Nonhomogenous bivariate fragmentation process: Asymptotic distribution via contraction method

International audience Two processes of random fragmentation of an interval are investigated. For each of them, there is a splitting probability at each step of the fragmentation process whose overall effect is to stabilize the global number of splitting events. More precisely, we consider two models. In the first model, the fragmentation stops which a probability $p$ witch can not depend on the fragment size. The number of stable fragments with sizes less than a given $t \geq 0$, denoted by $K(t)$, is introduced and studied. In the second one the probability to split a fragment of size $x$ is $p(x)=1-e^{-x}$. For this model we utilize the contraction method to show that the distribution of a suitably normalized version of the number of stable fragments converges in law. It's shown that the limit is the fixed-point solution (in the Wasserstein space) to a distributional equation. An explicit solution to the fixed-point equation is easily verified to be Gaussian.

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Properties of the beta coefficient of the global minimum variance portfolio

Mathematical Modeling and Computing ◽

10.23939/mmc2021.01.011 ◽

2020 ◽

Vol 8 (1) ◽

pp. 11-21

Author(s):

S. M. Yaroshko ◽

◽

M. V. Zabolotskyy ◽

T. M. Zabolotskyy ◽

◽

...

Keyword(s):

Confidence Interval ◽

Asymptotic Distribution ◽

Global Minimum ◽

Asset Returns ◽

Minimum Variance ◽

Benchmark Portfolio ◽

Beta Coefficient ◽

Minimum Variance Portfolio ◽

Global Minimum Variance Portfolio ◽

Daily Returns

The paper is devoted to the investigation of statistical properties of the sample estimator of the beta coefficient in the case when the weights of benchmark portfolio are constant and for the target portfolio, the global minimum variance portfolio is taken. We provide the asymptotic distribution of the sample estimator of the beta coefficient assuming that the asset returns are multivariate normally distributed. Based on the asymptotic distribution we construct the confidence interval for the beta coefficient. We use the daily returns on the assets included in the DAX index for the period from 01.01.2018 to 30.09.2019 to compare empirical and asymptotic means, variances and densities of the standardized estimator for the beta coefficient. We obtain that the bias of the sample estimator converges to zero very slowly for a large number of assets in the portfolio. We present the adjusted estimator of the beta coefficient for which convergence of the empirical variances to the asymptotic ones is not significantly slower than for a sample estimator but the bias of the adjusted estimator is significantly smaller.

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Joint Asymptotic Distribution of Marginal Quantiles and Quantile Functions in Samples from a Multivariate Population.

10.21236/ada193385 ◽

1987 ◽

Author(s):

C. J. Babu ◽

C. R. Rao

Keyword(s):

Asymptotic Distribution ◽

Quantile Functions

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Chapter Five: The First Asymptotic Distribution

Statistics of Extremes ◽

10.7312/gumb92958-007 ◽

1958 ◽

pp. 156-200 ◽

Cited By ~ 1

Keyword(s):

Asymptotic Distribution

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Comparison of powers for the sphericity tests using both the asymptotic distribution and the bootstrap

Communication in Statistics- Theory and Methods ◽

10.1080/03610928808829648 ◽

1988 ◽

Vol 17 (3) ◽

pp. 671-690 ◽

Cited By ~ 9

Author(s):

Y.M. Chan ◽

M.S. Srivastava

Keyword(s):

Asymptotic Distribution

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A matrix-less method to approximate the spectrum and the spectral function of Toeplitz matrices with real eigenvalues

Numerical Algorithms ◽

10.1007/s11075-021-01130-9 ◽

2021 ◽

Author(s):

Sven-Erik Ekström ◽

Paris Vassalos

Keyword(s):

Distribution Function ◽

Asymptotic Distribution ◽

Numerical Experiments ◽

Toeplitz Matrices ◽

Future Research ◽

Research Directions ◽

Numerical Examples ◽

Working Hypothesis ◽

Wide Range ◽

Future Research Directions

AbstractIt is known that the generating function f of a sequence of Toeplitz matrices {Tn(f)}n may not describe the asymptotic distribution of the eigenvalues of Tn(f) if f is not real. In this paper, we assume as a working hypothesis that, if the eigenvalues of Tn(f) are real for all n, then they admit an asymptotic expansion of the same type as considered in previous works, where the first function, called the eigenvalue symbol $\mathfrak {f}$ f , appearing in this expansion is real and describes the asymptotic distribution of the eigenvalues of Tn(f). This eigenvalue symbol $\mathfrak {f}$ f is in general not known in closed form. After validating this working hypothesis through a number of numerical experiments, we propose a matrix-less algorithm in order to approximate the eigenvalue distribution function $\mathfrak {f}$ f . The proposed algorithm, which opposed to previous versions, does not need any information about neither f nor $\mathfrak {f}$ f is tested on a wide range of numerical examples; in some cases, we are even able to find the analytical expression of $\mathfrak {f}$ f . Future research directions are outlined at the end of the paper.

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