On the Breiman conjecture

Transcription

1 Péter Kevei 1 David Mason 2 1 TU Munich 2 University of Delaware 12th GPSD Bochum

2 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

3 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

4 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

5 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

6 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

7 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

8 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

9 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

10 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

11 Motivation Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

12 Motivation Breiman 1965 Coin tossing random walk S 1, S 2,.... Put Y 1, Y 2,... the interarrival times between the zeros of S 1, S 2,.... X, X 1, X 2,... iid P{X = 0} = 1 2 = P{X = 1}. T n = i=1 X iy i i=1 Y i is the proportion of the time that the random walk spends in [0, ).

13 Motivation Breiman 1965 Coin tossing random walk S 1, S 2,.... Put Y 1, Y 2,... the interarrival times between the zeros of S 1, S 2,.... X, X 1, X 2,... iid P{X = 0} = 1 2 = P{X = 1}. T n = i=1 X iy i i=1 Y i is the proportion of the time that the random walk spends in [0, ).

14 Motivation Arc-sine law In this case: lim P {T n x} = 2 n π arcsin x

15 Motivation In general Y, Y 1, Y 2,... non-negative iid rv s with df G X, X 1, X 2,... iid with df F, independent from Y, Y 1, Y 2,..., E X < T n = i=1 X iy i i=1 Y i

16 Motivation In general Y, Y 1, Y 2,... non-negative iid rv s with df G X, X 1, X 2,... iid with df F, independent from Y, Y 1, Y 2,..., E X < T n = i=1 X iy i i=1 Y i

17 Motivation In general Y, Y 1, Y 2,... non-negative iid rv s with df G X, X 1, X 2,... iid with df F, independent from Y, Y 1, Y 2,..., E X < T n = i=1 X iy i i=1 Y i

18 Motivation Remark If EY <, then i=1 X iy i i=1 Y i = i=1 X i Y i n i=1 Y i n a.s. EX. E X < implies (T n ) is tight.

19 Motivation Remark If EY <, then i=1 X iy i i=1 Y i = i=1 X i Y i n i=1 Y i n a.s. EX. E X < implies (T n ) is tight.

20 Earlier results Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

21 Earlier results Theorem (Breiman, 1965) If T n converges in distribution for every F, and the limit is non-degenerate for at least one F, then Y D(β), for some β [0, 1). Conjecture (Breiman) If T n has a non-degenerate limit for some F, then Y D(β) for some β [0, 1). Breiman, L. On some limit theorems similar to the arc-sin law Teor. Verojatnost. i Primenen , 1965.

22 Earlier results Theorem (Breiman, 1965) If T n converges in distribution for every F, and the limit is non-degenerate for at least one F, then Y D(β), for some β [0, 1). Conjecture (Breiman) If T n has a non-degenerate limit for some F, then Y D(β) for some β [0, 1). Breiman, L. On some limit theorems similar to the arc-sin law Teor. Verojatnost. i Primenen , 1965.

23 Earlier results D(β) Domain of attraction of an β-stable law: Y D(β) 1 G(x) = l(x) x β, where l is slowly varying (l(λx)/l(x) 1 for any λ > 0 as x ).

24 Earlier results D(0) Y D(0) if 1 G(x) is slowly varying in which case (Darling, 1952) max{y i : i = 1, 2,..., n} i=1 Y i P 1 and so i=1 X iy i i=1 Y i D X

25 Earlier results D(0) Y D(0) if 1 G(x) is slowly varying in which case (Darling, 1952) max{y i : i = 1, 2,..., n} i=1 Y i P 1 and so i=1 X iy i i=1 Y i D X

26 Earlier results Limits Theorem (Breiman) Assume that Y D(β), β (0, 1), and E X β+ε <, for some ε > 0. Then T n D T, where P {T x} = [ u x β πβ arctan sgn(x u)f(du) u x β tan πβ F(du) 2 ]. P{T > x} P{X > x}

27 Earlier results Limits Theorem (Breiman) Assume that Y D(β), β (0, 1), and E X β+ε <, for some ε > 0. Then T n D T, where P {T x} = [ u x β πβ arctan sgn(x u)f(du) u x β tan πβ F(du) 2 ]. P{T > x} P{X > x}

28 Earlier results E X 2+δ < Theorem (Mason & Zinn, 2005) Assume that E X 2+δ <. Then T n R, where R is non-degenerate, iff Y D(β), β [0, 1).

