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For sets Q and Y, the generalized Fréchet mean m ∈ Q of a random variable Y , which has values in Y, is any minimizerThere are little restrictions to Q and Y. In particular, Q can be a non-Euclidean metric space. We provide convergence rates for the empirical generalized Fréchet mean. Conditions for rates in probability and rates in expectation are given. In contrast to previous results on Fréchet means, we do not require a finite diameter of the Q or Y. Instead, we assume an inequality, which we call quadruple inequality. It generalizes an otherwise common Lipschitz condition on the cost function. This quadruple inequality is known to hold in Hadamard spaces. We show that it also holds in a suitable way for certain powers of a Hadamard-metric.

The nature of dielectric echoes in amorphous solids at low temperatures is investigated. It is shown that at long delay times the echo amplitude is determined by a small subset of two level systems (TLS) having negligible relaxation and decoherence because of their weak coupling to phonons. The echo decay can then be described approximately by power law time dependencies with different powers at times longer and shorter than the typical TLS relaxation time. The theory is applied to recent measurements of two and three pulse dipolar echo in borosilicate glass BK7 and provides a perfect data fit in the broad time and temperature ranges under the assumption that there exist two TLS relaxation mechanisms due to TLS-phonons and TLS-TLS interaction. This interpretation is consistent with the previous experimental and theoretical investigations. Further experiments verifying the theory predictions are suggested.

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