By James E. Gentle

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32) 1/2j j=0 µj diverges. A Companion for Mathematical Statistics c 2010 James E. 33) where M, λ, γ > 0 and ρ ≥ 1. 34) where M, λ, γ > 0 and ρ ≥ 1/2. 12 are called the Carleman criteria (after Torsten Carleman). 29). Within a specific family of distributions, no matter how broad the family, the full set of moments can usually be expected to identify the distribution. For more narrowly-defined families of distributions, such as the normal or gamma families, often only one or two moments completely identify the family.

Proof. It will be convenient to define a function related to the CF: let h(t) = eµt ϕX (t); hence h(0) = 1, h (0) = 0, and h (0) = σ 2 . Now expand h in a Taylor series about 0: 1 h(t) = h(0) + h (0)it − h (ξ)t2 , 2 for some ξ between 0 and t. Substituting for h(0) and h (0), and adding and subtracting σ 2 t/ 2 to this, we have h(t) = 1 − σ 2 t2 (h (ξ) − σ 2 )t2 − . 2 2 A Companion for Mathematical Statistics c 2010 James E. 1 Some Important Probability Facts 31 This is the form we will find useful.

15). It is the integral of a function, X, over Ω with respect to a measure, P , over the σ-field that together with Ω forms the measurable space. To emphasize the meaning more precisely, we could write the integral in the definition as E(X) = X(ω) dP (ω). 15) is over an abstract domain Ω. We can also write the expectation over the real range of the random variable and an equivalent measure on that range. 16) IR d or in the more precise form, E(X) = x dF (x). IRd If the PDF exists and is f, we also have E(X) = xf(x) dx.