## Topics in ProbabilityRecent research in probability has been concerned with applications such as data mining and finance models. Some aspects of the foundations of probability theory have receded into the background. Yet, these aspects are very important and have to be brought back into prominence. |

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### Contents

1 Probability Distributions | 1 |

2 Characteristic Functions | 11 |

3 Analytic Characteristic Functions | 27 |

4 Infinitely Divisible Distributions | 43 |

5 SelfDecomposable Distributions Triangular Arrays | 69 |

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absolutely continuous analytic c.f. atoms bution c.f. is given c.f. of F centering function characteristic functions compound Poisson distribution continuity of F continuity theorem continuous function converges in distribution converges properly convolution F Corollary degenerate distribution dF(a distri distribution concentrated distribution F distribution is infinitely distribution measure domain of attraction dominated convergence theorem e”d F(r Feller measure follows independent random variables infinitely divisible c.f. Infinitely Divisible Distributions integral intentionally left blank interval of continuity Lebesgue measure Lemma Let F Lévy process lim Sup log p(0 M{dt normal distribution numbers obtain points of increase probability distribution Problems for Solution Proof prove right-continuous Self-Decomposable Distributions sequence F sequence of independent singular distribution ſº stable distribution Suppose Theorem 4.5 Triangular Arrays uniquely determined variance write X1 and X2 X2 are independent zero