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I am interested under what conditions Cramer's theorem applies in random variables taking values in Hilbert spaces. Following these lecture notes, but using a Hilbert space:

Let $X_1,X_2,\cdots$, be i.i.d. random vectors taking values in a Hilbert space $(\mathcal{H},(\cdot,\cdot))$ and $\mathcal{B}$ the Borel algebra. Denote $\mu=E[X_1]$ and $S_n=\frac{1}{n}[X_1+\cdots+X_n]$. Let $\{\mu_n\}_{n=1}^\infty$ be the distribution of $S_n$.

The cumulant generating function and its domain are given by \begin{align*} \Lambda(\lambda) &= \log E[\exp(\lambda,X_1)] \quad \text{and}\\ \mathcal{D}_\Lambda &= \{\lambda \in \mathcal{H} : \Lambda(\lambda) < \infty\} \end{align*} respectively. The Fenchel-Legendre transform and its domain are defined as \begin{align*} \Lambda^*(x) &= \sup\limits_{\lambda \in \mathcal{H}}\{(\lambda, x)-\Lambda(\lambda)\}\\ \mathcal{D}_\Lambda^* &= \{x \in \mathcal{H} : \Lambda^*(x) < \infty\} \end{align*} respectively.

Does the following theorem (2.1.6 in the lecture notes) still hold?

If $0\in \text{interior }\mathcal{D}_\Lambda$, then $\{\mu_n\}_{n=1}^\infty$ satisfies the large deviations principle with good rate function $\Lambda^*$: For all $\Gamma\in\mathcal{B}$,

$-\inf\limits_{x\in\text{interior }\Gamma} \Lambda^*(x) \leq \lim\limits_{\epsilon\to\infty}\inf\epsilon\log \mu_\epsilon(\Gamma) \leq \lim\limits_{\epsilon\to 0}\sup\epsilon\log\mu_\epsilon(\Gamma)\leq-\inf\limits_{x\in\text{closure }\Gamma} \Lambda^*(x)$

A reference would be really helpful.

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  • 2
    $\begingroup$ Theorem 6.1.3 in Dembo-Zeitouni covers this case I think. $\endgroup$ – Bati Oct 30 '14 at 16:57

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