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Seminar Counterexamples in Probability Presenter : Joung In Kim Seminar | |

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

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

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Seite 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

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Seite 5 Definition (Characteristic function)

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Seite 6 Properties of a characteristic function

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Seite 7 Properties of a characteristic function

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Seite 8 Fourier expansion of a periodic function

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

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

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

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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.

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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 :

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

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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 :

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

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

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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 X n1,..., X nn such that X X n1 + ∙∙∙ + X nn Equivalent : ∙ Ǝ d.f. F n with F=(F n ) *n ∙ Ǝ ch.f. ϕ n with ϕ =( ϕ n ) n

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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 bX + γ Equivalent :

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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.

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Seite 21 Example 4. A non-vanishing characteristic function which is not infinitely divisible random variable X => => ϕ does not vanish. X01 P(X=x)1/83/41/8

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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

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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 X n ~Poi(λ/n) : => =>X is infinitely divisible

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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.

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

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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.

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Seite 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

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Seite 28 Thank you very much !!!

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