Extreme value statistics

Transcription

1 Extreme value statistics Density of near-extreme events Sanjib Sabhapandit Laboratoire de Physique Théorique et Modèles Statistiques CNRS UMR 8626 Université Paris-Sud Orsay cedex, France Collaborator Satya N. Majumdar Ref. Phys. Rev. Lett. 98, (2007).

2 Extreme value statistics: The statistics of the maximum or the minimum value of a set of random observations {X 1, X 2,..., X N }. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 1 / 26

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

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

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

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

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

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 René Fréchet (1927): Obtained two of the three kinds of extreme-value distribution. [Fréchet (1927)] (not printed until 1928) S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 3 / 26

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 René Fréchet (1927): Obtained two of the three kinds of extreme-value distribution. [Fréchet (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

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 René Fréchet (1927): Obtained two of the three kinds of extreme-value distribution. [Fréchet (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

11 A brief history... (II): applications Emil Julius Gumbel [Gumbel (1958)] Les intervalles extrêmes entre les émissions radio-actives. (Extreme intervals between radioactive emissions) J. Phys. Radium. 8, , (1937). Les intervalles extrêmes entre les émissions radioactives. II J. Phys. Radium 8, (1937) The return period of flood flows. Ann. Math. Statistics 12, (1941). Probability-interpretation of the observed return-periods of floods. Trans. Amer. Geophys. Union 1941, (1941). On the frequency distribution of extreme values in meteorological data. Bull. Amer. Meteorol. Soc. 23, (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

12 A brief history... (II): applications Emil Julius Gumbel [Gumbel (1958)] Les intervalles extrêmes entre les émissions radio-actives. (Extreme intervals between radioactive emissions) J. Phys. Radium. 8, , (1937). Les intervalles extrêmes entre les émissions radioactives. II J. Phys. Radium 8, (1937) The return period of flood flows. Ann. Math. Statistics 12, (1941). Probability-interpretation of the observed return-periods of floods. Trans. Amer. Geophys. Union 1941, (1941). On the frequency distribution of extreme values in meteorological data. Bull. Amer. Meteorol. Soc. 23, (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, (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

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. Mézard, JPA 30, 7997 (1997). D.S. Dean & S.N. Majumdar, PRE 64, (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, (2006). Fluctuating interfaces Györgyi, Holdsworth, Portelli, & Rácz, PRE 68, (2003). S.N. Majumdar & A. Comtet PRL 92, (2004). Computer science Search tree problems S.N. Majumdar & P.L. Krapivsky, PRE 65, (2002). Optimization problems S.N. Majumdar & P.L. Krapivsky, PRE 62, 7735 (2000). Finance e.g. see P. Embrechts, C. Klüppelberg, & T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer, Berlin, 1997). S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 5 / 26

14 Extreme value statistics of i.i.d. random variables {X 1, X 2,..., X }{{ N } set of i.i.d. random variables } each drawn from p(x ) parent distribution S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 6 / 26

15 Extreme value statistics of i.i.d. random variables {X 1, X 2,..., X }{{ N } set of i.i.d. random variables } each drawn from p(x ) parent distribution X max := max(x 1, X 2,..., X N ) random variable S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 6 / 26

16 Extreme value statistics of i.i.d. random variables {X 1, X 2,..., X }{{ N } set of i.i.d. random variables } each drawn from p(x ) parent distribution X max := max(x 1, X 2,..., X N ) random variable Q N (x) := Prob[X max x] = Prob[X 1 x, X 2 x,..., X N x] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 6 / 26

17 Extreme value statistics of i.i.d. random variables {X 1, X 2,..., X }{{ N } set of i.i.d. random variables } each drawn from p(x ) parent distribution X max := max(x 1, X 2,..., X N ) random variable Q N (x) := Prob[X max x] = Prob[X 1 x, X 2 x,..., X N x] Independence [ x ] N [ ] N Q N (x) = p(x ) dx = 1 p(x ) dx x S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 6 / 26

