Multisensory decisions provide support for probabilistic number representations.

Previous article Next article On the Extension of the Class of Stable DistributionsV. M. KruglovV. M. Kruglovhttps://doi.org/10.1137/1117081PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] B. V. Gnedenko and , A. N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables, Addison-Wesley, Cambridge, Mass., 1962 Google Scholar[2] Yu. V. Linnik, Decomposition of probability distributions, Edited by S. J. Taylor, Dover Publications Inc., New York, 1964xii+242 MR0189085 (32:6512) 0121.35502 Google Scholar[3] B. V. Gnedenko, Some theorems on distribution functions, Učenie Zapiski Moskovskogo Univ., 30 (1939), 61–82, (In Russian.) Google Scholar[4] V. M. Kruglov, A note on infinitely divisible distributions, Theory Prob. Applications, 15 (1970), 319–324 10.1137/1115038 0329.60011 LinkGoogle Scholar[5] I. A. Ibragimov and , Yu. V. Linnik, Independent and stationary sequences of random variables, Wolters-Noordhoff Publishing, Groningen, 1971, 443– MR0322926 (48:1287) 0219.60027 Google Scholar Previous article Next article FiguresRelatedReferencesCited byDetails Functional L.I.L. for the Upper Extremes of i.i.d. Random Variarbles from a Sub-class of Heavy Tailed Distributions31 October 2021 | Journal of the Indian Society for Probability and Statistics, Vol. 22, No. 2 Cross Ref Functional law of the iterated logarithm for partial sum processes of non-negative valued iid random variables belonging to the domain of partial attraction of a semi-stable lawStatistics & Probability Letters, Vol. 137 Cross Ref A Log Log Law for Subsequences of Delayed Random Sums27 July 2017 | Journal of the Indian Society for Probability and Statistics, Vol. 18, No. 2 Cross Ref Merging to Semistable ProcessesI. Fazekas15 November 2012 | Theory of Probability & Its Applications, Vol. 56, No. 4AbstractPDF (209 KB)A Precise Asymptotic Behaviour of the Large Deviation Probabilities for Weighted SumsApplied Mathematics, Vol. 02, No. 09 Cross Ref A note on asymptotics of linear combinations of iid random variables18 March 2010 | Periodica Mathematica Hungarica, Vol. 60, No. 1 Cross Ref Merging of Linear Combinations to Semistable Laws3 January 2008 | Journal of Theoretical Probability, Vol. 22, No. 3 Cross Ref Merging asymptotic expansions for semistable random variables11 March 2009 | Lithuanian Mathematical Journal, Vol. 49, No. 1 Cross Ref Law of iterated logarithm and almost sure limit points for random sumsStatistical Methodology, Vol. 3, No. 4 Cross Ref On a functional equation generalizing the class of semistable distributionsAnnals of the Institute of Statistical Mathematics, Vol. 57, No. 4 Cross Ref Merging to Semistable LawsS. Csorgo and Z. Megyesi25 July 2006 | Theory of Probability & Its Applications, Vol. 47, No. 1AbstractPDF (220 KB)On Lévy–Fréchet processes and related self-similar and semi-self-similar onesChaos, Solitons & Fractals, Vol. 14, No. 5 Cross Ref ON LÉVY STABLE AND SEMISTABLE DISTRIBUTIONS20 November 2011 | Fractals, Vol. 09, No. 03 Cross Ref Rarely Observed Sample MaximaL. Canto E Castro, L De Haan, and M. G. Temido25 July 2006 | Theory of Probability & Its Applications, Vol. 45, No. 4AbstractPDF (90 KB)A law of the iterated logarithm for semistable laws on vector spaces and homogeneous groupsJournal of Mathematical Sciences, Vol. 89, No. 5 Cross Ref Characterization of multivariate distributions through a functional equation of their characteristic functionsJournal of Statistical Planning and Inference, Vol. 63, No. 2 Cross Ref Series representation for semistable laws and their domains of semistable attractionJournal of Theoretical Probability, Vol. 9, No. 4 Cross Ref The Domains of Attraction of Semistable LawsI. V. Grinevich and Yu. S. Khokhlov12 July 2006 | Theory of Probability & Its Applications, Vol. 40, No. 2AbstractPDF (670 KB)Moments of measures attracted to operator semi-stable lawsStatistics & Probability Letters, Vol. 24, No. 3 Cross Ref Recurrence and transience of operator semi-stable processesProceedings of the Japan Academy, Series A, Mathematical Sciences, Vol. 71, No. 5 Cross Ref XVI Seminar on Stability Problems of Stochastic Models17 July 2006 | Theory of Probability & Its Applications, Vol. 37, No. 4AbstractPDF (5240 KB)A characterization of gamma mixtures of stable laws motivated by limit theoremsStatistica Neerlandica, Vol. 46, No. 2-3 Cross Ref ON CHARACTERIZATIONS THROUGH MIXED SUMS28 June 2008 | Australian Journal of Statistics, Vol. 34, No. 2 Cross Ref The Suzdal International Seminar of 1991 on Stability Problems for Stochastic Models17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 4AbstractPDF (5581 KB)Law of iterated logarithm for random subsequencesStatistics & Probability Letters, Vol. 12, No. 3 Cross Ref Limit theorems for probability measures on simply connected nilpotent lie groupsJournal of Theoretical Probability, Vol. 4, No. 2 Cross Ref On a Scheme of Partial SummationE. D. Chibisova17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 3AbstractPDF (466 KB)Operator-Semistable Distributions on ${\bf R}^d$V. Chorny17 July 2006 | Theory of Probability & Its Applications, Vol. 31, No. 4AbstractPDF (368 KB)On a Problem of B. V. Gnedenko in a Scheme of Partial SummationE. D. Chibisova17 July 2006 | Theory of Probability & Its Applications, Vol. 31, No. 4AbstractPDF (386 KB)Stable processes and measures; A survey15 November 2006 Cross Ref Operator semistable probability measures on a Hilbert space17 April 2009 | Bulletin of the Australian Mathematical Society, Vol. 22, No. 3 Cross Ref Semistable measures on a Hilbert spaceJournal of Multivariate Analysis, Vol. 10, No. 1 Cross Ref Support and seminorm integrability theorems for r-semistable probability measures on LCTVS22 October 2006 Cross Ref The renormalization group: A probabilistic view22 November 2007 | Il Nuovo Cimento B Series 11, Vol. 26, No. 1 Cross Ref Volume 17, Issue 4| 1973Theory of Probability & Its Applications History Submitted:02 July 1970Published online:28 July 2006 InformationCopyright © 1973 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1117081Article page range:pp. 685-694ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

