 1) variables are distributed independently according to the common conditional pdf pcond(X|y) =

y

−∞

p(w) dw

−1

p(X) θ(y − X).

3

The conditional mean is

ρcond(r, N|y) =

N − 1

N

pcond(y − r|y).

4

In terms of the conditional mean

ρ(r, N) = ∞

−∞

ρcond(r, N|y) pmax(y, N)

= Np(y)

y

−∞

p(w) dw

N−1

dy.

5

Finally putting everything together

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx (valid for all N).

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 15 / 26

SLIDE 53

Analyzing the limiting behavior for large N

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx (valid for all N) bN pmax(x = aN + bNz, N)

N→∞

− − − → f (z)

where f (z) ≡ (Fr´ echet) f1(z), (Gumbel) f2(z) or (Weibull) f3(z)

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 16 / 26

SLIDE 54

Analyzing the limiting behavior for large N

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx (valid for all N) bN pmax(x = aN + bNz, N)

N→∞

− − − → f (z)

where f (z) ≡ (Fr´ echet) f1(z), (Gumbel) f2(z) or (Weibull) f3(z) Example If p(X) ∼ exp(−X δ) for large X, then: f (z) ≡ f2(z), aN ∼ (ln N)1/δ and bN ∼ δ−1(ln N)1/δ−1 for large N.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 16 / 26

SLIDE 55

Analyzing the limiting behavior for large N

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx (valid for all N) bN pmax(x = aN + bNz, N)

N→∞

− − − → f (z)

where f (z) ≡ (Fr´ echet) f1(z), (Gumbel) f2(z) or (Weibull) f3(z) Example If p(X) ∼ exp(−X δ) for large X, then: f (z) ≡ f2(z), aN ∼ (ln N)1/δ and bN ∼ δ−1(ln N)1/δ−1 for large N.

1

If p(X) has slower than exp(−x) tail (e.g. δ < 1), bN → ∞ as N → ∞. x = aN + bNz ⇒ bNp(bNz + aN − r)

N→∞

− − − → δ

z − r − aN

bN

.

N → ∞

ρ(r, N)

N→∞

− − − →

1 bN f

r − aN

bN

.
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 16 / 26

SLIDE 56

Analyzing the limiting behavior for large N

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx (valid for all N) bN pmax(x = aN + bNz, N)

N→∞

− − − → f (z)

where f (z) ≡ (Fr´ echet) f1(z), (Gumbel) f2(z) or (Weibull) f3(z) Example If p(X) ∼ exp(−X δ) for large X, then: f (z) ≡ f2(z), aN ∼ (ln N)1/δ and bN ∼ δ−1(ln N)1/δ−1 for large N.

1

If p(X) has slower than exp(−x) tail (e.g. δ < 1), bN → ∞ as N → ∞. x = aN + bNz ⇒ bNp(bNz + aN − r)

N→∞

− − − → δ

z − r − aN

bN

.

N → ∞

ρ(r, N)

N→∞

− − − →

1 bN f

r − aN

bN

.

2

If p(X) has faster than exp(−x) tail (e.g. δ > 1), bN → 0 as N → ∞.

⇒ pmax(x, N) → δ(x − aN). ρ(r, N)

N→∞

− − − → p(aN − r).

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 16 / 26

SLIDE 57

Next: examples with explicit forms of p(X)

1

slower than pure exponential p(X) = α exp(−X −α) X 1+α

, α > 0,

X ∈ [0, ∞)

A

p(X) = δX δ−1 exp(−X δ),

δ < 1,

X ∈ [0, ∞)

B 2

pure exponential p(X) = exp(−X), X ∈ [0, ∞)

3

faster than pure exponential p(X) = δX δ−1 exp(−X δ),

δ > 1,

X ∈ [0, ∞)

A

p(X) = βa−β(a − X)β−1, β > 0, X ∈ [0, a]

B

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 17 / 26

SLIDE 58

Next: examples with explicit forms of p(X)