18 Extreme value statistics of i.i.d. random variables {X 1, X 2,..., X }{{ N } set of i.i.d. random variables } each drawn from p(x ) parent distribution X max := max(x 1, X 2,..., X N ) random variable Q N (x) := Prob[X max x] = Prob[X 1 x, X 2 x,..., X N x] Independence [ x ] N [ ] N Q N (x) = p(x ) dx = 1 p(x ) dx x Scaling limit: X large, N large ( ) x, N x an Q N (x) F (x a N )/b N fixed b N S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 6 / 26

19 Extreme value statistics of i.i.d. random variables {X 1, X 2,..., X }{{ N } set of i.i.d. random variables } each drawn from p(x ) parent distribution X max := max(x 1, X 2,..., X N ) random variable Q N (x) := Prob[X max x] = Prob[X 1 x, X 2 x,..., X N x] Independence [ x ] N [ ] N Q N (x) = p(x ) dx = 1 p(x ) dx x Scaling limit: X large, N large ( ) x, N x an Q N (x) F (x a N )/b N fixed b N or lim Q N(a N + b N z) = F (z) N S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 6 / 26

20 Extreme value statistics of i.i.d. random variables {X 1, X 2,..., X }{{ N } set of i.i.d. random variables } each drawn from p(x ) parent distribution X max := max(x 1, X 2,..., X N ) random variable Q N (x) := Prob[X max x] = Prob[X 1 x, X 2 x,..., X N x] Independence [ x ] N [ ] N Q N (x) = p(x ) dx = 1 p(x ) dx x Scaling limit: X large, N large ( ) x, N x an Q N (x) F (x a N )/b N fixed b N or lim Q N(a N + b N z) = F (z) N a N, b N non-universal scale factors dependent on p(x ) S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 6 / 26

21 Extreme value statistics of i.i.d. random variables {X 1, X 2,..., X }{{ N } set of i.i.d. random variables } each drawn from p(x ) parent distribution X max := max(x 1, X 2,..., X N ) random variable Q N (x) := Prob[X max x] = Prob[X 1 x, X 2 x,..., X N x] Independence [ x ] N [ ] N Q N (x) = p(x ) dx = 1 p(x ) dx x Scaling limit: X large, N large ( ) x, N x an Q N (x) F (x a N )/b N fixed b N or lim Q N(a N + b N z) = F (z) N a N, b N non-universal scale factors dependent on p(x ) F (z) universal scaling function: only of three possible types F 1 (z) or F 2 (z) or F 3 (z) depending on the tails of p(x ) [Fréchet (1927), Fisher & Tippett (1928), Gnedenko (1943), Gumbel (1958)] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 6 / 26

22 The three limiting extreme value distributions 1 Fréchet: If p(x ) has power-law tail p(x ) X (1+α). { [ exp z α ] for z 0, F 1 (z) = 0 for z 0. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 7 / 26

23 The three limiting extreme value distributions 1 Fréchet: If p(x ) has power-law tail p(x ) X (1+α). F 1 (z) = pdf: { exp [ z α ] for z 0, 0 for z 0. f 1 (z) = α exp [ z α] z 1+α, z [0, ). 0.6 α = 3/ Z S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 7 / 26

24 The three limiting extreme value distributions 1 Fréchet: If p(x ) has power-law tail p(x ) X (1+α). F 1 (z) = pdf: { exp [ z α ] for z 0, 0 for z 0. f 1 (z) = α exp [ z α] z 1+α, z [0, ). 2 Gumbel: If p(x ) has faster than power-law but unbounded tail p(x ) exp( X δ ). F 2 (z) = exp [ e z] α = 3/2 4 Z S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 7 / 26

25 The three limiting extreme value distributions 1 Fréchet: If p(x ) has power-law tail p(x ) X (1+α). F 1 (z) = pdf: { exp [ z α ] for z 0, 0 for z 0. f 1 (z) = α exp [ z α] z 1+α, z [0, ). 2 Gumbel: If p(x ) has faster than power-law but unbounded tail p(x ) exp( X δ ). F 2 (z) = exp [ e z]. pdf: f 2 (z) = exp [ z e z], z (, ) α = 3/2 4 Z S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 7 / 26

