On the rate of convergence for extremes of mean square differentiable stationary normal processes

1997 ◽  
Vol 34 (4) ◽  
pp. 908-923 ◽  
Author(s):  
Marie F. Kratz ◽  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Covariance Function ◽  
Mean Square ◽  
Normal Process ◽  
Distribution Of The Maximum ◽  
Image Position ◽  
High Level

Let ξ (t); t ≧ 0 be a normalized continuous mean square differentiable stationary normal process with covariance function r(t). Further, let and set . We give bounds which are roughly of order Τ –δ for the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level by ξ (t) in the interval [0, T]. The results assume that r(t) and r′(t) decay polynomially at infinity and that r″ (t) is suitably bounded. For the number of upcrossings it is in addition assumed that r(t) is non-negative.

On the rate of convergence for extremes of mean square differentiable stationary normal processes

1997 ◽  
Vol 34 (04) ◽  
pp. 908-923 ◽  
Author(s):  
Marie F. Kratz ◽  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Covariance Function ◽  
Mean Square ◽  
Normal Process ◽  
Distribution Of The Maximum ◽  
Image Position ◽  
High Level

Let ξ (t); t ≧ 0 be a normalized continuous mean square differentiable stationary normal process with covariance function r(t). Further, let and set . We give bounds which are roughly of order Τ –δ for the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level by ξ (t) in the interval [0, T]. The results assume that r(t) and r′(t) decay polynomially at infinity and that r ″ (t) is suitably bounded. For the number of upcrossings it is in addition assumed that r(t) is non-negative.

Intersession reliability of GPS-based and accelerometer-based physical variables in small-sided games with and without the offside rule

2021 ◽  
pp. 175433712098764
Author(s):  
Igor Junio de Oliveira Custódio ◽  
Gibson Moreira Praça ◽  
Leandro Vinhas de Paula ◽  
Sarah da Glória Teles Bredt ◽  
Fabio Yuzo Nakamura ◽  
...  
Keyword(s):  
Root Mean Square ◽  
Global Positioning System ◽  
Intraclass Correlation ◽  
Correlation Coefficients ◽  
Mean Square ◽  
Physical Demands ◽  
Intraclass Correlation Coefficients ◽  
Total Distance ◽  
Global Positioning ◽  
High Level

This study aimed to analyze the intersession reliability of global positioning system (GPS-based) distances and accelerometer-based (acceleration) variables in small-sided soccer games (SSG) with and without the offside rule, as well as compare variables between the tasks. Twenty-four high-level U-17 soccer athletes played 3 versus 3 (plus goalkeepers) SSG in two formats (with and without the offside rule). SSG were performed on eight consecutive weeks (4 weeks for each group), twice a week. The physical demands were recorded using a GPS with an embedded triaxial accelerometer. GPS-based variables (total distance, average speed, and distances covered at different speeds) and accelerometer-based variables (Player Load™, root mean square of the acceleration recorded in each movement axis, and the root mean square of resultant acceleration) were calculated. Results showed that the inclusion of the offside rule reduced the total distance covered (large effect) and the distances covered at moderate speed zones (7–12.9 km/h – moderate effect; 13–17.9 km/h – large effect). In both SSG formats, GPS-based variables presented good to excellent reliability (intraclass correlation coefficients – ICC > 0.62) and accelerometer-based variables presented excellent reliability (ICC values > 0.89). Based on the results of this study, the offside rule decreases the physical demand of 3 versus 3 SSG and the physical demands required in these SSG present high intersession reliability.

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].

