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.

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. Part II

1976 ◽  
Vol 13 (04) ◽  
pp. 733-740
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Random Walk ◽  
Rate Of Convergence ◽  
Real Line ◽  
Exponential Convergence ◽  
Exponential Rate ◽  
The Real ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence ◽  
Successive Maximum

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn (x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G ∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

The exponential rate of convergence of the distribution of the maximum of a random walk. Part II

1976 ◽  
Vol 13 (4) ◽  
pp. 733-740 ◽  
Author(s):  
N. Veraverbeke ◽  
J. L. Teugels
Keyword(s):  
Random Walk ◽  
Rate Of Convergence ◽  
Real Line ◽  
Exponential Convergence ◽  
Exponential Rate ◽  
The Real ◽  
Distribution Of The Maximum ◽  
Exponential Rate Of Convergence ◽  
Successive Maximum

Let Gn (x) be the distribution of the nth successive maximum of a random walk on the real line. Under conditions typical for complete exponential convergence, the decay of Gn (x) – limn→∞ Gn(x) is asymptotically equal to H(x) γn n–3/2 as n → ∞where γ < 1 and H(x) a function solely depending on x. For the case of drift to + ∞, G∞(x) = 0 and the result is new; for drift to – ∞we give a new proof, simplifying and correcting an earlier version in [9].

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 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)

scholarly journals Exponential rate of convergence in current reservoirs

Bernoulli ◽  
2015 ◽  
Vol 21 (3) ◽  
pp. 1844-1854 ◽  
Author(s):  
Anna De Masi ◽  
Errico Presutti ◽  
Dimitrios Tsagkarogiannis ◽  
Maria Eulalia Vares
Keyword(s):  
Rate Of Convergence ◽  
Exponential Rate ◽  
Exponential Rate Of Convergence

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.

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