26 The three limiting extreme value distributions 1 Fréchet: If p(x ) has power-law tail p(x ) X (1+α). F 1 (z) = pdf: { exp [ z α ] for z 0, 0 for z 0. f 1 (z) = α exp [ z α] z 1+α, z [0, ). 2 Gumbel: If p(x ) has faster than power-law but unbounded tail p(x ) exp( X δ ). F 2 (z) = exp [ e z]. pdf: f 2 (z) = exp [ z e z], z (, ) α = 3/2 3 Weibull: If p(x ) is bounded p(x ) (a X ) β 1 as X a. { [ exp ( z) β] for z 0, F 3 (z) = 1 for z 0. 4 Z S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 7 / 26

27 The three limiting extreme value distributions 1 Fréchet: If p(x ) has power-law tail p(x ) X (1+α). F 1 (z) = pdf: { exp [ z α ] for z 0, 0 for z 0. f 1 (z) = α exp [ z α] z 1+α, z [0, ). 2 Gumbel: If p(x ) has faster than power-law but unbounded tail p(x ) exp( X δ ). F 2 (z) = exp [ e z]. pdf: f 2 (z) = exp [ z e z], z (, ) α = 3/2 3 Weibull: If p(x ) is bounded p(x ) (a X ) β 1 as X a. { [ exp ( z) β] for z 0, F 3 (z) = 1 for z 0. pdf: f 3 (z) = β( z) β 1 exp [ ( z) β], 4 β = 3/2 z (, 0]. Z S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 7 / 26

28 ...in Bose gas or integer partition problem Q(N E) := Prob[N ex N, given energy E] [A. Comtet, P. Leboeuf & S.N. Majumdar (2007)] the cumulative distribution of the no. of excited particles N ex particles of an ideal gas of bosons with fixed total energy E and with single particle density of states ρ(ɛ) = νɛ ν 1. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 8 / 26

29 ...in Bose gas or integer partition problem Q(N E) := Prob[N ex N, given energy E] [A. Comtet, P. Leboeuf & S.N. Majumdar (2007)] the cumulative distribution of the no. of excited particles N ex particles of an ideal gas of bosons with fixed total energy E and with single particle density of states ρ(ɛ) = νɛ ν 1. C(N E) # partitions of E into N parts := E = n i i 1/ν Ω(E) # partitions of E i the cumulative distribution of the no. of parts in partitions of E. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 8 / 26

30 ...in Bose gas or integer partition problem Q(N E) := Prob[N ex N, given energy E] [A. Comtet, P. Leboeuf & S.N. Majumdar (2007)] the cumulative distribution of the no. of excited particles N ex particles of an ideal gas of bosons with fixed total energy E and with single particle density of states ρ(ɛ) = νɛ ν 1. C(N E) # partitions of E into N parts := E = n i i 1/ν Ω(E) # partitions of E i the cumulative distribution of the no. of parts in partitions of E. In terms of a suitable rescaled variable z z(n, E, ν): F 1 (z) if 0 < ν < 1 (Fréchet) scaling limit Q(N E) F 2 (z) if ν = 1 (Gumbel) 1 N E F 3 (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

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) ] ρ ( ) Dt 1 x a(t) x F 2 b(t) [ ( )] 1/2 ρ Dt a(t) = 4Dt ln 2 + π [ ( )] 1/2 ρ Dt b(t) = Dt ln 2 + π F 2 (z) = exp ( exp( z) ) [SS, J. Stat. Mech. (2007) L05002] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 9 / 26

32 Maximum of weakly correlated random variables M 1 M 2 M 3 M 4 M5 M 6 X i ξ ξ ξ ξ ξ ξ X max = max(x 1, X 2,..., X }{{ N ) = max(m } 1, M 2,..., M n ), n = N }{{} ξ 1. correlated uncorrelated S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 10 / 26

33 Maximum of weakly correlated random variables M 1 M 2 M 3 M 4 M5 M 6 X i ξ ξ ξ ξ ξ ξ X max = max(x 1, X 2,..., X }{{ N ) = max(m } 1, M 2,..., M n ), n = N }{{} ξ 1. correlated uncorrelated limit laws of i.i.d. random variables Extreme values in samples from m-dependent stationary stochastic processes [Watson (1954)] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 10 / 26