1

slower than pure exponential p(X) = α exp(−X −α) X 1+α

, α > 0,

X ∈ [0, ∞) Fr´ echet

A

p(X) = δX δ−1 exp(−X δ),

δ < 1,

X ∈ [0, ∞)

B 2

pure exponential p(X) = exp(−X), X ∈ [0, ∞) Gumbel

3

faster than pure exponential p(X) = δX δ−1 exp(−X δ),

δ > 1,

X ∈ [0, ∞)

A

p(X) = βa−β(a − X)β−1, β > 0, X ∈ [0, a] Weibull

B

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 17 / 26

SLIDE 59

Example: power-law tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = α exp(−X −α) X 1+α , X ∈ [0, ∞).

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 18 / 26

SLIDE 60

Example: power-law tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = α exp(−X −α) X 1+α , X ∈ [0, ∞).

aN = 0

bN = N1/α

N→∞

− − − → ∞

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 18 / 26

SLIDE 61

Example: power-law tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = α exp(−X −α) X 1+α , X ∈ [0, ∞).

aN = 0

bN = N1/α

N→∞

− − − → ∞

Therefore,

B

ρ(r, N)

N→∞

− − − →

1 bN f

r

bN

,

where f (z) ≡ f1(z) = α exp

−z−α

z1+α

,

z ≥ 0.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 18 / 26

SLIDE 62

Example: power-law tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = α exp(−X −α) X 1+α , X ∈ [0, ∞).

aN = 0

bN = N1/α

N→∞

− − − → ∞

Therefore,

B

ρ(r, N)

N→∞

− − − →

1 bN f

r

bN

,

where f (z) ≡ f1(z) = α exp

−z−α

z1+α

,

z ≥ 0.

1 2 3 4 5 0.0 0.2 0.4 0.6 0.8

r/bN bN ρ(r, N)

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 18 / 26

    

N = 102 N = 103 N = 104

A

(Fr´ echet)

B

SLIDE 63

Example: power-law tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = α exp(−X −α) X 1+α , X ∈ [0, ∞).

aN = 0

bN = N1/α

N→∞

− − − → ∞

Therefore,

B

ρ(r, N)

N→∞

− − − →

1 bN f

r

bN

,

where f (z) ≡ f1(z) = α exp

−z−α

z1+α

,

z ≥ 0.

1 2 3 4 5 0.0 0.2 0.4 0.6 0.8

r/bN bN ρ(r, N) For r = 0 :

ρ(0, N)

N→∞

− − − → αΓ(2 + 1/α)

N1+1/α

.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 18 / 26

    

N = 102 N = 103 N = 104

A

(Fr´ echet)

B

SLIDE 64

Example: Faster than power-law, but unbounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = δX δ−1 exp(−X δ), X ∈ [0, ∞).

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 19 / 26

SLIDE 65

Example: Faster than power-law, but unbounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = δX δ−1 exp(−X δ), X ∈ [0, ∞).

  

aN = (ln N)1/δ bN = δ−1(ln N)1/δ−1

N→∞

− − − →    ∞

for δ < 1 1 for δ = 1 for δ > 1

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 19 / 26

SLIDE 66

Example: Faster than power-law, but unbounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = δX δ−1 exp(−X δ), X ∈ [0, ∞).

  

aN = (ln N)1/δ bN = δ−1(ln N)1/δ−1

N→∞

− − − →    ∞

for δ < 1 1 for δ = 1 for δ > 1 For r = 0 :

ρ(0, N)

N→∞

− − − → p(aN) = δ

N (ln N)1−1/δ.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 19 / 26

SLIDE 67

Example: Faster than power-law, but unbounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = δX δ−1 exp(−X δ), X ∈ [0, ∞).

  

aN = (ln N)1/δ bN = δ−1(ln N)1/δ−1

N→∞

− − − →    ∞

for δ < 1 1 for δ = 1 for δ > 1 For r = 0 :

ρ(0, N)

N→∞

− − − → p(aN) = δ

N (ln N)1−1/δ.

