Notes 9 : Infinitely divisible and stable laws

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1 Notes 9 : Infinitely divisible and stable laws Math Fall 203 Lecturer: Sebastien Roch References: [Dur0, Section 3.7, 3.8], [Shi96, Section III.6]. Infinitely divisible distributions Recall: EX 9. (Normal distribution) Let Z, Z 2 N(0, ) independent then and Z + Z 2 N(0, 2). φ Z +Z 2 (t) = φ Z (t)φ Z2 (t) = e t2, EX 9.2 (Poisson distribution) Similarly, if Y, Y 2 Poi(λ) independent then and Y + Y 2 Poi(2). φ Y +Y 2 (t) = φ Y (t)φ Y2 (t) = exp ( 2λ(e it ) ), DEF 9.3 (Infinitely divisible distributions) A DF F is infinitely divisible if there for all n, there is a DF F n such that Z = d X n, + + X n,n, where Z F and the X n,k s are IID with X n,k F n for all k n. THM 9.4 A DF F can be a limit in distribution of IID triangular arrays if and only if F is infinitely divisible. Z n = X n + + X n,n, Proof: One direction is obvious. If F is infinitely divisible, then we can make Z = d Z n for all n with Z F.

2 Lecture 9: Infinitely divisible and stable laws 2 For the other direction, assume Z n Z. Write Z 2n = (X 2n, + + X 2n,n ) + (X 2n,n+ + + X 2n,2n ) = Y n + Y n. Clearly Y n and Y n are independent and identically distributed. We claim further that they are tight. Indeed P[Y n > y] 2 = P[Y n > y]p[y n > y] P [Z 2n > 2y] and Z n converges to a probability distribution by assumption. Now take a subsequence of {Y n } so that Y nk Y (and hence Y n k Y ) then, using CFs, Y nk + Y n k Y + Y, where Y, Y are independent. Since Z n Z it follows that Z = d Y + Y. EX 9.5 (Normal distribution) To test whether a DF F is infinitely divisible, it suffices to check that (φ F (t)) /n is a CF, n. In the N(0, ) case, which is the CF of a N(0, n). (φ F (t)) /n = e t2 /(2n), EX 9.6 (Poisson distribution) Similarly, in the F Poi(λ) case, is the CF of a Poi(λ/n). (φ F (t)) /n = exp(λ(e it )/n), EX 9.7 (Gamma distribution) Let F be a Gamma(α, β) distribution, that is, with density f(x) = xα e x/β Γ(α)β α x 0, for α, β > 0, and CF Hence φ(t) = (φ(t)) /n = which is the CF of a Gamma(α/n, β). ( iβt) α. ( iβt) α/n,

3 Lecture 9: Infinitely divisible and stable laws 3 It is possible to characterize all infinitely divisble distributions. We quote the following without proof. See Breiman s or Feller s books. THM 9.8 (Lévy-Khinchin Theorem) A DF F is infinitely divisible if and only if φ F (t) is of the form e ψ(t) with ψ(t) = itβ t2 σ 2 + ( 2 + e itx itx ) + x 2 µ(dx), where β R, σ 2 0, and µ is a measure on R with µ({0}) = 0 and x 2 +x 2 µ(dx) < +. (The previous ratio is so any finite measure will work.) EX 9.9 (Normal distribution) Taking β = 0, σ 2 =, and µ 0 gives a N(0, ). EX 9.0 (Poisson distribution) Taking σ 2 = 0, µ({}) = λ, and β = x µ(dx) +x 2 gives a Poi(λ). EX 9. (Compound Poisson distribution) Let W be a RV with CF φ(t). A compound Poisson RV with parameters λ and φ is Z = N W i, where the W i s are IID with CF φ and N Poi(λ). By direct calculation, e λ λ n φ Z (t) = φ(t) n = exp(λ(φ(t) )). n! n=0 i= Z is infinitely divisible for the same reason that the Poisson distribution is. Taking σ 2 = 0, µ({}) = λ PM of Z, and β = x µ(dx) gives φ +x 2 Z. (The previous integral exists for PMs because when x is large the ratio is essentially /x.) 2 Stable laws An important special case of infinitely divisible distributions arises when the triangular array takes the special form X k, n, k,, up to ormalization that depends on n. E.g., the simple form of the CLT. DEF 9.2 A DF F is stable if for all n Z = d n i= W i b n, where Z, {W i } i F and the W i s are independent and > 0, b n R are constants.