29 Earlier results Studentization Other type of self-normalization (Logan & Mallows & Rice & Shepp, 1973): i=1 X i n, i=1 X i 2 X, X 1, X 2,... iid. Student s T -statistic: i=1 X i n n i=1 (X i X) 2 n 1 The two ratios are asymptotically the same.

30 Earlier results Studentization Other type of self-normalization (Logan & Mallows & Rice & Shepp, 1973): i=1 X i n, i=1 X i 2 X, X 1, X 2,... iid. Student s T -statistic: i=1 X i n n i=1 (X i X) 2 n 1 The two ratios are asymptotically the same.

31 Earlier results Conjecture (Logan & Mallows & Rice & Shepp, 1973) i=1 X i D n W, i=1 X i 2 where P{ W = 1} < 1, iff X D(α), α (0, 2]; if α > 1, EX = 0; if α = 1, X D(Cauchy). Giné & Götze & Mason (1997): W is standard normal iff X D(2) and EX = 0 Chistyakov & Götze (2004): in general

32 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

33 Recall Y, Y 1, Y 2,... non-negative iid rv s with df G X, X 1, X 2,... iid with df F, independent from Y, Y 1, Y 2,..., E X <. φ X (t) = Ee itx

34 Theorem (K Mason) Assume that for some EX = 0, 1 < α 2, positive slowly varying function L at zero and c > 0, log (Rφ X (t)) t α L ( t ) c, as t 0. Whenever i=1 X iy i i=1 Y i D W (W nondegenerate) then Y D(β) for some β [0, 1).

35 What does this condition mean? As log Rφ X (t) 1 Rφ X (t), t 0, log (Rφ X (t)) t α L ( t ) c 1 Rφ X (t) t α L( t ) c. For α < 2 this holds iff (Pitman) P { X > x} L(1/x)x α cγ(α) 2 π sin ( πα 2 ) If EX = 0 and X D(α) then this condition is satisfied. Also if EX = 0, EX 2 < then the condition of the theorem is satisfied (α = 2, c = σ 2 /2).

36 What does this condition mean? As log Rφ X (t) 1 Rφ X (t), t 0, log (Rφ X (t)) t α L ( t ) c 1 Rφ X (t) t α L( t ) c. For α < 2 this holds iff (Pitman) P { X > x} L(1/x)x α cγ(α) 2 π sin ( πα 2 ) If EX = 0 and X D(α) then this condition is satisfied. Also if EX = 0, EX 2 < then the condition of the theorem is satisfied (α = 2, c = σ 2 /2).

37 What does this condition mean? As log Rφ X (t) 1 Rφ X (t), t 0, log (Rφ X (t)) t α L ( t ) c 1 Rφ X (t) t α L( t ) c. For α < 2 this holds iff (Pitman) P { X > x} L(1/x)x α cγ(α) 2 π sin ( πα 2 ) If EX = 0 and X D(α) then this condition is satisfied. Also if EX = 0, EX 2 < then the condition of the theorem is satisfied (α = 2, c = σ 2 /2).

38 Proposition Assume that the assumptions of the theorem hold. Then for some 0 < γ 1 i=1 E Y i α ( i=1 Y α γ. ( ) i)

39 Proposition If ( ) holds with some γ (0, 1] then Y D(β), for some β [0, 1), where β ( 1, 0] is the unique solution of Beta(α 1, 1 β) = Γ(α 1)Γ(1 β) Γ(α β) = 1 γ(α 1). In particular, Y D(0) for γ = 1. Conversely, if Y D(β), 0 β < 1, then ( ) holds with γ = Γ(α β) Γ(α)Γ(1 β) = 1 (α 1)Beta(α 1, 1 β). Extension of a result by Fuchs, Joffe and Teugels (2001), where α = 2.

40 Proposition If ( ) holds with some γ (0, 1] then Y D(β), for some β [0, 1), where β ( 1, 0] is the unique solution of Beta(α 1, 1 β) = Γ(α 1)Γ(1 β) Γ(α β) = 1 γ(α 1). In particular, Y D(0) for γ = 1. Conversely, if Y D(β), 0 β < 1, then ( ) holds with γ = Γ(α β) Γ(α)Γ(1 β) = 1 (α 1)Beta(α 1, 1 β). Extension of a result by Fuchs, Joffe and Teugels (2001), where α = 2.

