Kestens counterexample to the Cram er-Wold device for regular - PowerPoint PPT Presentation

1 Kesten’s counterexample to the Cram´ er-Wold device for regular variation F ILIP L INDSKOG R oyal I nstitute of T echnology, S tockholm 2005 based on joint work with Henrik Hult www.math.kth.se/ ∼ lindskog

2 The Cram´ er-Wold device Let X , X 1 , X 2 , . . . be R d -valued random (column) vectors. It holds that d → X as n → ∞ X n if and only if x T X n → x T X d as n → ∞ for all x ∈ R d , where x T X n = x (1) X (1) + · · · + x ( d ) X ( d ) n . n Let h x : R d → R be given by h x ( z ) = x T z . If µ denotes the distribution of X , then µh − 1 is the distribution of x T X : x P( x T X ≤ y ) = P( h x ( X ) ∈ ( −∞ , y ]) = P( X ∈ h − 1 x ( −∞ , y ]) .

3 x ( −∞ , y ] = { z ∈ R d : x T z ≤ y } is a half space. Notice that h − 1 The characteristic function of x T X is � � � R d e is x T z µ (d z ) = � e isy µh − 1 R d e ish x ( z ) µ (d z ) = x (d y ) = µ ( s x ) . R ⇒ If we know the distribution µh − 1 of x T X for all x , then we know the x characteristic function � µ of X in every point x . Since � µ uniquely determines µ we find that: A probability measure µ is uniquely determined by the values it gives to half spaces. Notice that the set of half spaces is not a π -system, the intersection of two half spaces is not a half space.

4 Two half spaces 10 5 0 −5 −10 −10 −5 0 5 10 h − 1 (1 , 1) ( −∞ , 1] = { ( x, y ) : x + y ≤ 1 } and h − 1 (2 , 1) ( −∞ , 1] = { ( x, y ) : 2 x + y ≤ 1 }

5 Regular variation An R d -valued random vector X is regularly varying with index α ∈ (0 , ∞ ) and spectral measure σ on B ( S d − 1 ) , S d − 1 = { z ∈ R d : | z | = 1 } , if P( | X | > xt, X / | X | ∈ · ) → x − α σ ( · ) a B ( S d − 1 ) as t → ∞ for x > 0 . w p˚ P( | X | > t ) 1 1 0.5 0.5 0 0 1 2 3 4 5 6 1 2 3 4 5 6 Spectral measures with respect to the 2-norm and max-norm for bivariate t ( α, Σ) -distributions, α = 0 , 2 , 4 , 8 , 16 , Σ 11 = Σ 22 = 1 and Σ 12 = Σ 21 = 1 / 2 .

6 Regular variation A function L is slowly varying if lim t →∞ L ( ut ) /L ( t ) = 1 for alla u > 0 . An R d -valued random vector is regularly varying with index α ∈ (0 , ∞ ) if and only if there exist a slowly varying function L and a measure µ � = 0 such that t →∞ t α L ( t ) P( X ∈ tA ) = µ ( A ) lim for all A ∈ B ( R d ) bounded away from 0 with µ ( ∂A ) = 0 . It follows that µ ( uA ) = u − α µ ( A ) for all u > 0 and A ∈ B ( R d ) . We write X ∈ RV( α, µ ) .

7 Regular variation and linear combinations Suppose that X ∈ RV( α, µ ) , i.e. lim t →∞ t α L ( t ) P( X ∈ tA ) = µ ( A ) . Let W x = { z ∈ R d : x T z > 1 } - a half space which does not contain 0 . Then it holds that for all x � = 0 t →∞ t α L ( t ) P( x T X > t ) = lim t →∞ t α L ( t ) P( X ∈ tW x ) = µ ( W x ) . lim We have shown that (1) X ∈ RV( α, µ ) implies  lim t →∞ t α L ( t ) P( x T X > t ) = h ( x ) exists , for all x � = 0 ,  (2)  h ( x ) > 0 for some x � = 0 , with h ( x ) = µ ( W x ) . But does it hold that (2) ⇒ (1) ? i.e. Does the Cram´ er-Wold device for regular variation hold?

8 Is the measure µ determined by its values on half spaces? A sufficient condition for (1) ⇔ (2) is that µ ( uW x ) = u − α µ ( W x ) u > 0 , x � = 0 and µ ( W x ) = � µ ( W x ) x � = 0 implies µ = � µ , i.e. that µ is determined by the values it gives to half spaces. Problem: Since µ is not a finite measure we cannot use characteristic functions. (Basrak, Davis, Mikosch 2002) If α is not an integer, then µ is determined by the values it gives to half spaces. Hence, er-Wold device for regular variation holds if α is not an integer. the Cram´

9 Problem if α is an integer Consider R 2 -valued stochastic vectors X 1 och X 2 with d d = R (cos Θ 1 , sin Θ 1 ) ′ = R (cos Θ 2 , sin Θ 2 ) ′ , and X 1 X 2 where R is Pareto-distributed, P( R > r ) = r − α for r > 1 , and independent of Θ 1 , Θ 2 . We see that X k , k = 1 , 2 , are regularly varying with index α and spectral measures P((cos Θ k , sin Θ k ) ∈ · ) . Let α be an integer and Θ 1 uniformly distributed on [0 , 2 π ) . Let Θ 2 have density f 2 ( θ ) = 1 2 π + c sin(( α + 2) θ ) , c ∈ (0 , 1 / 2 π ) .

10 Problem if α is an integer Since Θ 1 and Θ 2 have different distributions it holds that X 1 ∈ RV( α, µ 1 ) and X 2 ∈ RV( α, µ 2 ) with µ 1 � = µ 2 . It can be shown that µ 1 ( W x ) = µ 2 ( W x ) for all x � = 0 , i.e. µ 1 and µ 2 agree on half spaces. Conclusion: µ is not determined by the values it gives to half spaces if α is an integer! Notice that this does not answer the question: Does regular variation with index α of x T X for all x � = 0 imply that X is regularly varying with index α ?

11 Harry Kesten’s remark In (Kesten 1973) a remark says that for α = 1 regular variation of x T X for all x � = 0 is not a sufficient condition for regular variation of X . It turns out that unpublished notes by Kesten contains the idea for showing that: (Hult, Lindskog 2005) If α is an integer, then one can find a random vector X which is not regularly varying for which x T X is regularly varying with index α for all x � = 0 . Hence, The Cram´ er-Wold device for regular variation does not hold!

12 Stochastic recurrence equations If ( A 1 , B 1 ) , ( A 2 , B 2 ) , . . . are independent and identically distributed, A k ∈ R d × d and B k ∈ R d , then the stationary solution X ∞ to X n +1 = A n X n + B n , under mild conditions on ( A 1 , B 1 ) , satisfies  lim t →∞ t α P( x T X ∞ > t ) = h ( x ) exists , for all x � = 0 ,   h ( x ) > 0 for some x � = 0 , Hence, from the above condition it does not follow that X ∞ is regularly varying if α is an integer.

13 References Basrak, Davis, Mikosch 2002. A characterization of multivariate regular variation. Ann. Appl. Probab. 12, 908-920. Hult, Lindskog 2005. On Kesten’s counterexample to the Cram´ er-Wold device for regular variation. To appear in Bernoulli. Kesten 1973. Random difference equations and renewal theory for products of random matrices. Acta Math. 131, 207-248.