Strong Large Deviation Research Articles

Large deviations of multichordal $\operatorname{SLE}_{0+}$, real rational functions, and zeta-regularized determinants of Laplacians

We prove a strong large deviation principle (LDP) for multiple chordal \operatorname{SLE}_{0+} curves with respect to the Hausdorff metric. In the single-chord case, this result strengthens an earlier partial result by the second author. We also introduce a Loewner potential, which in the smooth case has a simple expression in terms of zeta-regularized determinants of Laplacians. This potential differs from the LDP rate function by an additive constant depending only on the boundary data, which satisfies PDEs arising as a semiclassical limit of the Belavin–Polyakov–Zamolodchikov equations of level 2 in conformal field theory with central charge c \to -\infty . Furthermore, we show that every multichord minimizing the potential in the upper half-plane for given boundary data is the real locus of a rational function and is unique, thus coinciding with the \kappa \to 0+ limit of the multiple \operatorname{SLE}_\kappa . As a by-product, we provide an analytic proof of the Shapiro conjecture in real enumerative geometry, first proved by Eremenko and Gabrielov: if all critical points of a rational function are real, then the function is real up to post-composition with a Möbius transformation.

Third-Order Asymptotics of Variable-Length Compression Allowing Errors

This study investigates the fundamental limits of variable-length compression in which prefix-free constraints are not imposed (i.e., one-to-one codes are studied) and non-vanishing error probabilities are permitted. Due in part to a crucial relation between the variable-length and fixed-length compression problems, our analysis requires a careful and refined analysis of the fundamental limits of fixed-length compression in the setting where the error probabilities are allowed to approach either zero or one polynomially in the blocklength. To obtain the refinements, we employ tools from moderate deviations and strong large deviations. Finally, we provide the third-order asymptotics for the problem of variable-length compression with non-vanishing error probabilities. We show that unlike several other information-theoretic problems in which the third-order asymptotics are known, for the problem of interest here, the third-order term depends on the permissible error probability.

Exact Asymptotics for Learning Tree-Structured Graphical Models With Side Information: Noiseless and Noisy Samples

Given side information that an Ising tree-structured graphical model is homogeneous and has no external field, we derive the exact asymptotics of learning its structure from independently drawn samples. Our results, which leverage the use of probabilistic tools from the theory of strong large deviations, refine the large deviation (error exponents) results of Tan et al. (2011) and strictly improve those of Bresler and Karzand (2020). In addition, we extend our results to the scenario in which the samples are observed in random noise. In this case, we show that they strictly improve on the recent results of Nikolakakis et al. (2019). Our theoretical results demonstrate keen agreement with experimental results for sample sizes as small as that in the hundreds.

Higher order asymptotics for large deviations – Part I

For sequences of non-lattice weakly dependent random variables, we obtain asymptotic expansions for Large Deviation Principles. These expansions, commonly referred to as strong large deviation results, are in the spirit of Edgeworth Expansions for the Central Limit Theorem. We show that the results are applicable to Diophantine iid sequences, finite state Markov chains, strongly ergodic Markov chains and Birkhoff sums of smooth expanding maps & subshifts of finite type.

Moderate Deviation Analysis for Classical-Quantum Channels and Quantum Hypothesis Testing

In this work, we study the tradeoffs between the error probabilities of classical-quantum channels and the blocklength $n$ when the transmission rates approach the channel capacity at a rate slower than $1/\sqrt{n}$, a research topic known as moderate deviation analysis. We show that the optimal error probability vanishes under this rate convergence. Our main technical contributions are a tight quantum sphere-packing bound, obtained via Chaganty and Sethuraman's concentration inequality in strong large deviation theory, and asymptotic expansions of error-exponent functions. Moderate deviation analysis for quantum hypothesis testing is also established. The converse directly follows from our channel coding result, while the achievability relies on a martingale inequality.

Multidimensional strong large deviation results

We establish strong large deviation results for an arbitrary sequence of random vectors under some assumptions on the normalized cumulant generating function. In other words, we give asymptotic approximations for a multivariate tail probability of the same kind as the one obtained by Bahadur and Rao (Ann Math Stat 31:1015–1027, 1960) for the sample mean (in the one-dimensional case). The proof of our results follows the same lines as in Chaganty and Sethuraman (J Stat Plan Inference, 55:265–280, 1996). We also present three statistical applications to illustrate our results, the first one dealing with a vector of independent sample variances, the second one with a Gaussian multiple linear regression model and the third one with the multivariate Nadaraya–Watson estimator. Some numerical results are also presented for the first two applications.

