Application of a Weierstrass Theorem to the Convergence of Moments

The Lévy distance, L(F,G), between two distribution functions F and G has the important property that convergence of L(Fn,F) is equivalent to convergence in distribution. The fact that L(F,G) is not invariant under a change of scale has been thought to be a disadvantage. However, simple bounds on the Lévy distance between the transformed distribution functions can be found.

Download Full-text

A note on the Lévy distance

Journal of Applied Probability ◽

10.1017/s0021900200048154 ◽

1975 ◽

Vol 12 (02) ◽

pp. 412-414

Author(s):

J. W. Thompson

Keyword(s):

Distribution Functions ◽

Convergence In Distribution ◽

Change Of Scale ◽

Simple Bounds ◽

Lévy Distance ◽

Transformed Distribution ◽

Levy Distance

The Lévy distance, L(F,G), between two distribution functions F and G has the important property that convergence of L(Fn,F) is equivalent to convergence in distribution. The fact that L(F,G) is not invariant under a change of scale has been thought to be a disadvantage. However, simple bounds on the Lévy distance between the transformed distribution functions can be found.

Download Full-text

Distributional Convergence for the Number of Symbol Comparisons Used by QuickSort (Extended Abstract)

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.2800 ◽

2010 ◽

Vol DMTCS Proceedings vol. AM,... (Proceedings) ◽

Author(s):

james Allen fill

Keyword(s):

Classical Model ◽

Key Comparisons ◽

Sorting Algorithm ◽

Convergence In Distribution ◽

Convergence Of Moments ◽

International Audience ◽

The Mean ◽

Periodic Fluctuations ◽

The Cost ◽

Infinite String

International audience Most previous studies of the sorting algorithm $\mathtt{QuickSort}$ have used the number of key comparisons as a measure of the cost of executing the algorithm. Here we suppose that the $n$ independent and identically distributed (iid) keys are each represented as a sequence of symbols from a probabilistic source and that $\mathtt{QuickSort}$ operates on individual symbols, and we measure the execution cost as the number of symbol comparisons. Assuming only a mild "tameness'' condition on the source, we show that there is a limiting distribution for the number of symbol comparisons after normalization: first centering by the mean and then dividing by $n$. Additionally, under a condition that grows more restrictive as $p$ increases, we have convergence of moments of orders $p$ and smaller. In particular, we have convergence in distribution and convergence of moments of every order whenever the source is memoryless, i.e., whenever each key is generated as an infinite string of iid symbols. This is somewhat surprising: Even for the classical model that each key is an iid string of unbiased ("fair'') bits, the mean exhibits periodic fluctuations of order $n$.

Download Full-text

Computer processing of high-resolution images of periodic specimens

Proceedings, annual meeting, Electron Microscopy Society of America ◽

10.1017/s0424820100141834 ◽

1986 ◽

Vol 44 ◽

pp. 2-5

Author(s):

W. Chiu ◽

M.F. Schmid ◽

T.-W. Jeng

Keyword(s):

High Resolution ◽

Fourier Transforms ◽

Complex Structure ◽

Distribution Functions ◽

Multiple Image ◽

Cryo Electron Microscopy ◽

Structure Factors ◽

Unit Cells ◽

High Resolution Images ◽

Phase Probability Distribution

Cryo-electron microscopy has been developed to the point where one can image thin protein crystals to 3.5 Å resolution. In our study of the crotoxin complex crystal, we can confirm this structural resolution from optical diffractograms of the low dose images. To retrieve high resolution phases from images, we have to include as many unit cells as possible in order to detect the weak signals in the Fourier transforms of the image. Hayward and Stroud proposed to superimpose multiple image areas by combining phase probability distribution functions for each reflection. The reliability of their phase determination was evaluated in terms of a crystallographic “figure of merit”. Grant and co-workers used a different procedure to enhance the signals from multiple image areas by vector summation of the complex structure factors in reciprocal space.

Download Full-text

Does a Graduate Course in Fluency Disorders Make a Difference?

Perspectives on Fluency and Fluency Disorders ◽

10.1044/ffd20.1.10 ◽

2010 ◽

Vol 20 (1) ◽

pp. 10-14 ◽

Cited By ~ 2

Author(s):

Evelyn R. Klein ◽

Barbara J. Amster

Keyword(s):

Undergraduate Students ◽

Graduate Programs ◽

The United States ◽

Cross Sectional Study ◽

Sectional Study ◽

Graduate Course ◽

Cross Sectional ◽

Fluency Disorders ◽

Introductory Course ◽

Communicative Disorders

Abstract A study by Yaruss and Quesal (2002), based on responses from 134 of 239 ASHA accredited graduate programs, indicated that approximately 25% of graduate programs in the United States allow students to earn their degree without having coursework in fluency disorders and 66% of programs allow students to graduate without clinical experience treating people who stutter (PWS). It is not surprising that many clinicians report discomfort in treating PWS. This cross-sectional study compares differences in beliefs about the cause of stuttering between freshman undergraduate students enrolled in an introductory course in communicative disorders and graduate students enrolled and in the final weeks of a graduate course in fluency disorders.

Download Full-text