Comportement d un échantillon sous conditionnement extrême

Transcription

1 Comportement d un échantillon sous conditionnement extrême Séminaire Statistiques à l IRMA, Université de Strasbourg Zhansheng CAO LSTA-UPMC Paris 6 Joint work with: M. BRONIATOWSKI Avril 09, 2013

2 General introduction 2 / 64

3 General introduction Notation and hypotheses Random walks for log-concave Density under extreme deviation Application 1 Breakdown point 2 3 / 64

4 General introduction and Context Gibbs conditional principles under extreme events 1 Local Results 2 Strenghtening of the local Gibbs conditional principle Marginally Conditional Density and its application 1 Marginally Conditional Density 2 One application 4 / 64

5 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Part I : Stretched random walks and behavior of their summands 5 / 64

6 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application A basic fact on an example If X 1,X 2 are i.i.d. random variables with L(X 1 ) = L(X 2 ) = N(0,1), then when a, we have (( X1 L a, X ) 2 X 1 + X 2 a 2 ) > a Dirac at (1,1). Thus for fixed n, in some cases, it holds (( X1 L a,..., X ) ) n X X n > a Dirac at (1,...,1). (1) a n 6 / 64

7 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Some previous results Barbe and Broniatowski (2004) : show that for fixed n, (1) holds under some regularity conditions Frisch and Sornette (1997) : obtain one density version of (1) (without conditioning) and apply to fragmentation processes and turbulence Question What if a = a n as n. 7 / 64

8 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application The objective of Part 1 Let S n 1 := X X n, where X 1,...,X n are i.i.d. r.v s distributed as X, X is unbounded upwards Let a n be some positive sequence lim n a n = + (2) Assuming that C n := (S n 1 /n > a n) (3) Let ε n denote a positive sequence and I n := n i=1 (X i (a n ε n,a n + ε n )). (4) 8 / 64

9 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application The objective of Part 1 Reformulate the Question we have lim n P(I n C n ) = 1. (5) Which is the acceptable growth of the sequence a n and the possible behaviours of the sequence ε n such that and is it possible under a large class of choices for P X? ε n = o(a n ) (6) lim n ε n = 0 (7) 9 / 64

10 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Notation and hypotheses For positive functions h and some positive normalized constant c, X has density p(x) := cexp( h(x)) For x R n define For A a Borel set in R n denote I h (x) := h(x i ), 1 i n I h (A) = inf x A I h (x). 10 / 64

11 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Random walks for log-concave Density under ED Theorem Let X 1,...,X n be i.i.d. copies of a r.v. X with density p(x) = cexp( g(x)), where g(x) is a positive convex differentiable function on R +. Assume that g is increasing on some interval [Y, ) and satisfies Let a n satisfy lim g(x)/x =. x and that for some positive sequence ε n logg(a n ) lim inf > 0 (8) n logn 11 / 64

12 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Theorem nlogg(a n + ε n ) lim = 0, n H(a n,ε n ) (9) ng(a n ) lim = 0, n H(a n,ε n ) (10) where H(a n,ε n ) = min ( F g + (a n,ε n ),Fg (a n,ε n ) ) ng(a n ), ( G(a n ) = g a n + 1 ) g(a n ), g(a n ) 12 / 64

13 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Theorem F + g (a n,ε n ) and F g (a n,ε n ) are defined as and Then it holds F + g (a n,ε n ) = g(a n + ε n ) + (n 1)g ( a n 1 ) n 1 ε n, ( Fg (a n,ε n ) = g(a n ε n ) + (n 1)g a n + 1 ) n 1 ε n. lim P(I n C n ) = 1. n 13 / 64

14 Corollary Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Let X 1,...,X n be independent r.v. s with common Weibull density { kx k 1 e xk when x > 0 p(x) = 0 otherwise, where k > 2. Let a n = n α, (11) for any α > 1/(k 2), let ε n be a positive sequence tending to 0 such that Then lim n P(I n C n ) = 1. nloga n lim n a k 2 n ε 2 n = 0. (12) 14 / 64