4 Lecture 9: Infinitely divisible and stable laws 4 Note that this applies to Gaussians but not to Poisson variables (which are integervalued). THM 9.3 Let F be a DF and Z F. A necessary and sufficient condition for the existence of constants > 0, b n such that Z n = S n b n where S n = n i= X i with X i s IID is that F is stable. Proof: We prove the non-trivial direction. Assume as above. Letting and Then Z n = S n b n S () n = i n S (2) n = n+ i 2n ({ } S n () b n Z 2n = a + n a 2n X i, X i. { S (2) n b n } 2b n b 2n The LHS and both terms in bracket converge weakly. One has to show that the constants converge. LEM 9.4 (Convergence of types) If W n W and there are constants α n > 0, β n so that W n = α n W n + β n W where W and W are nondegenerate, then there are constants α and β so that α n α and β n β. See [D] for proof. It requires an exercise done in homework. Finally there is also a characterization for stable laws. THM 9.5 (Lévy-Khinchin Representation) A DF F is stable if and only if the CF φ F (t) is of the form e ψ(t) with ψ(t) = itβ d t α ( + iκ sgn(t)g(t, α)), where 0 < α 2, β R, d 0, κ, and { tan G(t, α) = 2πα if α, 2 π log t if α =. The parameter α is called the index of the stable law. ).

5 Lecture 9: Infinitely divisible and stable laws 5 EX 9.6 Taking α = 2 gives a Gaussian. Note that a DF is symmetric if and only if or, in other words, φ(t) is real. φ(t) = E[e itx ] = E[e itx ] = φ(t), THM 9.7 (Lévy-Khinchin representation: Symmetric case) A DF F is symmetric and stable if and only if its CF is of the form e d t α where 0 < α 2 and d 0. The parameter α is called the index of the stable law. α = 2 is the Gaussian case. α = is the Cauchy case. 2. Domain of attraction DEF 9.8 (Domain of attraction) A DF F is in the domain of attraction of a DF G if there are constants > 0 and b n such that S n b n where Z G and S n = k n X k, with X k F and independent. DEF 9.9 (Slowly varying function) A function L is slowly varying if for all t > 0 L(tx) lim x + L(x) =. EX 9.20 If L(t) = log t then L(tx) L(x) = log t + log x log x. On the other hand, if L(t) = t ε with ε > 0 then L(tx) L(x) = tε x ε x ε t ε. THM 9.2 Suppose (X n ) n are IID with a distribution that satisfies

6 Lecture 9: Infinitely divisible and stable laws 6. lim x P[X > x]/p[ X > x] = θ [0, ] 2. P[ X > x] = x α L(x), where 0 < α < 2 and L is slowly varying. Let S n = k n X k, = inf{x : P[ X > x] n } (Note that is roughly of order n /α.) and Then where Z has an α-stable distribution. b n = ne[x X ]. S n b n This condition is also necessary. Note also that this extends in some sense the CLT to some infinite variance cases when α = 2. See the infinite variance example in Section 3.4 on [D]. EX 9.22 (Cauchy distribution) Let (X n ) n be IID with a density symmetric about 0 and continuous and positive at 0. Then ( + + ) a Cauchy distribution. n X X n Clearly θ = /2. Moreover P[/X > x] = P[0 < X < x ] = x 0 f(y)dy f(0)/x, and similarly for P[/X < x]. So α =. Clearly b n = 0 by symmetry. Finally 2f(0)n. EX 9.23 (Centering constants) Suppose (X n ) n are IID with for x > P[X > x] = θx α, P[X < x] = ( θ)x α, where 0 < α < 2. In this case = n /α and n /α cn α > b n = n (2θ )αx α dx cn log n α = cn /α α <.

7 Lecture 9: Infinitely divisible and stable laws 7 References [Dur0] Rick Durrett. Probability: theory and examples. Cambridge Series in Statistical and Probabilistic Mathematics. Cambridge University Press, Cambridge, 200. [Shi96] A. N. Shiryaev. Probability, volume 95 of Graduate Texts in Mathematics. Springer-Verlag, New York, 996.