34 For a stationary Gaussian sequence p(x 1, X 2,..., X N ) = 1 ( (2π) N det(σ) exp 1 ) 2 XT Σ 1 X X 1 C 0 C 1... C N 1 X 2 X =, C 1 C 0... C N 2 Σ = , X N C N 1 C N 2... C 0 C n X i X i+n X max := max(x 1, X 2,..., X N ) If either lim C n ln n = 0 n or Cn 2 < then n=1 lim Prob[X max a N + b N z] = exp( e z ) N }{{} Gumbel [Berman (1964)] ln ln N + ln 4π a N = 2 ln N 2 2 ln N 1 b N = 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

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 others, or 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

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 others, or 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

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 others, or 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

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 others, or 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

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 others, or 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

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 others, or 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

41 A measure of the crowding of near-extreme events The density of states with respect to the maximum: ρ(r, N) = 1 N Normalization: N 1 {X i X max} 0 [ ] δ r ({ X max X i ) ρ(r, N) dr = 1 1 N = max(x 1, X 2,..., X }{{ N ) } i.i.d. from p(x ) Note: even though the random variables are independent, the different terms become correlated through their common maximum X max. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 13 / 26

42 A measure of the crowding of near-extreme events The density of states with respect to the maximum: ρ(r, N) = 1 N Normalization: N 1 {X i X max} 0 [ ] δ r ({ X max X i ) ρ(r, N) dr = 1 1 N = max(x 1, X 2,..., X }{{ N ) } i.i.d. from p(x ) Note: even though the random variables are independent, the different terms become correlated through their common maximum X max. ρ(r, N) fluctuates from one realization of {X 1, X 2,..., X N } 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

43 A measure of the crowding of near-extreme events The density of states with respect to the maximum: ρ(r, N) = 1 N Normalization: N 1 {X i X max} 0 [ ] δ r ({ X max X i ) ρ(r, N) dr = 1 1 N = max(x 1, X 2,..., X }{{ N ) } i.i.d. from p(x ) Note: even though the random variables are independent, the different terms become correlated through their common maximum X max. ρ(r, N) fluctuates from one realization of {X 1, X 2,..., X N } 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

44 Limiting behavior of mean density of states ρ(r, N) = 1 N N 1 {X i X max} [ ] δ r ({ X max X i ) = max(x 1, X 2,..., X }{{ N ) } i.i.d. from p(x ) 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

45 Limiting behavior of mean density of states ρ(r, N) = 1 N N 1 {X i X max} [ ] δ r ({ X max X i ) = max(x 1, X 2,..., X }{{ N ) } i.i.d. from p(x ) 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 ( ) r an f b N b N where f (z) f 1 (z) or f 2 (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

46 Limiting behavior of mean density of states ρ(r, N) = 1 N N 1 {X i X max} [ ] δ r ({ X max X i ) = max(x 1, X 2,..., X }{{ N ) } i.i.d. from p(x ) 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 ( ) r an f b N b N where f (z) f 1 (z) or f 2 (z) 2 p(x ) has faster than exp( x) tail: ρ(r, N) N p(a N r ) 3 p(x ) has exp( x) tail S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 14 / 26

47 Limiting behavior of mean density of states ρ(r, N) = 1 N N 1 {X i X max} [ ] δ r ({ X max X i ) = max(x 1, X 2,..., X }{{ N ) } i.i.d. from p(x ) 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 ( ) r an f b N b N where f (z) f 1 (z) or f 2 (z) 2 p(x ) has faster than exp( x) tail: ρ(r, N) N p(a N r ) 3 p(x ) has exp( x) tail: ρ(r, N) = g(r a N ), where g(z) = e z [ 1 ( 1 + e z) e e z ] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 14 / 26

48 Derivation of the mean density of states 1 First consider ρ(r, N) = 1 N N 1 {X i X max} [ ] δ r ({ X max X i ) for a given value of the maximum at X max = y. = max(x 1, X 2,..., X }{{ N ) } i.i.d. from p(x ) S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 15 / 26