B

ρ(r, N)

N→∞

− − − →     

1 bN f

r−aN

bN

for δ < 1

g(r − aN) for δ = 1 p(aN − r) for δ > 1 where f (z) ≡ f2(z) = exp [−z − exp(−z)] (Gumbel) g(z) = ez 1 −

1 + e−z

e−e−z

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 19 / 26

SLIDE 68

Example: Faster than power-law, but unbounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = δX δ−1 exp(−X δ), X ∈ [0, ∞).

  

aN = (ln N)1/δ bN = δ−1(ln N)1/δ−1

N→∞

− − − →    ∞

for δ < 1 1 for δ = 1 for δ > 1 For r = 0 :

ρ(0, N)

N→∞

− − − → p(aN) = δ

N (ln N)1−1/δ.

B

ρ(r, N)

N→∞

− − − →     

1 bN f

r−aN

bN

for δ < 1

g(r − aN) for δ = 1 p(aN − r) for δ > 1 where f (z) ≡ f2(z) = exp [−z − exp(−z)] (Gumbel) g(z) = ez 1 −

1 + e−z

e−e−z

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 19 / 26

  

N = 103 N = 105 N = 107

A

(Gumbel)

B

4 2 2 4 6 8 0.1 0.2 0.3 0.4 (r − aN)/bN bN ρ(r, N)

δ = 1/2

SLIDE 69

Example: Faster than power-law, but unbounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = δX δ−1 exp(−X δ), X ∈ [0, ∞).

  

aN = (ln N)1/δ bN = δ−1(ln N)1/δ−1

N→∞

− − − →    ∞

for δ < 1 1 for δ = 1 for δ > 1 For r = 0 :

ρ(0, N)

N→∞

− − − → p(aN) = δ

N (ln N)1−1/δ.

B

ρ(r, N)

N→∞

− − − →     

1 bN f

r−aN

bN

for δ < 1

g(r − aN) for δ = 1 p(aN − r) for δ > 1 where f (z) ≡ f2(z) = exp [−z − exp(−z)] (Gumbel) g(z) = ez 1 −

1 + e−z

e−e−z

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 19 / 26

N = 10

N = 50 N = 100

A

g(r − aN)

B

  

N = 103 N = 105 N = 107

A

(Gumbel)

B

4 2 2 4 6 8 0.1 0.2 0.3 0.4 6 4 2 2 4 0.1 0.2 0.3 (r − aN)/bN bN ρ(r, N)

δ = 1/2

r − aN

ρ(r, N) δ = 1

SLIDE 70

Example: Faster than power-law, but unbounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = δX δ−1 exp(−X δ), X ∈ [0, ∞).

  

aN = (ln N)1/δ bN = δ−1(ln N)1/δ−1

N→∞

− − − →    ∞

for δ < 1 1 for δ = 1 for δ > 1 For r = 0 :

ρ(0, N)

N→∞

− − − → p(aN) = δ

N (ln N)1−1/δ.

B

ρ(r, N)

N→∞

− − − →     

1 bN f

r−aN

bN

for δ < 1

g(r − aN) for δ = 1 p(aN − r) for δ > 1 where f (z) ≡ f2(z) = exp [−z − exp(−z)] (Gumbel) g(z) = ez 1 −

1 + e−z

e−e−z

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 19 / 26

  

N = 103 N = 106 N = 109

A

p(aN − r)

B

N = 10

N = 50 N = 100

A

g(r − aN)

B

  

N = 103 N = 105 N = 107

A

(Gumbel)

B

4 2 2 4 6 8 0.1 0.2 0.3 0.4 6 4 2 2 4 0.1 0.2 0.3 3 2 1 0.0 0.2 0.4 0.6 0.8 (r − aN)/bN bN ρ(r, N)

δ = 1/2

r − aN

ρ(r, N) δ = 1

r − aN

ρ(r, N) δ = 2

SLIDE 71

Bounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = βa−β(a − X)β−1, X ∈ [0, a].

