Optimal Summation and Integration by Deterministic, Randomized, and Quantum Algorithms

We survey old and new results about optimal algorithms for summation of finite sequences and for integration of functions from Hoelder or Sobolev spaces. First we discuss optimal deterministic and randomized algorithms. Then we add a new aspect, which has not been covered before on conferences about (quasi-) Monte Carlo methods: quantum computation. We give a short introduction into this setting and present recent results of the authors on optimal quantum algorithms for summation and integration. We discuss comparisons between the three settings. The most interesting case for Monte Carlo and quantum integration is that of moderate smoothness k and large dimension d which, in fact, occurs in a number of important applied problems. In that case the deterministic exponent is negligible, so the n^{-1/2} Monte Carlo and the n^{-1} quantum speedup essentially constitute the entire convergence rate. We observe that -- there is an exponential speed-up of quantum algorithms over deterministic (classical) algorithms, if k/d tends to zero; -- there is a (roughly) quadratic speed-up of quantum algorithms over randomized classical algorithms, if k/d is small.Comment: 13 pages, contribution to the 4th International Conference on Monte Carlo and Quasi-Monte Carlo Methods, Hong Kong 200

Pair Correlation Function of Wilson Loops

We give a path integral prescription for the pair correlation function of Wilson loops lying in the worldvolume of Dbranes in the bosonic open and closed string theory. The results can be applied both in ordinary flat spacetime in the critical dimension d or in the presence of a generic background for the Liouville field. We compute the potential between heavy nonrelativistic sources in an abelian gauge theory in relative collinear motion with velocity v = tanh(u), probing length scales down to r_min^2 = 2 \pi \alpha' u. We predict a universal -(d-2)/r static interaction at short distances. We show that the velocity dependent corrections to the short distance potential in the bosonic string take the form of an infinite power series in the dimensionless variables z = r_min^2/r^2, uz/\pi, and u^2.Comment: 16 pages, 1 figure, Revtex. Corrected factor of two in potential. Some changes in discussio

On the accuracy of poisson approximation

The problem of evaluating the accuracy of Poisson approximation to the distribution of a sum of independent integer-valued random variables has attracted a lot of attention in the past six decades. From a practical point of view, it has important applications in insurance, reliability theory, extreme value theory, etc.; from a theoretical point of view, the topic provides insights into Kolmogorov's problem. The task of establishing an estimate with the best possible constant at the leading term remained open for decades. The paper presents a solution to that problem. A first-order asymptotic expansion is established as well. We generalise and sharpen the corresponding inequalities of Prokhorov, LeCam, Barbour, Hall, Deheuvels, Pfeifer, and Roos. A new result is established for the intensively studied topic of Poisson approximation to the binomial distribution

Measures of financial risk

The paper compares a number of available measures of financial risk and presents arguments in favor of a dynamic measure of risk. We argue that traditional measures are static, while the dynamic measure of risk lacks statistical scrutiny. The main obstacle to building a body of empirical evidence in support of the dynamic risk measure is computational difficulty of identifying local extrema as price charts appear objects of fractal geometry. We overview approaches to financial risk measurement and formulate a number of open questions. The arguments are illustrated on examples of real data

Lower bounds to the accuracy of inference on heavy tails

The paper suggests a simple method of deriving minimax lower bounds to the accuracy of statistical inference on heavy tails. A well-known result by Hall and Welsh (Ann. Statist. 12 (1984) 1079-1084) states that if α^n is an estimator of the tail index αP and {zn} is a sequence of positive numbers such that supP∈DrP(|α^n−αP|≥zn)→0, where Dr is a certain class of heavy-tailed distributions, then zn≫n−r. The paper presents a non-asymptotic lower bound to the probabilities P(|α^n−αP|≥zn). We also establish non-uniform lower bounds to the accuracy of tail constant and extreme quantiles estimation. The results reveal that normalising sequences of robust estimators should depend in a specific way on the tail index and the tail constant

On the accuracy of inference on heavy-tailed distributions

This paper suggests a simple method of deriving nonparametric lower bounds of the accuracy of statistical inference on heavy-tailed distributions. We present lower bounds of the mean squared error of the tail index, the tail constant, and extreme quantiles estimators. The results show that the normalizing sequences of robust estimators must depend in a specific way on the tail index and the tail constant

Poisson approximation. Addendum

An important feature of a Poisson limit theorem in [4] is the absence of the traditional assumption (A.2). The purpose of this addendum is to explain why assumption (A.2) is not required, and compare the assumptions of the Poisson limit theorem in [4] with traditional ones. We present details of the argument behind Theorem 3 in [4]

On the T-test

The aim of this article is to show that the T-test can be misleading. We argue that normal or Student's approximation to the distribution L(tn) of Student's statistic tn does not hold uniformly over the class Pn of samples fX1; :::;Xng from zero-mean unit-variance bounded distributions. We present lower bounds to the corresponding error. We suggest a generalisation of the T-test that allows for variability of possible approximating distributions to L(tn)

Non-parametric lower bounds and information functions

We argue that common features of non-parametric estimation appear in parametric cases as well if there is a deviation from the classical regularity condition. Namely, in many non-parametric estimation problems (as well as some parametric cases) unbiased finite-variance estimators do not exist; neither estimator converges locally uniformly with the optimal rate; there are no asymptotically unbiased with the optimal rate estimators; etc.. We argue that these features naturally arise in particular parametric subfamilies of non-parametric classes of distributions. We generalize the notion of regularity of a family of distributions and present a general regularity condition, which leads to the notions of the information index and the information function. We argue that the typical structure of a continuity modulus explains why unbiased finite-variance estimators cannot exist if the information index is larger than two, while in typical non-parametric situations neither estimator converges locally uniformly with the optimal rate. We present a new result on impossibility of locally uniform convergence with the optimal rate