41 Proposition If ( ) holds with some γ (0, 1] then Y D(β), for some β [0, 1), where β ( 1, 0] is the unique solution of Beta(α 1, 1 β) = Γ(α 1)Γ(1 β) Γ(α β) = 1 γ(α 1). In particular, Y D(0) for γ = 1. Conversely, if Y D(β), 0 β < 1, then ( ) holds with γ = Γ(α β) Γ(α)Γ(1 β) = 1 (α 1)Beta(α 1, 1 β). Extension of a result by Fuchs, Joffe and Teugels (2001), where α = 2.

42 Sketch of the proof Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

43 Sketch of the proof i=1 E Y i α ( i=1 Y α γ ( ) i) Proposition If ( ) holds with some γ (0, 1] then Y D(β), for some β [0, 1), where β ( 1, 0] is the unique solution of Beta(α 1, 1 β) = Γ(α 1)Γ(1 β) Γ(α β) = 1 γ(α 1). In particular, Y D(0) for γ = 1. Conversely,.... For α = 2 this gives 1 γ = β.

44 Sketch of the proof i=1 E Y i α Y1 α ( i=1 Y α = ne( i) i=1 Y α i) = n Γ(α) E = n Γ(α) n =: Γ(α) 0 0 Note that for α = 2 we have φ α = φ s 0 0 Y α 1 e t i=1 Y i t α 1 dt ( ) t α 1 E e ty 1 Y1 α (Ee ty 1 ) n 1 dt t α 1 φ α (t)φ 0 (t) n 1 dt. t α 1 φ α (t)e s log φ 0(t) dt γγ(α), s. 0.

45 Sketch of the proof i=1 E Y i α Y1 α ( i=1 Y α = ne( i) i=1 Y α i) = n Γ(α) E = n Γ(α) n =: Γ(α) 0 0 Note that for α = 2 we have φ α = φ s 0 0 Y α 1 e t i=1 Y i t α 1 dt ( ) t α 1 E e ty 1 Y1 α (Ee ty 1 ) n 1 dt t α 1 φ α (t)φ 0 (t) n 1 dt. t α 1 φ α (t)e s log φ 0(t) dt γγ(α), s. 0.

46 Sketch of the proof φ α (t) = Ee ty Y α, By Karamata s Tauberian theorem φ 0 (t) = Ee ty lim t 0 t 0 y α 1 φ α (y)dy 1 φ 0 (t) After some further calculation t α u 1 α e ut = G(u)uα 1 e ut du G(u)e ut du 1 Γ(α 1) 0 = γγ(α). γγ(α), as t 0. y α 2 e (y+t)u dy, which holds for u > 0 and α (1, 2]. Weyl-transform, or Weyl-fractional integral of the function e ut.

47 Sketch of the proof φ α (t) = Ee ty Y α, By Karamata s Tauberian theorem φ 0 (t) = Ee ty lim t 0 t 0 y α 1 φ α (y)dy 1 φ 0 (t) After some further calculation t α u 1 α e ut = G(u)uα 1 e ut du G(u)e ut du 1 Γ(α 1) 0 = γγ(α). γγ(α), as t 0. y α 2 e (y+t)u dy, which holds for u > 0 and α (1, 2]. Weyl-transform, or Weyl-fractional integral of the function e ut.

48 Sketch of the proof φ α (t) = Ee ty Y α, By Karamata s Tauberian theorem φ 0 (t) = Ee ty lim t 0 t 0 y α 1 φ α (y)dy 1 φ 0 (t) After some further calculation t α u 1 α e ut = G(u)uα 1 e ut du G(u)e ut du 1 Γ(α 1) 0 = γγ(α). γγ(α), as t 0. y α 2 e (y+t)u dy, which holds for u > 0 and α (1, 2]. Weyl-transform, or Weyl-fractional integral of the function e ut.

49 Sketch of the proof We obtain 1 (u 1) α 2 u α g (x/u) g (x) du = k M g (x) g (x) [γ(α 1)] 1 with g (x) = k M h(x) = 0 0 G(ux)u α 1 e u du. h(x/u)k(u)/udu Mellin-convolution of h and k. Drasin-Shea theorem implies that g (x) is regularly varying at infinity with index 0 ρ > 1.

50 Sketch of the proof We obtain 1 (u 1) α 2 u α g (x/u) g (x) du = k M g (x) g (x) [γ(α 1)] 1 with g (x) = k M h(x) = 0 0 G(ux)u α 1 e u du. h(x/u)k(u)/udu Mellin-convolution of h and k. Drasin-Shea theorem implies that g (x) is regularly varying at infinity with index 0 ρ > 1.

