The asymptotic variance and distribution of Spearman's rank correlation have previously been known only under independence. For variables with finite support, the population version of Spearman's rank correlation has been derived. Using this result, we show convergence to a normal distribution irrespectively of dependence, and derive the asymptotic variance. A small simulation study indicates t...

This paper presents some asymptotical results of the Hermite Normality test previously introduced. We show that the Hermite statistic S H is distributed under the null hypothesis as a quadratic form of normal variates and under the nonnull hypothesis as normal. The special case of tests with two polynomials is studied in details. Finally, we give some considerations for the choice of a best Her...

The asymptotic distribution of the Markowitz portfolio, Σ̂μ̂, is derived, for the general case (assuming fourth moments of returns exist), and for the case of multivariate normal returns. The derivation allows for inference which is robust to heteroskedasticity and autocorrelation of moments up to order four. As a side effect, one can estimate the proportion of error in the Markowitz portfolio du...

The first complete running time analysis of a stochastic divide and conquer algorithm was given for Quicksort, a sorting algorithm invented 1961 by Hoare. We analyse here the variant Random Median Quicksort. The analysis includes the expectation, the asymptotic distribution, the moments and exponential moments. The asymptotic distribution is characterized by a stochastic fixed point equation. T...

Let f(n) be an arbitrary arithmetical function and let A N and B N be sequences of real numbers with 0< B N . + oe with N. We give a sufficient condition for (f(n)-AN)/BN to have a limiting distribution. The case when f(n) is defined by f(n ) = Zg(d), where the summation is over all divisors d of n and g(d) is any given arithmetical function, is discussed in more detail . A concrete example is ...

Let ψ : [0, 1] → R be a strictly increasing continuous function. Let Pn be a polynomial of degree n determined by the biorthogonality conditions ∫ 1 0 Pn (x)ψ (x) j dx = 0, j = 0, 1, . . . , n− 1. We study the distribution of zeros of Pn as n → ∞, and related potential theory.

We study the asymptotic eigenvalue distribution of Toeplitz matrices generated by a singular symbol. It has been conjectured by Widom that, for a generic symbol, the eigenvalues converge to the image of the symbol. In this paper we ask how the eigenvalues converge to the image. For a given Toeplitz matrix Tn(a) of size n, we take the standard approach of looking at det(ζ − Tn(a)), of which the ...

In this correspondence we derive the asymptotic distribution of the MUSIC null spectrum, from which an exact expression of the asymptotic variance of the MUSIC null spectrum can be obtained. The result presented in this correspondence is a simpler alternative and a special case of a more general result recently obtained by Lee and Wengrovitz.