## Large Deviations of U-Empirical Measures in
Strong Topologies and Applications

**Abstract:**
We prove large deviation principles (LDP) for *m*-fold products of empirical
measures and for U-empirical measures, where the underlying i.i.d. random
variables take values in a measurable (not necessarily Polish) space *S*. The
results can be formulated on suitable subsets of all probability measures on the
product space *S*^{m}. We endow the spaces with topologies, which are
stronger than the usual tau-topology and which make integration with respect to
certain unbounded, Banach-space valued functions a continuous operation. A special
feature is the non-convexity of the rate function for *m*>2. Improved
versions of LDPs for Banach-space valued U- and V-statistics are obtained as a
particular application. Some further applications concerning the Gibbs conditioning
principle and a process level LDP are mentioned.

**Keywords:**
large deviations,
empirical measures,
Sanov's theorem,
strong topology,
U-statistics,
von Mises statistics,
Gibbs conditioning principle,
mean-field model,
process-level large deviations

**Reference:**
Annales de l'Institut Henri Poincaré,
Probabilités et Statistiques, Vol. 38, No. 5 (2002) 779-797.

DOI: 10.1016/S0246-0203(02)01116-0

**2010
Mathematics Subject Classification:**

- 60F10 Large deviations (primary)
- 62H10 Distribution of statistics
- 28A35 Measures and integrals in product spaces

The paper (16 pages, revised version
February 18, 2002) is available in:

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