49 Derivation of the mean density of states 1 First consider ρ(r, N) = 1 N 1 [ ] δ r ({ X max X i ) N {X i X max} = max(x 1, X 2,..., X }{{ N ) } for a given value of the maximum at X max = y. i.i.d. from p(x ) 2 Given X max = y, the rest of the (N 1) variables are distributed independently according to the common conditional pdf [ y 1 p cond (X y ) = p(w ) dw] p(x ) θ(y X ). S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 15 / 26

50 Derivation of the mean density of states 1 First consider ρ(r, N) = 1 N 1 [ ] δ r ({ X max X i ) N {X i X max} = max(x 1, X 2,..., X }{{ N ) } for a given value of the maximum at X max = y. i.i.d. from p(x ) 2 Given X max = y, the rest of the (N 1) variables are distributed independently according to the common conditional pdf [ y 1 p cond (X y ) = p(w ) dw] p(x ) θ(y X ). 3 The conditional mean is ( ) N 1 ρ cond (r, N y ) = p cond (y r y ). N S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 15 / 26

51 Derivation of the mean density of states 1 First consider ρ(r, N) = 1 N 1 [ ] δ r ({ X max X i ) N {X i X max} = max(x 1, X 2,..., X }{{ N ) } for a given value of the maximum at X max = y. i.i.d. from p(x ) 2 Given X max = y, the rest of the (N 1) variables are distributed independently according to the common conditional pdf [ y 1 p cond (X y ) = p(w ) dw] p(x ) θ(y X ). 3 The conditional mean is ( ) N 1 ρ cond (r, N y ) = p cond (y r y ). N 4 In terms of the conditional mean ρ(r, N) = ρ cond (r, N y ) p max (y, N) dy. }{{} [ y = Np(y ) p(w ) dw ] N 1 S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 15 / 26

52 Derivation of the mean density of states 1 First consider ρ(r, N) = 1 N 1 [ ] δ r ({ X max X i ) N {X i X max} = max(x 1, X 2,..., X }{{ N ) } for a given value of the maximum at X max = y. i.i.d. from p(x ) 2 Given X max = y, the rest of the (N 1) variables are distributed independently according to the common conditional pdf [ y 1 p cond (X y ) = p(w ) dw] p(x ) θ(y X ). 3 The conditional mean is ( ) N 1 ρ cond (r, N y ) = p cond (y r y ). N 4 In terms of the conditional mean ρ(r, N) = ρ cond (r, N y ) p max (y, N) dy. }{{} 5 Finally putting everything together ρ(r, N) = p(x r ) p max (x, N 1) dx [ y = Np(y ) p(w ) dw (valid for all N). ] N 1 S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 15 / 26

53 Analyzing the limiting behavior for large N ρ(r, N) = p(x r ) p max (x, N 1) dx (valid for all N) b N p max (x = a N + b N z, N) N f (z) where f (z) f 1 (z), f 2 (z) or f 3 (z) (Gumbel) (Fréchet) (Weibull) S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 16 / 26

54 Analyzing the limiting behavior for large N ρ(r, N) = p(x r ) p max (x, N 1) dx (valid for all N) b N p max (x = a N + b N z, N) N f (z) where f (z) f 1 (z), f 2 (z) or f 3 (z) Example If p(x ) exp( X δ ) for large X, then: f (z) f 2 (z), a N (ln N) 1/δ and b N δ 1 (ln N) 1/δ 1 for large N. (Gumbel) (Fréchet) (Weibull) S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 16 / 26

55 Analyzing the limiting behavior for large N ρ(r, N) = p(x r ) p max (x, N 1) dx (valid for all N) b N p max (x = a N + b N z, N) N f (z) where f (z) f 1 (z), f 2 (z) or f 3 (z) N Example If p(x ) exp( X δ ) for large X, then: f (z) f 2 (z), a N (ln N) 1/δ and b N δ 1 (ln N) 1/δ 1 for large N. (Gumbel) (Fréchet) (Weibull) 1 If p(x ) has slower than exp( x) tail (e.g. δ < 1), ( b N as N. x = a N + b N z b N p(b N z + a N r ) N δ z r a ) N. b N ρ(r, N) N 1 b N f ( ) r an. b N S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 16 / 26