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 20 / 26

SLIDE 72

Bounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = βa−β(a − X)β−1, X ∈ [0, a].

aN = a

bN = aN−1/β

N→∞

− − − → 0

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 20 / 26

SLIDE 73

Bounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = βa−β(a − X)β−1, X ∈ [0, a].

aN = a

bN = aN−1/β

N→∞

− − − → 0

B

ρ(r, N)

N→∞

− − − → p(aN − r).

r r

ρ(r, N)

β = 3 2 β = 1 2

2 4 6 8 10 0.00 0.04 0.08 0.12 0.16

2 4 6 8 10

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 20 / 26

N = 10

N = 102 N = 103

A B

  

N = 102 N = 103 N = 104

A B

SLIDE 74

Bounded tail

A

ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx for all N. Consider p(X) = βa−β(a − X)β−1, X ∈ [0, a].

aN = a

bN = aN−1/β

N→∞

− − − → 0

B

ρ(r, N)

N→∞

− − − → p(aN − r). ρ(0, N)

N→∞

− − − → (β/a)Γ(2 − 1/β)

N1−1/β

.

r r

ρ(r, N)

β = 3 2 β = 1 2

2 4 6 8 10 0.00 0.04 0.08 0.12 0.16

2 4 6 8 10

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 20 / 26

N = 10

N = 102 N = 103

A B

  

N = 102 N = 103 N = 104

A B

SLIDE 75

For a stationary Gaussian sequence...

Cn = 1 (1 + n2)γ/2

γ = 1/2 γ = 1 ρ(r, N) = ∞

−∞

p(x − r) pmax(x, N − 1) dx, where p(X) = e−X 2/2

√

2π

(c) (b) (a)

r ρ(r, N)

8 7 6 5 4 3 2 1 0.4 0.3 0.2 0.1

(a) N = 1025 (b) N = 16385 (c) N = 262145

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 21 / 26

SLIDE 76

Yamal peninsula multimillennial summer temperature reconstruction data

GEOGRAPHIC REGION: Northwestern Siberia PERIOD OF RECORD: 2067 BC - 1996 AD ORIGINAL REFERENCE: Holocene 12, 717 (2002). DATA FROM: IGBP PAGES/World Data Center for Paleoclimatology. http://www.ncdc.noaa.gov/paleo/pubs/hantemirov2002/hantemirov2002.html

(d) N = 1000

r

7 6 5

4 3 2 1

(c) N = 100

r ρ(r, N)

0.4 0.3 0.2 0.1 0.0

6 5

4 3 2 1

(a)

YEAR

∆T (oC) 4 2 −2 −4 2000 1500 1000 500 −500 −1000 −1500 −2000

p(∆T)

(b)

0.1 0.2 0.3 0.4

p(x) p(x) = 1

√

2π exp

− x2

2

aN =
2 log
N

√

4π log N

1/2

(c) 40 × 100 years. (d) 4 × 1000 years. p(aN − r) Exact numerical integration

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 22 / 26

SLIDE 77

Summary

Extreme value statistics: a brief introduction.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 23 / 26

SLIDE 78

Summary

Extreme value statistics: a brief introduction. A quantitative analysis of the phenomenon of crowding of near-extreme events in terms of the density of states with respect to the maximum of a set of independent and identically distributed random variables.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 23 / 26

SLIDE 79

Summary

Extreme value statistics: a brief introduction. A quantitative analysis of the phenomenon of crowding of near-extreme events in terms of the density of states with respect to the maximum of a set of independent and identically distributed random variables. The mean density of states converges to three different limiting forms depending on whether the tail of the parent distribution of the random variables decays:

◮ slower than pure exponential function ◮ faster than pure exponential function ◮ as a pure exponential function.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 23 / 26

SLIDE 80

Summary

Extreme value statistics: a brief introduction. A quantitative analysis of the phenomenon of crowding of near-extreme events in terms of the density of states with respect to the maximum of a set of independent and identically distributed random variables. The mean density of states converges to three different limiting forms depending on whether the tail of the parent distribution of the random variables decays:

◮ slower than pure exponential function ◮ faster than pure exponential function ◮ as a pure exponential function.

