Seminar 2012 - Counterexamples in Probability Presenter : Joung In Kim

1. Seminar 2012 -Counterexamples in Probability Presenter : Joung In Kim Seminar | 19.11.2012 |

2. Seminar – Counterexamples in Probability Ch8. Characteristic and Generating Functions Ch9. Infinitely divisible and stable distributions

3. Seminar – Counterexamples in Probability Ch8. Characteristic and Generating Functions Ch9. Infinitely divisible and stable distributions

4. Notation and Abbreviations • r.v. : random variable • ch. f. : characteristic function (ϕ(t)) • d.f. : distribution function (F) • i.i.d. : independent and identically distributed • : equality in distribution

5. Definition (Characteristic function)

6. Properties of a characteristic function

7. Properties of a characteristic function

8. Fourier expansion ofa periodic function

9. Example 1. Discrete and absolutely continuous distributions with the same characteristic functions on [-1, 1] continuous discrete

10. Example 1. Discrete and absolutely continuous distributions with the same chacteristic functions on [-1, 1]

11. Example 2. The absolute value of a characteristic function is not necessarily a characteristic function.

12. Decomposable and Indecomposable • Wesay that a ch.f. ϕ is decomposable • if it can be represented as a product of two non-trivial ch.f.s. ϕ1 and ϕ2, i.e. ϕ(t) = ϕ1(t) ϕ2(t) • and neither ϕ1 nor ϕ2 is the ch.f. of a probability measure which is concentrated at one point. • Otherwise ϕ is called indecomposable.

13. Example 3. The factorization of a characteristic function into indecomposable factors may not be unique. • discrete case • X : discrete uniform distribution on the set {0, 1, 2, 3, 4, 5}. • Characteristic function of X : • We can factorize the ch. f. in the following way :

14. Example 3. The factorization of a characteristic function into indecomposable factors may not be unique. • Need to check :

15. Example 3. The factorization of a characteristic function into indecomposable factors may not be unique. • (ii) Continuous case • ∙ Let X be a r.v. which is uniformly distributed on (-1,1). • ∙ Ch. f. of X :

16. Example 3. The factorization of a characteristic function into indecomposable factors may not be unique.

17. Seminar – Counterexamples in Probability Ch8. Characteristic and Generating Functions Ch9. Infinitely divisible and stable distributions

18. Definition (infinitely divisible distribution) • ∙ X : a r.v. with d.f. F • ∙ ϕ : ch.f. of X • ∙ X is called infinitely divisible if • for each n≥1 there exist i.i.d. r.v.s Xn1, ..., Xnn such that • X Xn1 + ∙∙∙ + Xnn • Equivalent : • ∙ Ǝ d.f. Fn with F=(Fn)*n • ∙ Ǝ ch.f. ϕn with ϕ =(ϕ n)n

19. Definition (stable distribution) • ∙ X : a r.v. with d.f. F • ∙ ϕ : ch.f. of X • ∙ X is called stable if • for X1 and X2 independent copies of X and any positive numbers b1 and b2, there is a positive number b and a real number γ s.t. : • b1X1+b2X2 bX + γ • Equivalent :

20. Properties of infinitely divisible and stable distributions • The ch.f. of an infinitely divisible r.v. does not vanish. • If a r.v. X is stable, then it is infinitely divisible.

21. Example 4. A non-vanishing characteristic function which is not infinitely divisible random variable X => => ϕ does not vanish.

22. Example 4. A non-vanishing characteristic function which is not infinitely divisible • Is X infinitely divisible? • Assume X X1+X2, (X1, X2 are iid r.v.s) • Since X has three possible values, each of X1 and X2 can take only two values, say a and b, a<b. • Let P[Xi=a]=p, P[Xi=b]=1-p for some p, 0<p<1, i=1,2 • 2a= -1, a+b=0, 2b=1, p2=1/8, 2p(1-p)=3/4, (1-p)2=1/8 => contradiction! • X X1+X2 is not possible. => X is not infinitely divisible.

23. Example 5. Infinitely divisible distribution, but not stable • X ~ Poi(λ) • , n=0, 1, 2, ∙∙∙, λ>0 • Characteristic funtion of X : • Characteristic funtion of Xn ~Poi(λ/n) : • => • => X is infinitely divisible

24. Example 5. Infinitely divisible distribution, but not stable • Is X a stable distribution? • If yes, for any b1 and b2 >0, there exist b>0 and γ∈ s.t.

25. Example 5. Infinitely divisible distribution, but not stable • ii) Let see the gamma distribution with parameter θ=1, k=1/2

26. Example 5. Infinitely divisible distribution, but not stable • Is X a stable distribution? • If yes, for any b1 and b2 >0, there exist b>0 and γ ∈ s.t.

27. REFERENCES • [1] J. Stoyanov. Counterexamples in probability (2nd edition). Wiley 1997 • [2] G. Samorodnitsky, M. S. Taqqu. Stable Non-Gaussian Random Processes. Chapman&Hall, 1994 • [3] K. L. Chung. A course in probability theory. Academic Press, 1974 • [4] E. Lukacs. Characteristic functions. Griffin, 1970

28. Thank you very much !!!