Large Deviation Theorem Research Articles

On the Integrated Density of States for Schrödinger Operators on ℤ2 with Quasi Periodic Potential

In this paper we consider discrete Schrodinger operators on the lattice ℤ2 with quasi periodic potential. We establish new regularity results for the integrated density of states, as well as a quantitative version of a “Thouless formula”, as previously considered by Craig and Simon, for real energies and with rates of convergence. The main ingredient is a large deviation theorem for the Green's function that was recently established by Bourgain, Goldstein, and the author. For the integrated density of states an argument of Bourgain is used. Finally, we establish certain fine properties of separately subharmonic functions of two variables that might be of independent interest.

Holder Continuity of the Integrated Density of States for Quasi-Periodic Schrodinger Equations and Averages of Shifts of Subharmonic Functions

In this paper we consider various regularity results for discrete quasiperiodic Schr6dinger equations --n+l - Pn-1 + V(9 + nw)on = EOn with analytic potential V. We prove that on intervals of positivity for the Lyapunov exponent the integrated density of states is Holder continuous in the energy provided w has a typical continued fraction expansion. The proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the avalanche-principle. The latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks. In the multi-frequency case the integrated density of states is shown to have a modulus of continuity of the form exp(- log tl) for some 0 < a < 1, but currently we do not obtain Holder continuity in the case of more than one frequency. We also present a mechanism for proving the positivity of the Lyapunov exponent for large disorders for a general class of equations. The only requirement for this approach is some weak form of a large deviation theorem for the Lyapunov exponents. In particular, we obtain an independent proof of the Herman-Sorets-Spencer theorem in the multi-frequency case. The approach in this paper is related to the recent nonperturbative proof of Anderson localization in the quasi-periodic case by J. Bourgain and M. Goldstein.

Refined exponential rates in Vapnik–Chervonenkis inequalities

Vapnik–Chervonenkis bounds on speeds of convergence of empirical means to their expectations have been continuously improved over the years. The result obtained by M. Talagrand in 1994 [11] seems to provide the final word as far as universal bounds are concerned. However, for fixed families of underlying probability distributions, the exponential rate in the deviation term can be fairly improved by the more adequate Cramér transform, as shown by theorems of large deviations.

FRACTAL DIMENSIONS AND MARTIN BOUNDARY OF GRAPHS

This pap r shows large deviation theorem for Markov chains with spatial symmetric Gr n ’s function.Spatial symm try and behaviour of the Martin boundary is inv stigat d.

Large deviation results for a U-statistical sum with product kernel

Large deviation theorems are proved for non-degenerate U-statistical sums of degree m with kernel h (x1, …, xm) = x1 … xm under the Cramér condition and under the Linnik condition. The method of proof uses truncation and the contraction technique.

An asymptotic formula for the Bayes risk in discriminating between two Markov chains

The problem of discriminating between two Markov chains is considered. It is assumed that the common state space of the chains is finite and all the finite dimensional distributions are mutually absolutely continuous. The Bayes risk is expressed through large deviation probabilities for sums of random variables defined on an auxiliary Markov chain. The proofs are based on a large deviation theorem recently established by Z. Szewczak.

An asymptotic formula for the Bayes risk in discriminating between two Markov chains

The problem of discriminating between two Markov chains is considered. It is assumed that the common state space of the chains is finite and all the finite dimensional distributions are mutually absolutely continuous. The Bayes risk is expressed through large deviation probabilities for sums of random variables defined on an auxiliary Markov chain. The proofs are based on a large deviation theorem recently established by Z. Szewczak.

On the Large-Deviation Asymptotics in an Allocation Scheme of Particles into Distinguishable Cells with Restrictions on the Size of the Cells

Equiprobable allocation schemes of allocation of n distinguishable or indistinguishable particles into N distinguishable cells are considered under the condition that the number of particles contained in any cell does not exceed a constant $p\in$ {\bf N}. Local and integral large-deviation theorems are obtained which estimate the tails of the distributions of the random variable equal to the number of empty cells. The asymptotic behavior of the expectation and variance of the random variable are investigated and a local normal limit theorem is proved for the probabilities of this random variable in the central domain of changing the parameters $n,N$, when $n,N\to\infty$ in such a way that $0 < \alpha_1\le\alpha=n/N\le\alpha_2 < p$ $(\alpha_1,\alpha_2$ are constants).