Verified also for a power-law correlated stationary Gaussian sequence.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 23 / 26

SLIDE 81

Summary

Extreme value statistics: a brief introduction. A quantitative analysis of the phenomenon of crowding of near-extreme events in terms of the density of states with respect to the maximum of a set of independent and identically distributed random variables. The mean density of states converges to three different limiting forms depending on whether the tail of the parent distribution of the random variables decays:

◮ slower than pure exponential function ◮ faster than pure exponential function ◮ as a pure exponential function.

Verified also for a power-law correlated stationary Gaussian sequence. Satisfactory agreement is found between the near-maximum crowding in the summer temperature reconstruction data of western Siberia and the theoretical prediction.

S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 23 / 26

SLIDE 82

References

Berman S M Limit theorems for the maximum term in stationary sequences

Ann. Math. Statist. 35, 502-516 (1964).

Comtet A, Leboeuf P and Majumdar S N Level Density of a Bose Gas and Extreme Value Statistics

Phys. Rev. Lett. 98, 070404 (2007).

Dodd E L The Greatest and the Least Variate Under General Laws of Error

Trans. Amer. Math. Soc. 25, 525-539 (1923).

Fisher R A and Tippett L H C (1928) Limiting forms of the frequency distribution of the largest and smallest member of a sample

Proc. Cambridge Philosophical Society 24, 180-190 (1928).

Fr´ echet M Sur la loi de probabilit´ e de l’´ ecart maximum

Ann. Soc. Math. Polon. 6, 93-116 ((1937).

Gnedenko B V Sur la distribution limite du terme maximum d’une s´ erie al´ eatoire

Ann. Math. 44, 423-453 (1943).
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 24 / 26

SLIDE 83

References

(cont.)

Gumbel E J Statistics of Extremes Columbia University Press, New York (1958). von Mises, R. (1923). Uber die Variationsbreite einer Beobachtungsreihe Sitzungsberichte der Berliner Mathematischen Gesellschaft 22, 3-8 (1923). von Bortkiewicz L Variationsbreite und mittlerer Fehler Sitzungsberichte der Berliner Mathematischen Gesellschaft 21, 3-11 (1922). Die Variationsbreite beim Gauss’schen Fehlergesetz Nordisk Statistisk Tidskrift 1, 11-38, 193-220 (1922). Watson G S Extreme Values in Samples from m-Dependent Stationary Stochastic Processes

Ann. Math. Statist. 25, 798-800 (1954).
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 25 / 26

SLIDE 84

Example

p(X) ∼

α

X 1+α , as X → ∞ QN(x) : = Prob

max(X1, X2, . . . , XN) ≤ x
= Prob
X1 ≤ x, X2 ≤ x, . . . , XN ≤ x
=
1 −

∞

x

p(x′) dx′

N ∼

1 − 1

xα

N

=

1 − 1

N

x

N1/α

−αN

N→∞

− − − − − − →

z=x/N1/α

exp

−z−α

aN = 0 bN = N1/α F(z) = exp

−z−α
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 26 / 26

SLIDE 85

Example

p(X) = exp(−X), X ∈ [0, ∞) QN(x) : = Prob

max(X1, X2, . . . , XN) ≤ x
= Prob
X1 ≤ x, X2 ≤ x, . . . , XN ≤ x
=
1 −

∞

x

p(x′) dx′

N

=

1 − exp(−x)

N

=

1 − 1

N exp

−(x − ln N)

N

N→∞

− − − − − − →

z=x−ln N exp

− exp(−z)
aN = ln N

bN = 1 F(z) = exp

− exp(−z)
S. Sabhapandit (LPTMS, Orsay, France)

Density of near-extreme events 26 / 26

SLIDE 86

Example

p(X) = βa−β(a − X)β−1, X ∈ [0, a] QN(x) : = Prob

max(X1, X2, . . . , XN) ≤ x
= Prob
X1 ≤ x, X2 ≤ x, . . . , XN ≤ x
=
1 −