The mediating role of number-to-magnitude mapping precision in the relationship between approximate number sense and math achievement depends on the domain of mathematics and age

In synchronized trampolining, two gymnasts perform the same routine at the same time. While trained gymnasts are thought to coordinate their own movements with the movements of another gymnast by detecting relevant movement information, the question arises how visual and auditory information contribute to the emergence of synchronicity between both gymnasts. Therefore the aim of this study was to examine the role of visual and auditory information in the emergence of coordinated behaviour in synchronized trampolining. Twenty female gymnasts were asked to synchronize their leaps with the leaps of a model gymnast, while visual and auditory information was manipulated. The results revealed that gymnasts needed more leaps to reach synchronicity when only either auditory (12.9 leaps) or visual information (10.8 leaps) was available, as compared to when both auditory and visual information was available (8.1 leaps). It is concluded that visual and auditory information play significant roles in synchronized trampolining, whilst visual information seems to be the dominant source for emerging behavioural synchronization, and auditory information supports this emergence.

Previous article Next article The Characteristic Properties of Normal and Stable DistributionsM. Sh. BravermanM. Sh. Bravermanhttps://doi.org/10.1137/1130059PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout Previous article Next article FiguresRelatedReferencesCited byDetails Robust object tracking with bidirectional corner matching and trajectory smoothness algorithm Cross Ref On characterization of multivariate stable distributions via random linear statisticsJournal of Theoretical Probability, Vol. 8, No. 1 Cross Ref Remarks on characterization of normal and stable distributionsJournal of Theoretical Probability, Vol. 6, No. 2 Cross Ref XVI Seminar on Stability Problems of Stochastic Models17 July 2006 | Theory of Probability & Its Applications, Vol. 37, No. 4AbstractPDF (5240 KB)A Least Squares Estimate, Nonquadratic Losses, and Gaussian DistributionsA. A. Zinger and A. M. Kagan17 July 2006 | Theory of Probability & Its Applications, Vol. 36, No. 1AbstractPDF (656 KB)A characterization of probability distributions by mean valuesJournal of Theoretical Probability, Vol. 4, No. 4 Cross Ref A Characterization of a Distribution by a Property of the Mean Values of Order StatisticsL. B. Klebanov and M. V. Neupokoeva17 July 2006 | Theory of Probability & Its Applications, Vol. 34, No. 4AbstractPDF (482 KB)On a Way of Characterizing Probability DistributionsM. Sh. Braverman17 July 2006 | Theory of Probability & Its Applications, Vol. 32, No. 3AbstractPDF (550 KB) Volume 30, Issue 3| 1986Theory of Probability & Its Applications History Submitted:27 February 1984Published online:17 July 2006 InformationCopyright © 1986 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1130059Article page range:pp. 465-474ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