15 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Theorem X 1,...,X n are i.i.d. real valued r.v. s with common density f (x) = cexp( (g(x) + q(x))), where g is some positive convex function on R + and twice differentiable. Assume that g(x) is increasing on some interval [Y, ) and satisfies lim g(x)/x =. x Let M(x) be some nonnegative continuous function on R + for which M(x) q(x) M(x) for all positive x together with as x M(x) = O(logg(x)). (13) 15 / 64

16 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Theorem Let a n be some positive sequence such that a n and ε n = o(a n ) be a positive sequence. Assume logg(a n ) lim inf > 0 n logn (14) nlogg(a n + ε n ) lim = 0, n H(a n,ε n ) (15) ng(a n ) lim = 0, n H(a n,ε n ) (16) Then it holds lim P(I n C n ) = 1, n 16 / 64

17 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Theorem where H(a n,ε n ) = min ( F g + (a n,ε n ),Fg (a n,ε n ) ) ng(a n ), ( G(a n ) = g a n + 1 ) g(a n ), g(a n ) F + g (a n,ε n ) and F g (a n,ε n ) are defined as in Theorem for log-concave density. 17 / 64

18 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Example (Almost Log-concave densities 1) f (x) = c(x)exp( g(x)), 0 < x < where g is a twice differentiable convex function with lim g(x)/x = x and where for some x 0 > 0 and constants 0 < c 1 < c 2 <, we have c 1 < c(x) < c 2 for x 0 < x <. This type of density includes the Normal,the hyperbolic density, etc. 18 / 64

19 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Example (Almost Log-concave densities 2) For constants x 0 > 0, α > 0, and A such that f (x) = Ax α 1 l(x)exp( g(x)) x > x 0 where l(x) is slowly varying, g is a twice differentiable convex function and lim g(x)/x =. x 19 / 64

20 Breakdown point Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Definition Breakdown point : Let X = (x 1,...,x n ) denote a finite sample of size n ; we can corrupt this sample by replacing an arbitrary subset of size m of X with arbitrary values. The proportion of such bad values in the new sample X is ε = m/n. The breakdown point ε is defined as the least ε such that { } ε (X,T n ) = inf ε sup T n (X) T n (X ) =, where T n is some statistics. X 20 / 64

21 What interests us is : set T n = X X n, does this Theorem still holds if n goes to and a depends on n? 21 / 64 General introduction Breakdown point Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Theorem He, Jureskova, Koenker and Portnoy (1990) : For fixed n, let X 1,...,X n be i.i.d. r.v. s with a common distribution F(x), and which have a symmetric density f ( x) = f (x) with x R. Let T n = X X n, suppose that for any fixed c > 0, it holds ln(1 F(x + c)) lim = 1, x ln(1 F(x)) then lnp( T n > na) lim a lnp( X 1 > a) = n m + 1, where m denote the least m such that ε attains its minimum.

22 Breakdown point Notation and hypotheses Random walks for log-concave Density under extreme deviation Application When n tends to, we prefer the following definition of the breakdown point Definition Let X = (x 1,...,x n ) be a infinite sample of size n, and corrupt this sample by replacing an arbitrary subset of size m of X with arbitrary values. Denote by X the corrupted sample. The breakdown point ε is defined as the least ε such that { ε (X,T n ) = inf ε lim sup n X By this definition, it is straightforward that m = 1. } T n (X )/T n (X) =. 22 / 64

23 Breakdown point Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Theorem With the same notation and hypotheses as in Theorem for non log-concave density, it holds lnp( S n > na n ) lim n lnp( X 1 > a n ε n ) n = / 64

24 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Improved Erdös-Rényi law and the forming of aggregates Consider a long run X 1,...,X N of i.i.d. r.v s with common c.d.f. F satisfying the hypotheses in Theorem 2.5. Let J(x) := suptx loge (exp(tx 1 )) t be the Legendre-Fenchel transform of the m.g.f. t loge (exp(tx 1 )) for t > 0. Denote γ(u) := J (u) := x the asymptotic inverse of J, such that This condition holds for many distributions. γ( log(1 F)(x)) lim = 1. (17) x x 24 / 64