56 Analyzing the limiting behavior for large N ρ(r, N) = p(x r ) p max (x, N 1) dx (valid for all N) b N p max (x = a N + b N z, N) N f (z) where f (z) f 1 (z), f 2 (z) or f 3 (z) N Example If p(x ) exp( X δ ) for large X, then: f (z) f 2 (z), a N (ln N) 1/δ and b N δ 1 (ln N) 1/δ 1 for large N. (Gumbel) (Fréchet) (Weibull) 1 If p(x ) has slower than exp( x) tail (e.g. δ < 1), ( b N as N. x = a N + b N z b N p(b N z + a N r ) N δ z r a ) N. b N ρ(r, N) N 1 b N f ( ) r an. b N 2 If p(x ) has faster than exp( x) tail (e.g. δ > 1), b N 0 as N. p max (x, N) δ(x a N ). ρ(r, N) N p(a N r ). S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 16 / 26

57 Next: examples with explicit forms of p(x ) 1 slower than pure exponential A B p(x ) = α exp( X α ) X 1+α, α > 0, X [0, ) p(x ) = δx δ 1 exp( X δ ), δ < 1, X [0, ) 2 pure exponential p(x ) = exp( X ), X [0, ) 3 faster than pure exponential A B p(x ) = δx δ 1 exp( X δ ), δ > 1, X [0, ) p(x ) = βa β (a X ) β 1, β > 0, X [0, a] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 17 / 26

58 Next: examples with explicit forms of p(x ) 1 slower than pure exponential A B p(x ) = α exp( X α ) X 1+α, α > 0, X [0, ) Fréchet p(x ) = δx δ 1 exp( X δ ), δ < 1, X [0, ) 2 pure exponential p(x ) = exp( X ), X [0, ) Gumbel 3 faster than pure exponential A p(x ) = δx δ 1 exp( X δ ), δ > 1, X [0, ) B p(x ) = βa β (a X ) β 1, β > 0, X [0, a] Weibull S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 17 / 26

59 Example: power-law tail A ρ(r, N) = p(x r ) p max (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

60 Example: power-law tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = α exp( X α ) X 1+α, X [0, ). { an = 0 b N = N 1/α N S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 18 / 26

61 Example: power-law tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = α exp( X α ) X 1+α, X [0, ). { an = 0 b N = N 1/α N Therefore, B where ρ(r, N) N 1 b N f ( r b N ) f (z) f 1 (z) = α exp [ z α], z 1+α, z 0. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 18 / 26

62 Example: power-law tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = α exp( X α ) b N ρ(r, N) X 1+α, X [0, ). 0.8 { an = 0 Therefore, B b N = N 1/α ρ(r, N) N 1 b N f N ( r b N ), A B N = 10 2 N = 10 3 N = 10 4 (Fréchet) where f (z) f 1 (z) = α exp [ z α] z 1+α, z r/b N S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 18 / 26

63 Example: power-law tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = α exp( X α ) b N ρ(r, N) X 1+α, X [0, ). 0.8 { an = 0 Therefore, B b N = N 1/α ρ(r, N) N 1 b N f N ( r b N ), A B N = 10 2 N = 10 3 N = 10 4 (Fréchet) where f (z) f 1 (z) = α exp [ z α] z 1+α, z r/b N For r = 0 : ρ(0, N) N αγ(2 + 1/α) N 1+1/α. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 18 / 26

64 Example: Faster than power-law, but unbounded tail A ρ(r, N) = p(x r ) p max (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

65 Example: Faster than power-law, but unbounded tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = δx δ 1 exp( X δ ), X [0, ). a N = (ln N) 1/δ b N = δ 1 (ln N) 1/δ 1 N for δ < 1 1 for δ = 1 0 for δ > 1 S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 19 / 26