51 Sketch of the proof We obtain 1 (u 1) α 2 u α g (x/u) g (x) du = k M g (x) g (x) [γ(α 1)] 1 with g (x) = k M h(x) = 0 0 G(ux)u α 1 e u du. h(x/u)k(u)/udu Mellin-convolution of h and k. Drasin-Shea theorem implies that g (x) is regularly varying at infinity with index 0 ρ > 1.

52 Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

53 Recall Y, Y 1, Y 2,... non-negative iid rv s with df G X, X 1, X 2,... iid with df F, independent from Y, Y 1, Y 2,..., E X < T n = i=1 X iy i i=1 Y i E X < implies (T n ) is tight.

54 Notation id(a, b, ν) infinitely divisible distribution on R d with characteristic exponent iu b 1 ( ) 2 u au + e iu x 1 iu xi( x 1) ν(dx), where b R d, a R d d is a positive semidefinite matrix and ν is the Lévy measure.

55 Theorem (K & Mason, 2012) If along a subsequence {n } 1 a n n i=1 Y i D W2, as n, where W 2 id(0, b, Λ), then ( n i=1 X ) n iy i i=1, Y i D (W 1, W 2 ), n, a n a n where (W 1, W 2 ) id(0, b, Π)

56 Theorem (K & Mason, 2012) If along a subsequence {n } 1 a n n i=1 Y i D W2, as n, where W 2 id(0, b, Λ), then ( n i=1 X ) n iy i i=1, Y i D (W 1, W 2 ), n, a n a n where (W 1, W 2 ) id(0, b, Π)

57 Theorem (K & Mason, 2012) i.e. its characteristic function Ψ(θ 1, θ 2 ) = Ee i(θ 1W 1 +θ 2 W 2 ) = exp + 0 { i(θ 1 b 1 + θ 2 b 2 ) ( ) e i(θ 1x+θ 2 y) 1 (iθ 1 x + iθ 2 y) 1 {x 2 +y 2 1} } F (dx/y) Λ (dy). { } W1 H(x) = P x = 1 W π ImΨ(u, ux) du. u

58 Theorem (K & Mason, 2012) i.e. its characteristic function Ψ(θ 1, θ 2 ) = Ee i(θ 1W 1 +θ 2 W 2 ) = exp + 0 { i(θ 1 b 1 + θ 2 b 2 ) ( ) e i(θ 1x+θ 2 y) 1 (iθ 1 x + iθ 2 y) 1 {x 2 +y 2 1} } F (dx/y) Λ (dy). { } W1 H(x) = P x = 1 W π ImΨ(u, ux) du. u

59 Feller class ξ, ξ 1,... iid with df F, S n = i=1 ξ i. F is in the centered Feller class, if there exists B n, such that every subsequence n has a further subsequence n, such that S n B n where W is non-degenerate. D W, Theorem (Feller (1966), Maller (1979)) Y is in the centered Feller class, iff lim sup x x 2 P{ Y > x} + x EYI( Y x) E[Y 2 I( Y x)] <.

60 Feller class ξ, ξ 1,... iid with df F, S n = i=1 ξ i. F is in the centered Feller class, if there exists B n, such that every subsequence n has a further subsequence n, such that S n B n where W is non-degenerate. D W, Theorem (Feller (1966), Maller (1979)) Y is in the centered Feller class, iff lim sup x x 2 P{ Y > x} + x EYI( Y x) E[Y 2 I( Y x)] <.

61 Surprising result Theorem (K & Mason, 2012) The subsequential limit distributions of T n = i=1 X iy i i=1 Y i are continuous for all X with finite expectation if and only if Y F c.

62 Further remarks Outline Introduction Motivation Earlier results Partial solution Sketch of the proof Subsequential limits Further remarks

63 Further remarks Towards Lévy processes (W 1, W 2 ) = D (a 1 + U, a 2 + V ), (( where (a 1, a 2 ) = b ) 1 0 xλ(dx) EX, b ) 1 0 xλ(dx) { Ee i(θ 1U+θ 2 V ) = exp 0 ( ) } e i(θ 1x+θ 2 y) 1 F (dx/y) Λ (dy) Under the conditions of the theorem ( 1 i n t X iy i 1 i n, t Y ) i D (a 1 t + U t, a 2 t + V t ) t>0, a n a n t>0 where (U t, V t ), t 0, is the corresponding Lévy process.

64 Further remarks U t V t D?, t 0 or t Kevei, P, Mason, D.M. Randomly Weighted Self-normalized Lévy Processes Stochastic Processes and their Applications, 123 (2) 2013,