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"We examine the asymptotic distribution of estimated coefficients and endogenous variables in a dynamic self-referential model when agents learn adaptively using a constant gain stochastic gradient algorithm. The model environment can represent a number of economic models, including asset pricing models, that have been studied recently in the adaptive learning framework. The asymptotic distributions of forecasts and endogenous variables are characterized using techniques from linear recursions with multiplicative noise and large deviations, and are shown to exhibit fat tails"--National Bureau of Economic Research web site

AbstractAsymptotic representations are found for the large deviation probabilities that the nα‐th order statistic exceeds δ, where δ>α. The probabilities are first expressed in terms of the empirical distribution function, and then the 1960 theorem of Bahadur and Ranga Rao is applied. The result is then shown to be more precise than a logarithmic statement in a 1969 paper of Sievers dealing with the asymptotic relative efficiency of the sample median test.

Topics covered in this volume (large deviations, differential geometry, asymptotic expansions, central limit theorems) give a full picture of the current advances in the application of asymptotic methods in mathematical finance, and thereby provide rigorous solutions to important mathematical and financial issues, such as implied volatility asymptotics, local volatility extrapolation, systemic risk and volatility estimation. This volume gathers together ground-breaking results in this field by some of its leading experts. Over the past decade, asymptotic methods have played an increasingly important role in the study of the behaviour of (financial) models. These methods provide a useful alternative to numerical methods in settings where the latter may lose accuracy (in extremes such as small and large strikes, and small maturities), and lead to a clearer understanding of the behaviour of models, and of the influence of parameters on this behaviour. Graduate students, researchers and practitioners will find this book very useful, and the diversity of topics will appeal to people from mathematical finance, probability theory and differential geometry.

The relative performance of two statistical tests of a hypothesis for large sample sizes is often investigated by means of asymptotic relative efficiencies in the sense of Pitman or Bahadur. Pitman efficiencies are directed at local alternatives, Bahadur efficiencies at fixed (non‐local) alternatives. The present paper reviews some methods and results on Bahadur efficiency with special attention to probabilities of large deviations, which play a key role in the computation of Bahadur efficiencies.

AbstractAn asymptotic representation for large deviation probabilities of the Winsorized mean of a sequence of independent, identically distributed exponential random variables is derived. The Winsorized mean, a linear combination of exponential order statistics, is first transformed into a weighted sum of exponential random variables, and then a large deviation theorem for weighted sums can be applied. The representation obtained is then compared with results already known for the mean and the median, the two extreme cases of the Winsorized mean.

Let (Xn, n ≥ 1) be an i.i.d. sequence of positive random variables with distribution function H. Let φH:={(n, Xn), n ≥ 1) be the associated observation process. We view φh as a measure on E:= [0, ∞) ∞ (0, φ] where φH (A) is the number of points of φH which lie in A. A family (Vs, s> 0) of transformations is defined on E in such a way that for suitable H the distributions of (VsφH, S > 0) satisfy a large deviation principle and that a related Strassen‐type law of the iterated logarithm also holds. Some consequent large deviation principles and loglog laws are derived for extreme values. Similar results are proved for φH replaced by certain planar Poisson processes.

AbstractLetXmandYndenote the medians of independent samples from continuous populations Fand Grespectively. In Theorem 1 of this note, a large deviation result for the difference dmn= Xm‐ Ynis established under the assumption that Fand Garc symmetric about 0. The result is then used to define a sequential rule based on medians for selecting the best of Ksymmetric populations.

This paper discusses the large-deviation principle of discriminant statistics for Gaussian locally stationary processes. First, large-deviation theorems for quadratic forms and the log-likelihood ratio for a Gaussian locally stationary process with a mean function are proved. Their asymptotics are described by the large deviation rate functions. Second, we consider the situations where processes are misspecified to be stationary. In these misspecified cases, we formally make the log-likelihood ratio discriminant statistics and derive the large deviation theorems of them. Since they are complicated, they are evaluated and illustrated by numerical examples. We realize the misspecification of the process to be stationary seriously affecting our discrimination.

We contribute an extension of large-deviation results obtained in [N.J.B. Aza, J.-B. Bru, W. de Siqueira Pedra, A. Ratsimanetrimanana, J. Math. Pures Appl. 125 (2019) 209] on conductivity theory at atomic scale of free lattice fermions in disordered media. Disorder is modeled by (i) a random external potential, like in the celebrated Anderson model, and (ii) a nearest-neighbor hopping term with random complex-valued amplitudes. In accordance with experimental observations, via the large deviation formalism, our previous paper showed in this case that quantum uncertainty of microscopic electric current densities around their (classical) macroscopic value is suppressed, exponentially fast with respect to the volume of the region of the lattice where an external electric field is applied. Here, the quantum fluctuations of linear response currents are shown to exist in the thermodynamic limit and we mathematically prove that they are related to the rate function of the large deviation principle associated with current densities. We also demonstrate that, in general, they do not vanish (in the thermodynamic limit) and the quantum uncertainty around the macroscopic current density disappears exponentially fast with an exponential rate proportional to the squared deviation of the current from its macroscopic value and the inverse current fluctuation, with respect to growing space (volume) scales. ; FAPESP (2017/22340-9); CNPq (309723/2020-5); by the Basque Government through the grant IT641-13; MTM2017-82160-C2-2-P