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Steve Gull, in unpublished work available on his Cambridge University homepage, has outlined a proof of Bell's theorem using Fourier theory. Gull's philosophy is that Bell's theorem (or perhaps a key lemma in its proof) can be seen as a no-go theorem for a project in distributed computing with classical, not quantum, computers. We present his argument, correcting misprints and filling gaps. In his argument, there were two completely separated computers in the network. We need three in order to fill all the gaps in his proof: a third computer supplies a stream of random numbers to the two computers representing the two measurement stations in Bell's work. One could also imagine that computer replaced by a cloned, virtual computer, generating the same pseudo-random numbers within each of Alice and Bob's computers. Either way, we need an assumption of the presence of shared i.i.d. randomness in the form of a synchronised sequence of realisations of i.i.d. hidden variables underlying the otherwise deterministic physics of the sequence of trials. Gull's proof then just needs a third step: rewriting an expectation as the expectation of a conditional expectation given the hidden variables., Comment: Version 8: minor corrections. arXiv admin note: text overlap with arXiv:2001.11338

We study asymptotically optimal statistical inference concerning the unknown state of $N$ identical quantum systems, using two complementary approaches: a "poor man's approach" based on the van Trees inequality, and a rather more sophisticated approach using the recently developed quantum form of LeCam's theory of Local Asymptotic Normality., 34 pages. to appear in `From Probability to Statistics and Back: High-Dimensional Models and Processes'. A Festschrift in Honor of Jon Wellner. Edited by Banerjee, M., Bunea, F., Huang, J., Maathuis, M. and Koltchinskii, V. extension of a previous paper arXiv:math/0512443

We present a novel tight bound on the quantum violations of the CGLMP inequality in the case of infinitely many outcomes. Like in the case of Tsirelson's inequality the proof of our new inequality does not require any assumptions on the dimension of the Hilbert space or kinds of operators involved. However, it is seen that the maximal violation is obtained by the conjectured best measurements and a pure, but not maximally entangled, state. We give an approximate state which, in the limit where the number of outcomes tends to infinity, goes to the optimal state for this setting. This state might be potentially relevant for experimental verifications of Bell inequalities through multi-dimenisonal entangled photon pairs.

I write out and discuss how one might try to prove the continuous time g-computation formula, in the simplest possible case: treatments (labelled a, for actions) and covariates (l: longitudinal data) form together a bivariate counting process.

An experimentally observed violation of Bell s inequality is supposed to show the failure of local realism to deal with quantum reality. However, finite statistics and the time sequential nature of real experiments still allows a loophole for local realism. We show that the randomised design of the Aspect experiment closes this loophole. Our main tool is van de Geer s (2000) supermartingale version of the classical Bernstein (1924) inequality guaranteeing, at the root n scale, a not-heavier-than-Gaussian tail of the distribution of a sum of bounded supermartingale differences. The results are used to specify a protocol for a public bet between the author and L. Accardi, who in recent papers (Accardi and Regoli, 2000a,b, 2001) has claimed to have produced a suite of computer programmes, to be run on a network of computers, which will simulate a violation of Bell s inequalites. At a sample size of thirty thousand, both error probabilities are guaranteed smaller than one in a million, provided we adhere to the sequential randomized design.

This is a brief survey of product-integration for statisticians.

This is a brief survey of product-integration for biostatisticians.

We use a Bayesian version of the Cramer-Rao lower bound due to van Trees to give an elementary proof that the limiting distibution of any regular estimator cannot have a variance less than the classical information bound, under minimal regularity conditions. We also show how minimax convergence rates can be derived in various non- and semi-parametric problems from the van Trees inequality. Finally we develop multivariate versions of the inequality and give applications.

Observations or measurements taken of a quantum system (a small number of fundamental particles) are inherently random. If the state of the system depends on unknown parameters, then the distribution of the outcome depends on these parameters too, and statistical inference problems result. Often one has a choice of what measurement to take, corresponding to dierent experimental set-ups or settings of measurement apparatus. This leads to a design problem|which measurement is best for a given statistical problem. This paper gives an introduction to this eld in the most simple of settings, that of estimating the state of a spin-half particle given n independent copies of the particle. We show how in some cases asymptotically optimal measurements can be constructed. Other cases present interesting open problems, connected to the fact that for some models, quantum Fisher information is in some sense non-additive. In physical terms, we have non-locality without entanglement.

