Seminar 2012 Counterexamples in Probability Seminar 19 11

Seminar 2012 - Counterexamples in Probability Seminar | 19. 11. 2012 | Presenter : Joung In Kim

Seite 2 Seminar – Counterexamples in Probability Ch 8. Characteristic and Generating Functions Ch 9. Infinitely divisible and stable distributions

Seite 3 Seminar – Counterexamples in Probability Ch 8. Characteristic and Generating Functions Ch 9. Infinitely divisible and stable distributions

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

Seite 5 Definition (Characteristic function)

Seite 6 Properties of a characteristic function

Seite 7 Properties of a characteristic function

Seite 8 Fourier expansion of a periodic function

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

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

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

Seite 12 Decomposable and Indecomposable We say that a ch. f. ϕ is decomposable if it can be represented as a product of two non-trivial ch. f. s. ϕ 1 and ϕ 2, i. e. ϕ(t) = ϕ 1(t) ϕ 2(t) and neither ϕ 1 nor ϕ 2 is the ch. f. of a probability measure which is concentrated at one point. Otherwise ϕ is called indecomposable.

Seite 13 Example 3. The factorization of a characteristic function into indecomposable factors may not be unique. (i) 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 :

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

Seite 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 :

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

Seite 17 Seminar – Counterexamples in Probability Ch 8. Characteristic and Generating Functions Ch 9. Infinitely divisible and stable distributions

Seite 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 Xn 1, . . . , Xnn such that X Equivalent : ∙ Ǝ d. f. Fn with F=(Fn)*n ∙ Ǝ ch. f. ϕn with ϕ =(ϕ n)n Xn 1 + ∙∙∙ + Xnn

Seite 19 Definition (stable distribution) ∙ X : a r. v. with d. f. F ∙ ϕ : ch. f. of X ∙ X is called stable if for X 1 and X 2 independent copies of X and any positive numbers b 1 and b 2, there is a positive number b and a real number γ s. t. : b 1 X 1+b 2 X 2 Equivalent : b. X + γ

Seite 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.

Seite 21 Example 4. A non-vanishing characteristic function which is not infinitely divisible random variable X X -1 0 1 P(X=x) 1/8 3/4 1/8 => => ϕ does not vanish.

Seite 22 Example 4. A non-vanishing characteristic function which is not infinitely divisible Is X infinitely divisible? Assume X X 1+X 2, (X 1, X 2 are iid r. v. s) Since X has three possible values, each of X 1 and X 2 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 X 1+X 2 2 a a+b 2 b P(X 1+X 2=x) p 2 2 p(1 -p)2 Þ 2 a= -1, a+b=0, 2 b=1, p 2=1/8, 2 p(1 -p)=3/4, (1 -p)2=1/8 => contradiction! Þ X X 1+X 2 is not possible. => X is not infinitely divisible.

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

Seite 24 Example 5. Infinitely divisible distribution, but not stable Is X a stable distribution? If yes, for any b 1 and b 2 >0, there exist b>0 and γ∈ s. t.

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

Seite 26 Example 5. Infinitely divisible distribution, but not stable Is X a stable distribution? If yes, for any b 1 and b 2 >0, there exist b>0 and γ ∈ s. t.

Seite 27 REFERENCES [1] J. Stoyanov. Counterexamples in probability (2 nd 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

Seite 28 Thank you very much !!!