25 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Improved Erdös-Rényi law and the forming of aggregates Let n(n), 1 n(n) N be an integer sequence and denote and assuming M (n(n)) := max 0 j N n(n) Sj+n(N) j+1, c(n(n)) := logn n(n) Under (17) the following result holds (Mason 1989) lim N lim c(n(n)) =. (18) N M (n(n)) = 1 a.s.. (19) n(n)γ(c(n(n))) 25 / 64

26 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Improved Erdös-Rényi law and the forming of aggregates Fix some sequence n(n) satisfying (18) and for any positive δ close to 0 a n(n),δ := (1 δ)γ(c(n(n))) =: (1 δ)a n(n). (20) Due to (19) the choice of n(n) makes M (n(n)) n(n) > a n(n),δ hold ultimately with probability 1, together with the hypotheses in Theorems 2.1 and 2.5,yields X i lim = 1 (21) n a n(n) in probability for any i between j + 1 and j + n(n). 26 / 64

27 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Aggregate forming when log(1 F) is a regularly varying function When g a regularly varying function with index k > 1, (17) holds. As an example consider X 1 has a Weibull distribution on R + with scale parameter 1 and shape parameter k > 1. Define n(n) and a n(n) with α > 0 through n(n) = (logn) 1 1+k/α. Then there exists a set of consecutive r.v s X j+1,..,x j+n(n) such that for i = j + 1,...,j + n(n) lim N X i a n(n) = 1 in probability. (22) 27 / 64

28 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Aggregate forming when log(1 F) is a regularly varying function Reciprocally we may fix a growth condition on a n(n) and determine n(n). Aggregates with high level : Define for 0 < γ 1 a n(n) = (γlogn) 1/k. Then n(n) tends to a constant with the order of magnitude of the upper quantile of order N γ. Aggregates with intermediate level : Define for positive α. Then a n(n) = (logn) 1 α+k 1 1+k/α 28 / 64

29 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Aggregate forming when log(1 F) is a regularly varying function Aggregates with low level : Define Then a n(n) = (γloglogn) 1/k. n(n) = which are long aggregates. logn (1 + o(1)) γloglogn 29 / 64

30 Class R : h R if h is increasing and strictly monotone and ψ R / 64 General introduction Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Aggregate forming when log(1 F) is a rapidly varying function Using the notation of Theorem 2.5, denote h(x) := g (x) and define Class R 0 : l R 0, if, in (32), l(x) as x and lim x xε (x)/ε(x) = 0, lim x 2 ε (x)/ε(x) = 0, (23) x and, for some η (0,1/4) Define the inverse function of h through lim inf x xη ε(x) > 0. (24) ψ(u) := h (u) := inf{x : h(x) u}. (25)

31 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Theorem If h R, then (17) holds. Aggregate forming when log(1 F) is a rapidly varying function Hence if h R, then there exists a set of consecutive r.v s X j+1,..,x j+n(n) such that for i = j + 1,...,j + n(n) lim N X i a n(n) = 1 in probability. (26) 31 / 64

32 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Law of large numbers for extreme values and properties of aggregates Consider n(n) = 1; then a 1 = γ(c(1)) is asymptotically equivalent to the 1/N upper quantile of F. Denote the maximum X N,N of the X i s in the sample X 1,..,X N, it is well known that (Mason 1989) a.s. under the current hypotheses in this paper. X N,N a 1 1 (27) 32 / 64

33 Notation and hypotheses Random walks for log-concave Density under extreme deviation Application Proposition Under the hypotheses of Theorems 2.1 or 2.5 and when (17) holds, then with a n(n) defined in (20) lim min N 0 j N n(n) ( max X i j i j+n(n) 1 a n(n) ) = 1 in probability. (28) Proposition In the case of the Weibull distribution with shape parameter k 2 the sequence ε n can be chosen such that lim n ε n = 0 which yields ( ) lim min max X i a n(n) = 0 in probability. N 0 j N n(n) j i j+n(n) 1 33 / 64

34 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Part II : A conditional limit theorem for random walks under Extreme deviation 34 / 64