A large deviation theorem for U-processes

This paper develops a large deviation theorem for families of sample means of U-statistic structure (i.e., U-processes). These results extend the work of Sethuraman (1964) and Wu (1994) on large deviation theory for families of ordinary sample means and the classical empirical process. Along the way we obtain an extension to U-statistics of an important isoperimetric inequality of Talagrand (1994) for ordinary means. Applications include the simplicial depth function of Liu (1990) and sup-norm statistics (e.g., Kolmogorov–Smirnov type goodness-of-fit statistics) defined over U-processes.

Large-deviation theorems in hypothesis-testing problems

Large-deviation theorems of Chernoff type for the logarithm of the likelihood ratio in general binary statistical experiments are obtained. The obtained limit theorems are used to investigate of the rate of decrease of the error probabilities for Neyman-Pearson tests, Bayes tests, and minimax tests in statistical experiments generated by normal autoregressive processes and Galton-Watson processes with immigration.

On large deviation theorem for data-driven Neyman's statistic

The aim of the paper is to show that for data-driven Neyman's statistic large deviation theorem does not hold. We derive an explicit estimate from below for probabilities of large and moderate deviations. The main tool is a version of a lower exponential inequality recently obtained by Mogulskii.

Limit theorems for Mandelbrot's multiplicative cascades

Let $W \geq 0$ be a random variable with $EW = 1$, and let $Z^{(r)} (r \geq 2)$ be the limit of a Mandelbrot’s martingale, defined as sums of product of independent random weights having the same distribution as $W$, indexed by nodes of a homogeneous $r$-ary tree. We study asymptotic properties of $Z^{(r)}$ as $r \to \infty$: we obtain a law of large numbers, a central limit theorem, a result for convergence of moment generating functions and a theorem of large deviations. Some results are extended to the case where the number of branches is a random variable whose distribution depends on a parameter $r$.

A large deviation theorem for the empirical eigenvalue distribution of random unitary matrices

It is shown that the empirical eigenvalue distribution of suitably distributed random unitary matrices satisfies the large deviation principle as the matrix size goes to infinity. The primary term of the rate function is the logarithmic energy (or the minus sign of Voiculescu's free entropy). Examples of random unitaries are also discussed, one of them is related to the work of Gross and Witten in quantum physics.

Large Deviation Theorems for Gaussian Processes and Their Applications in Information Theory

We discuss on the large deviation theorems for stationary Gaussian processes and their applications in information theory. The topics investigated here include error probability of string matching, error probabilities for random codings, and a conditional limit theorem which justifies the maximum entropy principle.

Rigorous solution of a mean field spin glass model

A separable spin glass model whose exchange integral takes the form Jij = J(ξi1ξj2 + ξi2ξj1) which was solved by van Hemmen et al. [12] using large deviation theory [14] is rigorously treated. The almost sure convergence criteria associated with the cumulant generating function C(t) with respect to the quenched random variables ξ is carefully investigated, and it is proved that the related excluded null set 𝒩 is independent of t. The free energy and hence the other thermodynamic quantities are rederived using Varadhan′s Large Deviation Theorem. A simulation is also presented for the entropy when ξ assumes a Gaussian distribution.

Large deviation theorems for extended random variables and some applications

Large deviation theorems of Chernoff type for extended random variables are proved. The large deviation principle for extended random variables is obtained, too. The obtained limit theorems are used to prove the large deviation theorems of Chernoff type for the logarithm of the likelihood ratio in general binary statistical experiments. The rate of decrease of the error probabilities is investigated for Neyman-Pearson tests, Bayes tests, and minimax tests.