Approximate number sense shares etiological overlap with mathematics and general cognitive ability

Most studies have been unable to identify reliable acoustic cues for the recognition of the English nonsibilant fricatives [see text]. The present study was designed to test the extent to which the perception of these fricatives by normal-hearing adults is based on other sources of information, namely, linguistic context and visual information. In Experiment 1, target words beginning with /f/, /theta/, /s/, or [see text] were preceded by either a semantically congruous or incongruous precursor sentence. Results showed an effect of linguistic context on the perception of the distinction between /f/ and /theta/ and on the acoustically more robust distinction between /s/ and [see text]. In Experiment 2, participants identified syllables consisting of the fricatives [see text] paired with the vowels /i, a, u/. Three conditions were contrasted: Stimuli were presented with (a) both auditory and visual information, (b) auditory information alone, or (c) visual information alone. When errors in terms of voicing were ignored in all 3 conditions, results indicated that perception of these fricatives is as good with visual information alone as with both auditory and visual information combined, and better than for auditory information alone. These findings suggest that accurate perception of nonsibilant fricatives derives from a combination of acoustic, linguistic, and visual information.

By M. Denny and S. Gaines Princeton University Press (2002) pp. 424. ISBN 0-691-09494-2 (pbk) £ 19.95 / $29.95. ![Figure][1] As a teacher of introductory biology, I try very hard to instill in my students the fact that quantitative reasoning is an integral tool

In mathematics and statistics, the desire to eliminate mathematical tedium and facilitate exploration has lead to computer algebra systems. These computer algebra systems allow students and researchers to perform more of their work at a conceptual level. The design of generic algorithms for tedious computations allows modelers to push current modeling boundaries outward more quickly. Probability theory, with its many theorems and symbolic manipulations of random variables is a discipline in which automation of certain processes is highly practical, functional, and efficient. There are many existing statistical software packages, such as SPSS, SAS, and S-Plus, that have numeric tools for statistical applications. There is a potential for a probability package analogous to these statistical packages for manipulation of random variables. The software package being developed as part of this dissertation, referred to as “A Probability Programming Language” (APPL) is a random variable manipulator and is proposed to fill a technology gap that exists in probability theory. My research involves developing algorithms for the manipulation of discrete random variables. By defining data structures for random variables and writing algorithms for implementing common operations, more interesting and mathematically intractable probability problems can be solved, including those not attempted in undergraduate statistics courses because they were deemed too mechanically arduous. Algorithms for calculating the probability density function of order statistics, transformations, convolutions, products, and minimums/maximums of independent discrete random variables are included in this dissertation.

Although speech perception is usually considered an auditory process, studies have shown that visual information provided by the movements of a talker’s mouth and face strongly influences what an observer perceives even when the auditory signal is clear and unambiguous. The now classic “McGurk effect” is an excellent example: the auditory syllable /bi/ presented in synchrony with a videotape of a talker saying the syllable /gi/ is usually perceived as /di/, a syllable not presented to either modality. This phenomenon has important implications for theories of speech perception which must account for how and why auditory and visual signals are integrated during phonetic processing. The purpose of the current talk is to describe some of the research that has contributed to our understanding of how visual and auditory information are used in phonetic perception. Specifically, this paper focuses on three issues: (1) what factors influence the integration of auditory and visual information; (2) the nature of the auditory and visual information used in phonetic perception; and (3) the interaction of the two sources of information during phonetic processing. The implications of these findings for theories of speech perception are discussed.

It has been reported that approximate number sense (ANS) task performance is impaired in individuals with Williams syndrome (WS) and Down syndrome (DS). Research with infants has suggested this impairment is caused by sticky fixation in WS and sustained attention deficits for those with DS. This study examined looking patterns of older children and adults with WS (n = 24) and DS (n = 23) during an ANS task compared to typically developing controls matched for chronological age and those matched for mental age. Results showed that, although there were no group differences, looking patterns changed with chronological age for both the WS and DS groups. Looking behaviour related to ANS performance only in the WS group. Implications for interventions are discussed.

Muth’s Rational Expectations Hypothesis, That “…The Subjective Probability Distributions Are Distributed Around an Objective Probability Distribution, for a given Information Set…” is an Oxymoron. It Can Be Corrected by Amending It to Read “…The Subjective Probability Distributions Are Distributed Around a Subjective Consensus Probability Distribution, for a given Information Set…”

Higher-order probabilistic programming languages allow programmers to write sophisticated models in machine learning and statistics in a succinct and structured way, but step outside the standard measure-theoretic formalization of probability theory. Programs may use both higher-order functions and continuous distributions, or even define a probability distribution on functions. But standard probability theory does not handle higher-order functions well: the category of measurable spaces is not cartesian closed. Here we introduce quasi-Borel spaces. We show that these spaces: form a new formalization of probability theory replacing measurable spaces; form a cartesian closed category and so support higher-order functions; form a well-pointed category and so support good proof principles for equational reasoning; and support continuous probability distributions. We demonstrate the use of quasi-Borel spaces for higher-order functions and probability by: showing that a well-known construction of probability theory involving random functions gains a cleaner expression; and generalizing de Finetti's theorem, that is a crucial theorem in probability theory, to quasi-Borel spaces.