35 and Context and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application The objective of this paper Let X 1,...,X n be n i.i.d. unbounded real r.v. s with some exponential type density, denote S n 1 := X X n. The purpose of this paper is to construct the limit distribution of p an (y k 1 ) = p(xk 1 = yk 1 Sn 1 = na n) =? where y k 1 = (y 1,...,y k ) lim a n =. n 35 / 64

36 g(x)/x, x. (31) 36 / 64 General introduction and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Density Function Define the uniformly bounded density function p(x), ( p(x) = cexp ( g(x) q(x) )) x R +, (29) where c is some positive normalized constant. Define h(x) := g (x), and assume there exists some positive constant ϑ, for large x, such that The function g is positive and satisfies 1 sup q(v) (30) v x <ϑx xh(x)

37 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Class R β Class R 0 : A function l R 0, if it can be represented as ( x ε(u) ) l(x) = exp 1 u du, x 1, (32) where ε(x) is twice differentiable and ε(x) 0 as x. Class R β : h(x) R β, with β > 0 and x large enough, if h(x) = x β l(x), where l(x) R 0 and in (32) ε(x) satisfies lim supx ε (x) <, x limsupx 2 ε (x) <. (33) x 37 / 64

38 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Class R Class R 0 : l R 0, if, in (32), l(x) as x and lim x xε (x)/ε(x) = 0, lim x 2 ε (x)/ε(x) = 0, (34) x and, for some η (0,1/4) Define the inverse function of h through lim inf x xη ε(x) > 0. (35) ψ(u) := h (u) := inf{x : h(x) u}. (36) Class R : h R if h is increasing and strictly monotone and ψ R 0. Denote R : = R β R. 38 / 64

39 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Examples Weibull Density. Let p be a Weibull density p(x) = kx k 1 exp( x k ), x 0 ( = k exp ( x k (k 1)logx )). Take g(x) = x k (k 1)logx and q(x) = 0, we have h R k 1. A rapidly varying density. Define p through p(x) = cexp( e x 1 ), x 0. Take g(x) = h(x) = e x, we have h R. 39 / 64

40 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Tilted Density Denote by Φ(t) the mgf which is finite in non void neighborhood N of 0 Φ(t) := E exptx. The r.v. X t has tilted density defined on R with parameter t π t (x) := exptx Φ(t) p(x). The expectation, the three first centered moments of X t defined on N m(t) := d dt logφ(t) s2 (t) := d dt m(t) µ j(t) := d dt s2 (t), j = 3,4. 40 / 64

41 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Theorem Let p(x) be defined as in (29) and h(x) R, then with ψ defined as in (36) it holds as t m(t) ψ(t), s 2 (t) ψ (t), µ 3 (t) M 6 3 ψ (t), 2 where M 6 is the sixth order moment of standard normal density. Corollary Let p(x) be defined as in (29) and h(x) R. Then it holds as t µ 3 (t) s 3 0. (37) (t) 41 / 64

42 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Edgeworth Expansion under Extreme Normalizing Factors With t determined by m(t) = a n define π a n through π a n (x) = e tx p(x)/φ(t). Define s := s(t) and the normalized density of π a n by π a n (x) = sπ a n (sx + a n ), and denote by ρ n the normalized density of n-convolution π a n n (x), ρ n (x) := n π a n n ( nx). We extend the local Edgeworth expansion to the setting of a triangular array. Theorem Denote by φ(x) the standard normal density, uniformly upon x it holds ( ρ n (x) = φ(x) 1 + µ 3 6 ( x 3 ns 3 3x )) ( 1 + o n ). 42 / 64

43 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Edgeworth Expansion under Extreme Normalizing Factors Some other results Juszczak and Nagaev (2004) : obtain one Abel type theorem for mgf and the asymptotic normality of ρ n (x) as t. Bhattacharya and Ranga Rao (1976) : Edgeworth expansion for random vectors for positive definite covariance matrix.(attention : our result s(t) 0 as t ) 43 / 64

44 Local Results and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Conditions and assumptions Fix y k 1 := (y 1,..,y k ) in R k and define s j i := y i y j for 1 i < j k. t is determined by m(t) = a n. Assume Define t i through ψ(t) 2 lim n nψ = 0, (38) (t) m(t i ) := na n s i 1. n i For the sake of brevity, we write m i instead of m(t i ). 44 / 64

