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I am studying some notes on exponential families and there is a section on the computation of moments. The exponential family has the form $$\exp(\sum_{j = 1}^k \phi_j B_j(x) + C(x) - D(\phi))$$

I understand that for $\lvert s \rvert < \delta$, $$M_{B(X)}(s) = \mathbb{E}[\exp(s^TB(X))] < \infty$$ but don't see how this implies that $\mathbb{E}[\lVert B(X) \rVert ^k] < \infty$. (I assume $\lVert .\rVert$ is the Euclidean norm)

I considered setting $s = \delta_0 \frac{B(X)}{\lVert B(X) \rVert}, \delta_0 < \delta$, to give

$$\mathbb{E}[\exp(s^TB(X))] = \mathbb{E}[\exp(\delta_0\lVert B(X) \rVert]$$

but I think $s$ needs to be a constant vector and so this approach is not allowed.

I have seen the following post (Does any moment generating function implies an existence of moments?) which demonstrates how finiteness of the mgf implies that $\mathbb{E}[\lvert X \rvert^k] < \infty$, but cannot see how to apply it to $\mathbb{E}[\lVert B(X) \rVert ^k] < \infty$.

I would appreciate it if I could be shown how

$$M_{B(X)}(s) < \infty \implies \mathbb{E}[\lVert B(X) \rVert ^k] < \infty.$$

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