Addendum: Asymptotic Normality in the Classical Ball Problem

We consider the problem of a hopping ball excited by a travelling harmonic wave on an elastic surface. The ball, considered as a particle, is assumed to interact with the surface through inelastic collisions. The surface motion due to the wave induces a horizontal drift in the ball. The problem is treated analytically under certain approximations. The phase space of the hopping motion is captured by constructing a phase-velocity return map. The fixed points of the return map and its compositions represent periodic hopping solutions. The linear stability of the obtained periodic solution is studied in detail. The minimum frequency for the onset of periodic hops, and the subsequent loss of stability at the bifurcation frequency, have been determined analytically. Interestingly, for small values of wave amplitude, the analytical solutions reveal striking similarities with the results of the classical bouncing ball problem.

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A model for random instantaneous growth on an interval

Journal of Applied Probability ◽

10.2307/3214489 ◽

1991 ◽

Vol 28 (3) ◽

pp. 529-538

Author(s):

M. P. Quine

Keyword(s):

Asymptotic Normality ◽

Instantaneous Growth ◽

Mean And Variance ◽

The Mean

Points arrive in succession on an interval and immediately ‘cover' a region of length ½ to each side (less if they are close to the boundary or to a covered part). The location of a new point is uniformly distributed on the uncovered parts. We study the mean and variance of the total number of points ever formed, in particular as a → 0, in which case we also establish asymptotic normality.

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Asymptotic normality of conditional distribution estimation in the single index model

Acta Universitatis Sapientiae Mathematica ◽

10.1515/ausm-2017-0010 ◽

2017 ◽

Vol 9 (1) ◽

pp. 162-175

Author(s):

Diaa Eddine Hamdaoui ◽

Amina Angelika Bouchentouf ◽

Abbes Rabhi ◽

Toufik Guendouzi

Keyword(s):

Distribution Function ◽

Confidence Interval ◽

Asymptotic Normality ◽

Conditional Distribution ◽

Conditional Distribution Function ◽

Single Index ◽

Index Model ◽

Single Index Model ◽

Distribution Estimation

AbstractThis paper deals with the estimation of conditional distribution function based on the single-index model. The asymptotic normality of the conditional distribution estimator is established. Moreover, as an application, the asymptotic (1 − γ) confidence interval of the conditional distribution function is given for 0 < γ < 1.

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