The asymptotic distribution of the rank for unimodal sequences

Asymptotic distribution of odd balanced unimodal sequences with rank congruent to a modulo c

Research in Number Theory ◽

10.1007/s40993-021-00298-2 ◽

2021 ◽

Vol 8 (1) ◽

Author(s):

Taylor Garnowski

Keyword(s):

Partition Function ◽

Modular Form ◽

Asymptotic Distribution ◽

Generating Function ◽

Modular Forms ◽

Asymptotic Estimate ◽

Asymptotic Properties ◽

Unimodal Sequences ◽

The World ◽

Mock Modular Form

AbstractKim et al. (Proc Am Math Soc 144:687–3700, 2016) introduced the notion of odd-balance unimodal sequences in 2016. Like was shown by Bryson et al. (Proc Natl Acad Sci USA 109:16063–16067, 2012) for the generating function of strongly unimodal sequences, the generating function for odd-balanced unimodal sequences also has quantum modular behavior. Odd-balanced unimodal sequences thus appear to be a fundamental piece in the world of modular forms and combinatorics, and understanding their asymptotic properties is important for understanding their place in this puzzle. In light of this, we compute an asymptotic estimate for odd balanced unimodal sequences for ranks congruent to $$a \pmod {c}$$ a ( mod c ) for $$c\ne 2$$ c ≠ 2 or a multiple of 4. We find the interesting result that the odd balanced unimodal sequences are asymptotically related to the overpartition function. This is in contrast to strongly unimodal sequences which, are asymptotically related to the partition function. Our proofs of the main theorems rely on the representation of the generating function in question as a mixed mock modular form.

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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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