We extend Robins theory of causal inference for complex longitudinal data to the case of continuously varying as opposed to discrete covariates and treatments In particular we establish versions of the key results of the discrete theory the gcomputation formula and a collection of powerful characterizations of the gnull hypothesis of no treatment eect This is accomplished under natural continuity hypotheses concerning the conditional distributions of the outcome variable and of the covariates given the past

Suppose that a random variable X of interest is grouped or censored or missing so that one only observes a coarsening of X ie a random set containing X with probability It is assumed that the coarsening mechanism has the coarsening at random property Suppose furthermore that the coarsening either equals X itself or that is a set with positive X-probability We modify the NPMLE of the distribution of X by demanding that its support is the set of observed data points We provide a general theorem giving sucient conditions for eciency of this NPMLE or eciency of the NPMLE after a small data reduction We apply the theorem to a number of examples

For applications in spatial statistics an important property of a random set X in Rk is its rst contact distribution This is the distribution of the distance from a xed point to the nearest point of X where distance is measured using scalar dilations of a xed test set B We show that if B is convex and contains a neighbourhood of the rst contact distribution function FB is absolutely continuous We give two explicit representations of FB and additional regularity conditions under which FB is continuously dierentiable A KaplanMeier estimator of FB is introduced and its basic properties examined

Intercept and deterministic trend functions are known to have a substantial e ect on cointegration analysis and notably on the asymptotic distributions of various test statistics In this paper we propose a unifying approach to the analysis of cointegrated vector autoregressions by allowing for a wide class of trend functions Next estimates of these trends are incorporated in the asymptotic distributions of the test statistics This approach allows incorporating elaborate drift functions in cointegration analysis while avoiding the issue of the signicance of the trend these functions give rise to Simulation techniques can yield the appropriate critical values

We analyse the thesis of Kummerer and Maassen that classical probability is unable to model the the stochastic nature of the Aspect experiment in which violation of Bells inequality was experimentally demonstrated According to these authors the experiment shows the need to introduce the extension of classical probability known as Quantum Probability We show that their argument depends on hidden assumptions and a highly restrictive view of the scope of classical probability A careful probabilistic analysis shows on the contrary that it is classical deterministic physical thinking which cannot cope with the Aspect experiment and therefore needs revision The ulterior aim of the paper is to help mathematical statisticians and probabilists to nd their way into the fascinating world of quantum probability thus the same aim as that of Kummerer and Maassen by dismantling the bamboo curtain between ordinary and quantum probability which over the years has been built up as physicists and pure mathematicians have repeated to one another Feynmans famous dictum quantum probability is a dierent kind of probability

This paper shows how heaping of duration data eg caused by rounding due to memory eects can be analyzed If the data are heaped Coxs partial likelihood approach which is often used in survival analysis is no longer appropriate We show how this problem can be overcome by considering the problem as a missing data problem A variant of Coxs Proportional Hazard Model is constructed that takes heaping into account and is estimated by maximum likelihood using the EM algorithm with many nuisance parameters simultaneously for all parameters Ingredients of our method are application of the EM algorithm Cox regression and nonparametric maximumlikelihood calculation with predicted data in each M step An example from practice where jackknife is used to estimate the variances illustrates the power of the new methodology

The speaker wants to make clear from the start that he is a mathematical statistician---i.e., a mathematician who studies and developes statistical theory---with certainly a great (layman's) interest in environmental matters, but with very little actual professional practical experience in this big and important field (though with plenty of professional practical experience in other fields of application of statistics).

Dit artikel is een aangepaste versie van een verslag wat geschreven is n.a.v. een onderzoek voor een casino. Dit onderzoek ging over de vraag of het spel Fantasie 24 (een vorm van observatieroulette) een behendigheidsspel dan wel kansspel is. We hebben een model opgesteld dat de behendigheid meet als gewogen gemiddelde over de inzetten van personen (wat gelijk blijkt te zijn aan de fractie van de gemiddelde winst over de dagen en het gemiddeld verlies over de dagen). De wet op de kansspelen kan daarmee een uitspraak kan geven over de vraag of het spel een behendigheidspel of een kansspel is. Om een betrouwbaarheidsinterval van deze fractie te bepalen wordt gestudentiseerde bootstrap gebruikt, waarmee een tweede orde correcte bovengrens is verkregen.