66 Example: Faster than power-law, but unbounded tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = δx δ 1 exp( X δ ), X [0, ). a N = (ln N) 1/δ b N = δ 1 (ln N) 1/δ 1 N for δ < 1 1 for δ = 1 0 for δ > 1 For r = 0 : ρ(0, N) N p(a N ) = δ N (ln N)1 1/δ. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 19 / 26

67 Example: Faster than power-law, but unbounded tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = δx δ 1 exp( X δ ), X [0, ). a N = (ln N) 1/δ b N = δ 1 (ln N) 1/δ 1 N for δ < 1 1 for δ = 1 0 for δ > 1 For r = 0 : ρ(0, N) N p(a N ) = δ N (ln N)1 1/δ. B where ρ(r, N) N 1 f b N ( r a N b N ) for δ < 1 g(r a N ) for δ = 1 p(a N r ) for δ > 1 f (z) f 2 (z) = exp [ z exp( z)] (Gumbel) [ g(z) = e z 1 ( 1 + e z) ] e e z S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 19 / 26

68 Example: Faster than power-law, but unbounded tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = δx δ 1 exp( X δ ), a N = (ln N) 1/δ b N = δ 1 (ln N) 1/δ 1 N X [0, ). for δ < 1 1 for δ = 1 0 for δ > 1 4 δ = 1/ (r a N)/b N b N ρ(r, N) N = 10 3 A N = 10 5 N = 10 7 B (Gumbel) For r = 0 : ρ(0, N) N p(a N ) = δ N (ln N)1 1/δ. B where ρ(r, N) N 1 f b N ( r a N b N ) for δ < 1 g(r a N ) for δ = 1 p(a N r ) for δ > 1 f (z) f 2 (z) = exp [ z exp( z)] (Gumbel) [ g(z) = e z 1 ( 1 + e z) ] e e z S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 19 / 26

69 Example: Faster than power-law, but unbounded tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = δx δ 1 exp( X δ ), a N = (ln N) 1/δ b N = δ 1 (ln N) 1/δ 1 For r = 0 : N X [0, ). for δ < 1 1 for δ = 1 0 for δ > 1 ρ(0, N) N p(a N ) = δ N (ln N)1 1/δ. A B 4 δ = 1/ { N = 10 N = 50 N = 100 g(r a N ) (r a N)/b N b N ρ(r, N) N = 10 3 A N = 10 5 N = 10 7 B (Gumbel) ρ(r, N) δ = 1 B where ρ(r, N) N 1 f b N ( r a N b N ) for δ < 1 g(r a N ) for δ = 1 p(a N r ) for δ > 1 f (z) f 2 (z) = exp [ z exp( z)] (Gumbel) [ g(z) = e z 1 ( 1 + e z) ] e e z 6 r a N S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 19 / 26

70 Example: Faster than power-law, but unbounded tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = δx δ 1 exp( X δ ), B a N = (ln N) 1/δ b N = δ 1 (ln N) 1/δ 1 For r = 0 : where ρ(r, N) N N X [0, ). for δ < 1 1 for δ = 1 0 for δ > 1 ρ(0, N) N p(a N ) = δ N (ln N)1 1/δ. 1 f b N ( r a N b N ) for δ < 1 g(r a N ) for δ = 1 p(a N r ) for δ > 1 f (z) f 2 (z) = exp [ z exp( z)] (Gumbel) [ g(z) = e z 1 ( 1 + e z) ] e e z { N = 10 N = 50 N = 100 g(r a N ) ρ(r, N) N = 10 3 A N = 10 6 N = 10 9 B p(a N r ) S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 19 / 26 A B δ = 1/ (r a N)/b N b N ρ(r, N) N = 10 3 A N = 10 5 N = 10 7 B (Gumbel) r a N r a N ρ(r, N) δ = 2 δ = 1

71 Bounded tail A ρ(r, N) = p(x r ) p max (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

72 Bounded tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = βa β (a X ) β 1, X [0, a]. { an = a b N = an 1/β N 0 S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 20 / 26

73 Bounded tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = βa β (a X ) β 1, X [0, a]. { an = a b N = an 1/β N 0 B ρ(r, N) N p(a N r ). β = β = ρ(r, N) N = 10 2 A N = 10 3 N = 10 4 B A B { N = 10 N = 10 2 N = 10 3 r r S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 20 / 26

