Seismic coda waves are a natural wonder. Because they are formed by scattered waves from numerous heterogeneities in the lithosphere, nature does the averaging over a large volume of the earth and leads to beautiful simplicity such as the separability of seismic source, propagation path and recording site effects. In this review, we shall focus on the decay rate of coda amplitudes, called coda Q, and present its significance as a geophysical parameter characterizing regional structures and earthquake processes in the lithosphere.

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The IMA volumes in mathematics and its applications

We give conditions for an O(1/n) rate of convergence of Fisher information and relative entropy in the Central Limit Theorem. We use the theory of projections in L
 2 spaces and Poincare inequalities, to provide a better understanding of the decrease in Fisher information implied by results of Barron and Brown. We show that if the standardized Fisher information ever becomes finite then it converges to zero.

Fisher Information inequalities and the Central Limit Theorem

Characterizations of distributions by variance bounds

The quality of information in financial asset markets is often hard to estimate. Reminiscent of the famous Ellsberg paradox, investors may be unable to form a single probability belief about asset returns conditional on information signals and may act on the basis of ambiguous (or multiple) probability beliefs. This paper analyzes information transmission in asset markets when agents’ information is ambiguous. We consider a market with risk-averse informed investors, risk-neutral competitive arbitrageurs, and noisy supply of the risky asset, first studied by Vives (Rev Financ Stud 8:3–40, 1995a, J Econ Theory 67:178–204, 1995b) with unambiguous information. Ambiguous information gives rise to the possibility of illiquid market where arbitrageurs choose not to trade in a rational expectations equilibrium. When market is illiquid, small informational or supply shocks have relatively large effects on asset prices.

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Liquidity and asset prices in rational expectations equilibrium with ambiguous information

An inequality due to Chernoff is generalized and a related Cramer-Rao type of inequality is studied.

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On an Inequality of Chernoff

Chernoff (1981) obtained an upper bound for the variance of a function of a standard normal random variable, using Hermite polynomials. Chen (1980) gave a different proof, using the Cauchy-Schwarz inequality, and extended the inequality to the case of a multivariate normal. Here it is shown how similar upper bounds can be obtained for other distributions, including discrete ones. Moreover, by using a variation of the Cramer-Rao inequality, analogous lower bounds are given for the variance of a function of a random variable which satisfies the usual regularity conditions. Matrix inequalities are also obtained. All these bounds involve the first two moments of derivatives or differences of the function.

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Theophilos Cacoullos

Papers

On Upper and Lower Bounds for the Variance of a Function of a Random Variable