Finite-size effect on optimal efficiency of heat engines

The optimal efficiency of quantum (or classical) heat engines whose heat baths are n-particle systems is given by the strong large deviation. We give the optimal work extraction process as a concrete energy-preserving unitary time evolution among the heat baths and the work storage. We show that our optimal work extraction turns the disordered energy of the heat baths to the ordered energy of the work storage, by evaluating the ratio of the entropy difference to the energy difference in the heat baths and the work storage, respectively. By comparing the statistical mechanical optimal efficiency with the macroscopic thermodynamic bound, we evaluate the accuracy of the macroscopic thermodynamics with finite-size heat baths from the statistical mechanical viewpoint. We also evaluate the quantum coherence effect on the optimal efficiency of the cycle processes without restricting their cycle time by comparing the classical and quantum optimal efficiencies.

EXACT ASYMPTOTICS OF SAMPLE-MEAN-RELATED RARE-EVENT PROBABILITIES

Relying only on the classical Bahadur–Rao approximation for large deviations (LDs) of univariate sample means, we derive strong LD approximations for probabilities involving two sets of sample means. The main result concerns the exact asymptotics (asn→∞) of$$ {\open P}\left({\max_{i\in\{1,\ldots,d_x\}}\bar X_{i,n} \les \min_{i\in\{1,\ldots,d_y\}}\bar Y_{i,n}}\right),$$with the${\bar X}_{i,n}{\rm s}$(${\bar Y}_{i,n}{\rm s}$, respectively) denotingdx(dy) independent copies of sample means associated with the random variableX(Y). Assuming${\open E}X \gt {\open E}Y$, this is a rare event probability that vanishes essentially exponentially, but with an additional polynomial term. We also point out how the probability of interest can be estimated using importance sampling in a logarithmically efficient way. To demonstrate the usefulness of the result, we show how it can be applied to compare the order statistics of the sample means of the two populations. This has various applications, for instance in queuing or packing problems.

On strong large deviation results for lightly trimmed sums and some applications

We obtain here the large deviation results for trimmed sums ((r)Sn) of i.i.d. random variables (Xn), with the distribution function belonging to the domain of attraction of a positive stable law. As an application, we establish a law of the iterated logarithm.

A strong large deviation theorem

We prove a strong large deviation theorem for an arbitrary sequence of random variables, that is, we establish a full asymptotic expansion of large deviation type for the tail probabilities. An Edgeworth expansion is required to derive the result. We illustrate our theorem with two statistical applications: the sample variance and the kernel density estimator.

The Solution Classical Feedback Optimal Control Problem for m-Persons Differential Game with Imperfect Information

A new approach, which is proposed in this paper allows one to construct the Bellman function V(t,x) and optimal control u(t) directly,i.e.,without any reference to the Bellman equation, by way of using strong large deviations principle for the solutions Colombeau-Ito's SDE.

Strong large deviations for arbitrary sequences of random variables

We establish strong large deviation results for an arbitrary sequence of random variables under some assumptions on the normalized cumulant generating function. In other words, we give asymptotic expansions for the tail probabilities of the same kind as those obtained by Bahadur and Rao (Ann. Math. Stat. 31:1015–1027, 1960) for the sample mean. We consider both the case where the random variables are absolutely continuous and the case where they are lattice-valued. Our proofs make use of arguments of Chaganty and Sethuraman (Ann. Probab. 21:1671–1690, 1993) who also obtained strong large deviation results and local limit theorems. We illustrate our results with the kernel density estimator, the sample variance, the Wilcoxon signed-rank statistic and the Kendall tau statistic.

Large deviation properties for empirical quantile-type production functions

In standard microeconomic theory of the firm, the production frontier is used to describe the geometric locus of the optimal production. Most limitations of the conventional estimators of this efficient frontier stem from its reliance on estimation of the upper surface of the joint support of (X, Y), where X∈ℝ p + represents the inputs-usage and Y∈ℝ+ is the produced output. Instead, a recent approach involving the estimation of a partial quantile-type frontier of the joint support, lying close to its full boundary, has emerged in the literature as an attractive alternative. In this paper, we present a new and simple proof of the asymptotic normality of the resulting nonparametric frontier estimators and provide an estimate for the accuracy of the normal approximation. We also propose a Kiefer-type asymptotic representation for these estimators. The main results concern the study of some weak, moderate and strong large deviation properties of the quantile-based frontier estimators.

