### A Berry-Esseen theorem on semisimple Lie groups

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A central limit theorem is proved on the space ${\mathcal{P}}_{n}$ of positive definite symmetric matrices. To do this, some natural analogs of the mean and dispersion on ${\mathcal{P}}_{n}$ are defined and investigated. One uses a Taylor expansion of the spherical functions on ${\mathcal{P}}_{n}$.

In this paper, a general convergence theorem of fuzzy random variables is considered. Using this result, we can easily prove the recent result of Joo et al, which gives generalization of a strong law of large numbers for sums of stationary and ergodic processes to the case of fuzzy random variables. We also generalize the recent result of Kim, which is a strong law of large numbers for sums of levelwise independent and levelwise identically distributed fuzzy random variables.

We give new and general sufficient conditions for a Gaussian upper bound on the convolutions ${K}_{m+n}\ast {K}_{m+n-1}\ast \cdots \ast {K}_{m+1}$ of a suitable sequence K₁, K₂, K₃, ... of complex-valued functions on a unimodular, compactly generated locally compact group. As applications, we obtain Gaussian bounds for convolutions of suitable probability densities, and for convolutions of small perturbations of densities.

In this note we continue a theme taken up in part I, see [Gzyl and Recht: The geometry on the class of probabilities (I). The finite dimensional case. Rev. Mat. Iberoamericana 22 (2006), 545-558], namely to provide a geometric interpretation of exponential families as end points of geodesics of a non-metric connection in a function space. For that we characterize the space of probability densities as a projective space in the class of strictly positive functions, and these will be regarded as a...