On Global Inverse Theorems of Szász and Baskakov Operators

1979 ◽  
Vol 31 (2) ◽  
pp. 255-263 ◽  
Author(s):  
Z. Ditzian
Keyword(s):  
Rate Of Convergence ◽  
Exponential Growth ◽  
Polynomial Growth ◽  
Continuous Functions ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Baskakov Operators ◽  
Inverse Theorems ◽  
Image Position ◽  
Necessary And Sufficient

The Szász and Baskakov approximation operators are given by1.11.2respectively. For continuous functions on [0, ∞) with exponential growth (i.e. ‖ƒ‖A ≡ supx\ƒ(x)e–Ax\ < M) the modulus of continuity is defined by1.3where ƒ ∈ Lip* (∝, A) for some 0 < ∝ ≦ 2 if w2(ƒ, δ, A) ≦ Mδ∝ for all δ < 1. We shall find a necessary and sufficient condition on the rate of convergence of An(ƒ, x) (representing Sn(ƒ, x) or Vn(ƒ, x)) to ƒ(x) for ƒ(x) ∈ Lip* (∝, A). In a recent paper of M. Becker [1] such conditions were found for functions of polynomial growth (where (1 + \x\N)−1 replaced e–Ax in the above). M. Becker explained the difficulties in treating functions of exponential growth.

The rate of convergence of extremes of stationary normal sequences

1983 ◽  
Vol 15 (01) ◽  
pp. 54-80 ◽  
Author(s):  
Holger Rootzén
Keyword(s):  
Rate Of Convergence ◽  
Joint Distribution ◽  
Point Processes ◽  
Rates Of Convergence ◽  
Joint Distribution Function ◽  
Maximal Correlation ◽  
Standard Normal ◽  
Image Position ◽  
The Right ◽  
Right Order

Let {ξ; t = 1, 2, …} be a stationary normal sequence with zero means, unit variances, and covariances let be independent and standard normal, and write . In this paper we find bounds on which are roughly of the order where ρ is the maximal correlation, ρ =sup {0, r 1 , r 2, …}. It is further shown that, at least for m-dependent sequences, the bounds are of the right order and, in a simple example, the errors are evaluated numerically. Bounds of the same order on the rate of convergence of the point processes of exceedances of one or several levels are obtained using a ‘representation' approach (which seems to be of rather wide applicability). As corollaries we obtain rates of convergence of several functionals of the point processes, including the joint distribution function of the k largest values amongst ξ1, …, ξn.

scholarly journals Mean square rate of convergence for random walk approximation of forward-backward SDEs

2020 ◽  
Vol 52 (3) ◽  
pp. 735-771
Author(s):  
Christel Geiss ◽  
Céline Labart ◽  
Antti Luoto
Keyword(s):  
Differential Equation ◽  
Random Walk ◽  
Rate Of Convergence ◽  
Backward Stochastic Differential Equation ◽  
Stochastic Equations ◽  
Terminal Condition ◽  
Probabilistic Methods ◽  
Mean Square ◽  
Finite Difference Equations ◽  
Backward Sdes

AbstractLet (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

VI.—Note on the High-level Deposits on the Chalk at Little Heath, near Berkhamsted

1919 ◽  
Vol 6 (3) ◽  
pp. 124-125
Author(s):  
Reginald A. Smith
Keyword(s):  
Geological Society ◽  
North East ◽  
Image Position ◽  
High Level

At a meeting of the Geological Society on January 22 a section at Little Heath, near Berkhamsted, was described and discussed; but both in the papers and in the discussion attention was focussed on the lower part of the section, and the gravel, which is separated by about 6 inches of bull-head from the Chalk, was assigned to the Pliocene and correlated with the Westleton Beds of Prestwich. But the upper beds have an interest of their own, especially for the archæologist who is acquainted with the work of the late Mr. Worthington Smith; and it seems worth while to point out some striking resemblances to deposits in a line running north-east of the site in question. The Little Heath strata referred to are:—

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.

scholarly journals On the rate of convergence of waiting times

1965 ◽  
Vol 5 (3) ◽  
pp. 365-373 ◽  
Author(s):  
C. K. Cheong ◽  
C. R. Heathcote
Keyword(s):  
Distribution Function ◽  
Rate Of Convergence ◽  
Waiting Times ◽  
Distribution Functions ◽  
Initial Condition ◽  
Hopf Equation ◽  
Unknown Distribution ◽  
Image Position

Let K(y) be a known distribution function on (−∞, ∞) and let {Fn(y), n = 0, 1,…} be a sequence of unknown distribution functions related by subject to the initial condition If the sequence {Fn(y)} converges to a distribution function F(y) then F(y) satisfies the Wiener-Hopf equation

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