scholarly journals The distribution of the maximum of partial sums of Kloosterman sums and other trace functions

2021 ◽  
Vol 157 (7) ◽  
pp. 1610-1651
Author(s):  
Pascal Autissier ◽  
Dante Bonolis ◽  
Youness Lamzouri
Keyword(s):  
Distribution Function ◽  
Recent Work ◽  
Character Sums ◽  
Kloosterman Sums ◽  
Partial Sums ◽  
Optimal Range ◽  
Uniform Estimates ◽  
The Third ◽  
Distribution Of The Maximum ◽  
Complex Valued

In this paper, we investigate the distribution of the maximum of partial sums of families of $m$ -periodic complex-valued functions satisfying certain conditions. We obtain precise uniform estimates for the distribution function of this maximum in a near-optimal range. Our results apply to partial sums of Kloosterman sums and other families of $\ell$ -adic trace functions, and are as strong as those obtained by Bober, Goldmakher, Granville and Koukoulopoulos for character sums. In particular, we improve on the recent work of the third author for Birch sums. However, unlike character sums, we are able to construct families of $m$ -periodic complex-valued functions which satisfy our conditions, but for which the Pólya–Vinogradov inequality is sharp.

The exponential rate of convergence of the distribution of the maximum of a random walk

1975 ◽  
Vol 12 (02) ◽  
pp. 279-288 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Exponential Convergence ◽  
Random Variables ◽  
Limiting Distribution ◽  
Partial Sums ◽  
Exponential Rate ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence

Let Gn (x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn (x) — G(x) is asymptotically equal to c.H(x)n −3/2 γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

The exponential rate of convergence of the distribution of the maximum of a random walk

1975 ◽  
Vol 12 (2) ◽  
pp. 279-288 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Exponential Convergence ◽  
Random Variables ◽  
Limiting Distribution ◽  
Partial Sums ◽  
Exponential Rate ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence

Let Gn(x) be the distribution function of the maximum of the successive partial sums of independent and identically distributed random variables and G(x) its limiting distribution function. Under conditions, typical for complete exponential convergence, the decay of Gn(x) — G(x) is asymptotically equal to c.H(x)n−3/2γn as n → ∞ where c and γ are known constants and H(x) is a function solely depending on x.

Supercharacters and mixed moments of Kloosterman sums

2018 ◽  
Vol 14 (04) ◽  
pp. 1023-1032 ◽  
Author(s):  
Ángel Chávez ◽  
George Todd
Keyword(s):  
Elliptic Curve ◽  
Recent Work ◽  
Kloosterman Sums ◽  
Fourth Degree ◽  
Mixed Power ◽  
Power Moments ◽  
Supercharacter Theory ◽  
Trace Of Frobenius

Recent work has realized Kloosterman sums as supercharacter values of a supercharacter theory on [Formula: see text]. We use this realization to express fourth degree mixed power moments of Kloosterman sums in terms of the trace of Frobenius of a certain elliptic curve.

scholarly journals Pascu-Type Harmonic Functions with Positive Coefficients Involving Salagean Operator

2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
K. Vijaya ◽  
G. Murugusundaramoorthy ◽  
M. Kasthuri
Keyword(s):  
Harmonic Functions ◽  
Extreme Points ◽  
Partial Sums ◽  
Open Unit ◽  
New Class ◽  
Sălăgean Operator ◽  
Salagean Operator ◽  
Convolution Properties ◽  
Positive Coefficients ◽  
Complex Valued

Making use of a Salagean operator, we introduce a new class of complex valued harmonic functions which are orientation preserving and univalent in the open unit disc. Among the results presented in this paper including the coeffcient bounds, distortion inequality, and covering property, extreme points, certain inclusion results, convolution properties, and partial sums for this generalized class of functions are discussed.

The Distribution of the Maximum of Partial Sums of Independent Random Variables

1950 ◽  
Vol 2 ◽  
pp. 375-384 ◽  
Author(s):  
Mark Kac ◽  
Harry Pollard
Keyword(s):  
Random Variables ◽  
Independent Random Variables ◽  
Partial Sums ◽  
Distribution Of The Maximum

1. The problem. It has been shown [1] that if Xi, + X2, … are independent random variables each of density then(1.1)

Limiting behaviour of the distributions of the maxima of partial sums of certain random walks

1972 ◽  
Vol 9 (3) ◽  
pp. 572-579 ◽  
Author(s):  
D. J. Emery
Keyword(s):  
Distribution Function ◽  
Random Walk ◽  
Asymptotic Property ◽  
Regular Variation ◽  
Domain Of Attraction ◽  
Hitting Times ◽  
Partial Sums ◽  
Symmetric Distributions ◽  
Stable Law ◽  
Limiting Behaviour

It is shown that, under certain conditions, satisfied by stable distributions, symmetric distributions, distributions with zero mean and finite second moment and other distributions, the distribution function of the maxima of successive partial sums of identically distributed random variables has an asymptotic property. This property implies the regular variation of the tail of the distribution of the hitting times of the associated random walk, and hence that these hitting times belong to the domain of attraction of a stable law.

The Oxford Handbook of Aesthetics

2009 ◽  
Keyword(s):  
Recent Work ◽  
Aesthetic Experience ◽  
The Third ◽  
Extensive Survey ◽  
General Overview ◽  
The Past ◽  
The Arts ◽  
History Of ◽  
Theory Application ◽  
Music Film

The Oxford Handbook of Aesthetics looks at a fascinating theme in philosophy and the arts. Leading figures in the field contribute forty-eight articles which detail the theory, application, history, and future of philosophy and all branches of the arts. The first article of the book gives a general overview of the field of philosophical aesthetics in two parts: the first is a quick sketch of the lay of the land, and the second an account of five central problems over the past fifty years. The second article gives an extensive survey of recent work in the history of modern aesthetics, or aesthetic thought from the seventeenth to the mid-twentieth centuries. There are three main parts to the book. The first part comprises sections dealing with problems in aesthetics, such as expression, fiction or aesthetic experience, considered apart from any particular artform. The second part contains articles on problems in aesthetics as they arise in connection with particular artforms, such as music, film, or dance. The third part addresses relations between aesthetics and other fields of enquiry, and explores viewpoints or concerns complimentary to those prominent in mainstream analytical aesthetics.

scholarly journals Lebesgue's Convergence Theorem of Complex-Valued Function

2009 ◽  
Vol 17 (2) ◽  
pp. 137-145 ◽  
Author(s):  
Keiko Narita ◽  
Noboru Endou ◽  
Yasunari Shidama
Keyword(s):  
Imaginary Part ◽  
Convergence Theorem ◽  
Partial Sums ◽  
Complex Valued

Lebesgue's Convergence Theorem of Complex-Valued Function In this article, we formalized Lebesgue's Convergence theorem of complex-valued function. We proved Lebesgue's Convergence Theorem of realvalued function using the theorem of extensional real-valued function. Then applying the former theorem to real part and imaginary part of complex-valued functional sequences, we proved Lebesgue's Convergence Theorem of complex-valued function. We also defined partial sums of real-valued functional sequences and complex-valued functional sequences and showed their properties. In addition, we proved properties of complex-valued simple functions.

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