This chapter introduces probability theory as a system of models, based on measure theory, of some real-world phenomena. The models are measure spaces of total measure 1 and usually have certain distinguished measurable functions defined on them. Section 1 begins by establishing the measure-theoretic framework and a short dictionary for passing back and forth between terminology in measure theory and terminology in probability theory. The latter terminology includes events, random variables, mean, probability distribution of a random variable, and joint probability distribution of several random variables. An important feature of probability is that it is possible to work with random variables without any explicit knowledge of the underlying measure space, the joint probability distributions of random variables being the objects of importance. Section 2 introduces conditional probability and uses that to motivate the mathematical definition of independence of events. In turn, independence of events leads naturally to a definition of independent random variables. Independent random variables are of great importance in the subject and play a much larger role than their counterparts in abstract measure theory. Examples at the end of the section indicate the extent to which functions of independent random variables can remain independent. The techniques in the examples are of use in the subject of statistical inference, which is introduced in Section 10. Section 3 states and proves the Kolmogorov Extension Theorem, a foundational result allowing one to create stochastic processes involving infinite sets of times out of data corresponding to finite subsets of those times. A special case of the theorem provides the existence of infinite sets of independent random variables with specified probability distributions. Section 4 establishes the celebrated Strong Law of Large Numbers, which says that the Cesaro sums of a sequence of identically distributed independent random variables with finite mean converge almost everywhere to a constant random variable, the constant being the mean. This is a theorem that is vaguely known to the general public and is widely misunderstood. The proof is based on Kolmogorov’s inequality. Sections 5–8 provide background for the Central Limit Theorem, whose statement and proof are in Section 9. Section 5 discusses three successively weaker kinds of convergence for random variables—almost sure convergence, convergence in probability, and convergence in distribution. Convergence in distribution will be the appropriate kind for the Central Limit Theorem. Section 6 contains the Portmanteau Lemma, which gives some equivalent formulations of convergence in distribution, Section 7 introduces characteristic functions as Fourier transforms of probability distributions, and Section 8 proves the Levy Continuity Theorem, which formulates convergence in distribution in terms of characteristic functions. Section 9 contains the statement and proof of the Central Limit Theorem, followed by some simple examples. This theorem is the most celebrated result in probability theory and has many applications in mathematics and other fields. Section 10 is a brief introduction to the subject of statistical inference, showing how the Central Limit theorem plays a role in practice through the $t$ test of W. S. Gosset.

One of the requirements for being a successful musical conductor is to be able to locate sounds instantaneously in time and space. Because this requires the integration of auditory and visual information, the purpose of this study was to examine multisensory processing in conductors and a matched set of control subjects. Subjects participated in a series of behavioral tasks, including pitch discrimination, temporal-order judgment (TOJ), and target localization. Additionally, fMRI scans were done on a subset of subjects who performed a multisensory TOJ task. Analyses of behavioral data indicate that, in the auditory realm, conductors were more accurate in both pitch discrimination and TOJs as well as in locating targets in space. Furthermore, these same subjects also demonstrated a benefit from the combination of auditory and visual information that was not observed in control subjects when locating visual targets. Finally, neural substrates in BA 37, 39/40 were identified as potential areas underlying the conductors' superior multisensory TOJs. Data collection and analyses are ongoing and will lead to an improved understanding of multisensory integration in a complex, musical behavior.

In the working memory model, phonological and visuospatial information are stored by separate and independent systems. However, sometimes binding of visual and verbal information must occur. This study investigated whether these memory systems cooperate in the recall of spatial location of a stimulus defined by both types of information. Participants memorized the spatial locations in which name-and-face pairs were presented and either recalled the position of a test stimulus in which the two types of information were always present (Experiment 1) or recalled the position of a test stimulus that non-predictively contained either the visual or verbal information (Experiment 2). The results showed no cooperation between visuospatial and phonological systems when both types of information were present in the test stimulus. Rather, a clear preference for verbal information was found (Experiment 1). When the test stimulus contained only one type of information, recall based on both verbal and visual information was impaired (Experiment 2). These results suggest that visual and verbal information are not automatically integrated into memory and that storage capacity is smaller for integrated information than for isolated information.