45 Local Results and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Theorem Assume that p(x) satisfies (29) and h(x) R. Assume that a n as n and that (38) holds. Then as n it holds p an (y k 1 ) = p(xk 1 = yk 1 Sn 1 = na n) = g m (y k 1 ) (1 + o ( 1 n ) ), with g m (y k 1 ) = k 1 i=0 ( ) π m i (X i+1 = y i+1 ). 45 / 64

46 Local Results and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application In the present case, namely for fixed k, an equivalent statement is Theorem Under the same notation and hypotheses as in the previous Theorem, it holds as n p an (y k 1 ) = p(xk 1 = yk 1 Sn 1 = na n) = g an (y k 1 ) (1 + o ( 1 n ) ), with g an (y k 1 ) = k i=1 ( ) π a n (X i = y i ). 46 / 64

47 Local Results and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application When a n = a, decide t by m(t) = a, we obtain the following corollary. (See also Broniatowski and Caron 2012 for one result with secondary term) Corollary X 1,...,X n are i.i.d. r.v. s with density p(x) defined in (29) and h(x) R. Then it holds as n ( p a (y k 1 ) = p(xk 1 = yk 1 Sn 1 = na) = g a(y k 1 ) 1 + o ( 1 ) ) n, with g a (y k 1 ) = k i=1 ( ) π a (X i = y i ). 47 / 64

48 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Strenghtening of the local Gibbs conditional principle Consider Y 1 with density p an and the resulting random variable p an (Y 1 ). We have proved Theorem With the same notation and hypotheses as in previous Theorems, it holds p an (Y 1 ) = g an (Y 1 )(1 + R n ) where g an = π a n the tilted density at point a n, and where R n is a function of Y1 n P an ( R n > δ n) 0 as n for any positive δ. such that 48 / 64

49 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Strenghtening of the local Gibbs conditional principle Denote the conditional probabilities by P an and G an which correspond on the density functions p an and g an, respectively. Define the variation norm where A R + k. P an G an := 2sup P an (A) G an (A), A Theorem Under all the notation and hypotheses above the total variation norm between P an and G an goes to 0 as n. 49 / 64

50 Marginally Conditional Density and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Proposition X 1,...,X n are i.i.d. random variables with density p(x) defined in (29) and h(x) R. Set m(t) = a n. Suppose as n ψ(t) 2 nψ 0, (39) (t) and η n 0 and nm 1 (a n )η n, (40) then ( p An (y 1 ) = p(x 1 = y 1 S1 n na n) = g An (y 1 ) 1 + o ( 1 ) ) n, 50 / 64

51 Marginally Conditional Density and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Proposition with an g An (y 1 ) = ts(t)e ni(a +η n n) a n g τ (y 1 )exp ( ni(τ) logs(t τ ) ) dτ, where g τ = π τ with t τ decided by m(t τ ) = τ and I(x) is defined by I(x) := xm 1 (x) logφ ( m 1 (x) ). (41) 51 / 64

52 Marginally Conditional Density and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application We extend the above Proposition to the case when the conditioning event is replaced for regularly varying function f (x) with index ρ by S f n = f (X 1 ) f (X n ) na n. Lemma When X is Weibull(k) with k > ρ > 0, f (X) = U has density p U (x) = k ρ x k ρ 1 l (x) k exp ( x k ρ l (x) k) when x, where l (x) is some slowly varying function. 52 / 64

53 Marginally Conditional Density and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Proposition Assume that X 1,...,X n are i.i.d. with Weibull density, set m(t) = a n, and it holds as n k > 1, k > ρ > 0, a k n n 0, (42) and η n 0 and na k 1 n η n, (43) then ( p f A n (y 1 ) = p(x 1 = y 1 Sn f na n ) = g f A n (y 1 ) 1 + o ( 1 ) ) n, 53 / 64