Theorems of Large Deviations in the Approximation by an Infinitely Divisible Law

The general lemma of large deviations has been proved in the approximation by an in finitely divisible law with a finite spectrum. The method of cumulants is used. First, vast possibilities of this nethod are surveyed when investigating large deviations in various approximations.

On the Cramér Series Coefficients

Previous article Next article On the Cramér Series CoefficientsL. V. RozovskyL. V. Rozovskyhttps://doi.org/10.1137/S0040585X97976763PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThis paper presents the formula for the Cramér series coefficients, which arise in large deviation theorems. As a consequence, some estimates for these coefficients are obtained.[1] H. Cramér, Sur un nouveau théorème limite de la théorie des probabilités, Actualités Sci. Indust., 736 (1938), Hermann, Paris. Google Scholar[2] Google Scholar[3] Google Scholar[4] W. Feller, Generalization of a probability limit theorem of Cramér, Trans. Amer. Math. Soc., 54 (1943), pp. 361–372. trq TAMTAM 0002-9947 Trans. Am. Math. Soc. Google Scholar[5] C. Lenart, On certain theorems of Berry and a limit theorem of Feller, Mat. Časopis Sloven. Akad. Vied., 18 (1968), pp. 59–75. Google Scholar[6] Google Scholar[7] Google ScholarKeywordsthe Cramér seriescumulantslarge deviations Previous article Next article FiguresRelatedReferencesCited ByDetails On Asymptotic Expansions in the “Interval” CLT for Sums of Independent Random VectorsJournal of Mathematical Sciences, Vol. 244, No. 5 | 8 January 2020 Cross Ref Toward the History of the Saint Petersburg School of Probability and Statistics. I. Limit Theorems for Sums of Independent Random VariablesVestnik St. Petersburg University, Mathematics, Vol. 51, No. 2 | 16 June 2018 Cross Ref ReferencesExtreme Value Methods with Applications to Finance | 7 February 2012 Cross Ref Uniform approximations for the M/G/1 queue with subexponential processing timesQueueing Systems, Vol. 68, No. 1 | 11 May 2011 Cross Ref An Estimate from Below of the Remainder in the Central Limit Theorem for a Sum of Independent Random Variables with Finite Moments of a High OrderL. V. RozovskyTheory of Probability & Its Applications, Vol. 47, No. 1 | 25 July 2006AbstractPDF (148 KB)On a Feller TheoremL. V. RozovskyTheory of Probability & Its Applications, Vol. 44, No. 1 | 25 July 2006AbstractPDF (114 KB) Volume 43, Issue 1| 1999Theory of Probability & Its Applications1-169 History Published online:25 July 2006 InformationCopyright © 1999 Society for Industrial and Applied MathematicsKeywordsthe Cramér seriescumulantslarge deviationsPDF Download Article & Publication DataArticle DOI:10.1137/S0040585X97976763Article page range:pp. 152-157ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Theorems of large deviations for sums of random variables connected in a Markov chain. II

Theorems of large deviations for sums of random variables connected in a Markov chain. II

A Portfolio Performance Index and Its Implications

Cramer's Large Deviation Theorem is used to formalize a modern time series variant of the Safety-First, loss aversion criterion, providing a behavioral foundation for a new portfolio performance index. When returns are normally distributed, the performance index is proportional to the squared Sharpe Index, so the mean-variance efficient tangency portfolio is optimal, and the CAPM results when all investors behave this way. When returns are non-normally distributed, the index generalizes to a different, preference parameter-free formula, which is easily estimable without prior knowledge of the distribution. The tangency portfolio is no longer optimal; what results instead is a non-CAPM relationship between an asset's expected excess return and the return's covariance with an exponential function of the performance index maximizing portfolio. The endogenous loss aversion induces a preference not only for higher mean and lower standard deviation, like the Sharpe Index, but also for higher skewness and lower kurtosis in non-normal cases, with the potential to explain the demand for portfolio insurance and/or other skewness enhancing strategies.