Large deviations for sums of partly dependent random variables

We use and extend a method by Hoeffding to obtain strong large deviation bounds for sums of dependent random variables with suitable dependency structure. The method is based on breaking up the sum...

Large deviations for sums of partly dependent random variables

AbstractWe use and extend a method by Hoeffding to obtain strong large deviation bounds for sums of dependent random variables with suitable dependency structure. The method is based on breaking up the sum into sums of independent variables. Applications are given to U‐statistics, random strings and random graphs. © 2004 Wiley Periodicals, Inc. Random Struct. Alg., 2004

On the large deviation principle for the almost sure CLT

Let S k be the kth partial sum of real-valued i.i.d. random variables X 1, X 2,… . Define the random “empirical” measures with logarithmic weights Q n= 1 log n ∑ k=1 n 1 k δ S k/ k . If E|X 1| m<∞ for all m>0, then Q n satisfies strong large deviations principle, as Heck (Stochastic. Process. Appl. 76 (1998) 61), March and Seppäläinen (J. Theoret. Probab. 10 (1997) 935) have recently proved. We show that the moment assumptions are optimal in this statement.

Large deviations in the van der Waals limit

In this paper we extend the analysis in Benois et al. (Markov Process. Rel. Fields (1997) 175–198) by proving a strong large deviation principle for the empirical distribution of Ising spins in d⩾2 dimensions when the interaction is determined by a Kac potential and the temperature is below the critical value.

Multidimensional strong large deviation theorems

In this paper we obtain some local limit theorems for arbitrary sequences of random vectors. The local limit theorems give conditions on the characteristic functions of random vectors for their pseudo-density function to converge uniformly on bounded sets. We then use these theorems to obtain strong large deviation results for an arbitrary sequence of random vectors. Thus our paper establishes the connection between the local limit theorems and the strong limit theorems. We apply our results to the multivariate F-distribution.

Strong Large Deviation and Local Limit Theorems

Most large deviation results give asymptotic expressions for $\log P(Y_n \geq y_n)$, where the event $\{Y_n \geq y_n\}$ is a large deviation event, that is, $P(Y_n \geq y_n)$ goes to 0 exponentially fast. We refer to such results as weak large deviation results. In this paper we obtain strong large deviation results for arbitrary random variables $\{Y_n\}$, that is, we obtain asymptotic expressions for $P(Y_n \geq y_n)$, where $\{Y_n \geq y_n\}$ is a large deviation event. These strong large deviation results are obtained for lattice valued and nonlattice valued random variables and require some conditions on their moment generating functions. These results strengthen existing results which apply mainly to sums of independent and identically distributed random variables. Since $Y_n$ may not possess a probability density function, we consider the function $q_n(y; b_n,S) = \lbrack(b_n/\mu(S))P(b_n(Y_n - y) \in S)\rbrack$, where $b_n \rightarrow \infty, \mu$ is the Lebesgue measure on $R$, and $S$ is a measurable subset of $R$ such that $0 < \mu(S) < \infty$. The function $q_n(y; b_n,S)$ is the p.d.f. of $Y_n + Z_n$, where $Z_n$ is uniform on $-S/b_n$, and will be called the pseudodensity function of $Y_n$. By a local limit theorem we mean the convergence of $q_n(y_n; b_n,S)$ as $n \rightarrow \infty$ and $y_n \rightarrow y^\ast$. In this paper we obtain local limit theorems for arbitrary random variables based on easily verifiable conditions on their characteristic functions. These local limit theorems play a major role in the proofs of the strong large deviation results of this paper. We illustrate these results with two typical applications.

The equivalence of the logarithmic Sobolev inequality and the Dobrushin-Shlosman mixing condition

Given a finite range lattice gas with a compact, continuous spin space, it is shown (cf. Theorem 1.2) that a uniform logarithmic Sobolev inequality (cf. 1.4) holds if and only if the Dobrushin-Shlosman mixing condition (cf. 1.5) holds. As a consequence of our considerations, we also show (cf. Theorems 3.2 and 3.6) that these conditions are equivalent to a statement about the uniform rate at which the associated Glauber dynamics tends to equilibrium. In this same direction, we show (cf. Theorem 3.19) that these ideas lead to a surprisingly strong large deviation principle for the occupation time distribution of the Glauber dynamics.