54 Marginally Conditional Density and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Proposition with g f an A n (y 1 ) = ts(t)e ni(a +η n n) a n g f τ(y 1 )exp ( ni(τ) logs(t τ ) ) dτ, where t τ decided by m(t τ ) = τ and I(x) is defined as in previous position, and g f a n (y 1 ) = p(y 1) p U (y 1 ) g a n,u(y 1 ), g an,u = π a n U, here πa n U is the tilted density for p U. 54 / 64

55 One Application and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Question : Solution of equation h(x) = x γ l(x) = a n, x =? when x is very large. 55 / 64

56 One Application and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application p h A n (y 1 ) = p(x 1 = y 1 Sn h na n ) ( n ) p X 1 = y 1 h(x i ) (a n ε n,a n + ε n ) i=1 ( ) = p X 1 = y 1 h(x 1 ) (a n ε n,a n + ε n ), the second step holds from non-convex Theorem in Chapter 1. Hence, we can sample from density p h A n (y 1 ), these sampling points are close to the solutions of equation h(x) = a n. 56 / 64

57 One Application and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application One example : h(x) = x 2. (with the help of V. Caron (2012)) Set l(x) = 1 and γ = 2, then h(x) = x 2, what is the solution of the equation ( S h n = n i=1 X2 i = na n ). x 2 = a n? x = ± a n of course! X p such that p(x) = ξexp ( 0.3x 2.5x 4) (see Cobb et al. (1983)). Mode : (0.03) 1/3. The density tilted is : π a exp ( 0.3x + tx 2 2.5x 4) where t is defined by m(t) = a n. Simulated three modes : 0, a n and a n. Hence the solutions of x 2 = a n are the new modes of the density p h A n. 57 / 64

58 One Application and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application a n = e+00 1e+17 2e+17 3e+17 4e / 64

59 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Projets en cours 59 / 64

60 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Lois conditionnelles pour des variables à queues lourdes Soit X,X 1,X 2,...,X n sont i.i.d. variables aléatoires de même loi F. Notons leur somme S n = X X n, n 1. Dans la littérature, de nombreux travaux ont été réalisés pour l obtention des expressions asymptotiques de la probabilité de queue F n (x) = P(S n > x). (44) Lorsque la condition de Cramer est satisfaite, Cramer a obtenu expressions asymptotiques pour F n (x). Petrov (1954,1994) a fourni les versions raffinées de ces résultats. 60 / 64

61 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Lois conditionnelles pour des variables à queues lourdes Fixons un certain a (0, ], et écrivons x + = (x,x + a]. Je suis intéressé à la mesure µ n,x conditionnée à S n x + µ n,x(a) := P(X A S n x + ). Une version discrète a été d obtenir par Pablo et al. (2007) pour la modélisation des phénomènes de condensation. Définir l opérateur T : n N R n n N R n qui échange la dernière et la composante maximale d une suite finie. Soit d n,l n deux suites, sous des hypothèses générales sur d n,l n et F (Armendáriz et Loulakis 2011 ont prouvé Theorem Si q n = d n l n, alors lim sup sup sup y n N x>q n y A B(R n 1 ) µ n,x T 1 (A R) F n 1 (A) = / 64

62 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Lois conditionnelles pour des variables à queues lourdes Je suis intéressé à la convergence beaucoup plus forte dans le sens lim sup sup sup y n N x>q n y A B(R n 1 ) µ n,x T 1 (A R) F n 1 1 (A) = 0. Cette convergence sera très intéressante lorsque F n 1 (A) est petit ou bien F n 1 (A) 0, n. C est le fait dans de nombreuses applications, tels que en assurance. 62 / 64

63 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application References Michel Broniatowski and Zhansheng Cao (2012) : Lights tails, All summands are large when the empirical mean is large (soumis à l Extremes en révision). Zhansheng Cao (2013) : An abelian theorem with application to Edgeworth expansions (soumis à Theory of Probability and Its Applications). Michel Broniatowski and Zhansheng Cao (2013) : A conditional limit theorem for random walks under extreme deviation (soumis à l Extremes). 63 / 64

64 and Context Gibbs conditional principles under extreme events Marginally Conditional Density and its application Merci de votre attention! 64 / 64