74 Bounded tail A ρ(r, N) = p(x r ) p max (x, N 1) dx for all N. Consider p(x ) = βa β (a X ) β 1, X [0, a]. { an = a b N = an 1/β N 0 B ρ(r, N) N p(a N r ). ρ(0, N) N (β/a)γ(2 1/β) N 1 1/β. β = β = ρ(r, N) N = 10 2 A N = 10 3 N = 10 4 B A B { N = 10 N = 10 2 N = 10 3 r r S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 20 / 26

75 For a stationary Gaussian sequence... Cn = γ = 1/2 γ=1 1 (1 + n2 )γ/2 Z 2 e X /2 where p(x ) = 2π p(x r ) pmax (x, N 1) dx, ρ(r, N) = (a) ρ(r, N) 0.4 (b) (c) (a) N = 1025 (b) N = (c) N = r S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 21 / 26

76 Yamal peninsula multimillennial summer temperature reconstruction data GEOGRAPHIC REGION: Northwestern Siberia PERIOD OF RECORD: 2067 BC AD ORIGINAL REFERENCE: Holocene 12, 717 (2002). DATA FROM: IGBP PAGES/World Data Center for Paleoclimatology. T ( o C) (a) (c) 500 YEAR 0 N = (d) p( T) N = 1000 (b) p(x) p(x) = 1 exp ( x ) 2 2π 2 [ ( )] 1/2 N a N = 2 log 4π log N ρ(r, N) (c) years. (d) years r r p(a N r ) Exact numerical integration S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 22 / 26

77 Summary Extreme value statistics: a brief introduction. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 23 / 26

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

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

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

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

82 References Berman S M Limit theorems for the maximum term in stationary sequences Ann. Math. Statist. 35, (1964). Comtet A, Leboeuf P and Majumdar S N Level Density of a Bose Gas and Extreme Value Statistics Phys. Rev. Lett. 98, (2007). Dodd E L The Greatest and the Least Variate Under General Laws of Error Trans. Amer. Math. Soc. 25, (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, (1928). Fréchet M Sur la loi de probabilité de l écart maximum Ann. Soc. Math. Polon. 6, ((1937). Gnedenko B V Sur la distribution limite du terme maximum d une série aléatoire Ann. Math. 44, (1943). S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 24 / 26

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, (1922). Watson G S Extreme Values in Samples from m-dependent Stationary Stochastic Processes Ann. Math. Statist. 25, (1954). S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 25 / 26

84 Example p(x ) α X 1+α, as X [ ] Q N (x) : = Prob max(x 1, X 2,..., X N ) x [ ] = Prob X 1 x, X 2 x,..., X N x [ ] N = 1 p(x ) dx [ = x [ N x α ] N ( x N 1/α ) α N exp [ z α] z=x/n 1/α ] N a N = 0 b N = N 1/α F (z) = exp [ z α] S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 26 / 26

85 Example p(x ) = exp( X ), X [0, ) [ ] Q N (x) : = Prob max(x 1, X 2,..., X N ) x [ ] = Prob X 1 x, X 2 x,..., X N x [ ] N = 1 p(x ) dx = x [ 1 exp( x) ] N = [1 1N exp( (x )] N ln N) [ ] N exp exp( z) z=x ln N [ ] a N = ln N b N = 1 F (z) = exp exp( z) S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 26 / 26

86 Example p(x ) = βa β (a X ) β 1, X [0, a] [ ] Q N (x) : = Prob max(x 1, X 2,..., X N ) x [ ] = Prob X 1 x, X 2 x,..., X N x [ a ] N = 1 p(x ) dx x a x [ ( ) ] β N a x = 1 a [ = 1 1 ( ) ] β N a x N an 1/β [ N exp ( z) β], z 0. z=(x a)/an 1/β [ a N = a b N = an 1/β F (z) = exp ( z) β], z 0. S. Sabhapandit (LPTMS, Orsay, France) Density of near-extreme events 26 / 26