Absorption probabilities for Gaussian polytopes and regular spherical simplices

The Gaussian polytope $\mathcal P_{n,d}$ is the convex hull of n independent standard normally distributed points in $\mathbb{R}^d$ . We derive explicit expressions for the probability that $\mathcal P_{n,d}$ contains a fixed point $x\in\mathbb{R}^d$ as a function of the Euclidean norm of x, and the probability that $\mathcal P_{n,d}$ contains the point $\sigma X$ , where $\sigma\geq 0$ is constant and X is a standard normal vector independent of $\mathcal P_{n,d}$ . As a by-product, we also compute the expected number of k-faces and the expected volume of $\mathcal P_{n,d}$ , thus recovering the results of Affentranger and Schneider (Discr. and Comput. Geometry, 1992) and Efron (Biometrika, 1965), respectively. All formulas are in terms of the volumes of regular spherical simplices, which, in turn, can be expressed through the standard normal distribution function $\Phi(z)$ and its complex version $\Phi(iz)$ . The main tool used in the proofs is the conic version of the Crofton formula.

ON THE GENERALISATION OF SIDEL’NIKOV’S THEOREM TO -ARY LINEAR CODES

We generalise Sidel’nikov’s theorem from binary codes to $q$-ary codes for $q>2$. Denoting by $A(z)$ the cumulative distribution function attached to the weight distribution of the code and by $\unicode[STIX]{x1D6F7}(z)$ the standard normal distribution function, we show that $|A(z)-\unicode[STIX]{x1D6F7}(z)|$ is bounded above by a term which tends to $0$ when the code length tends to infinity.

Optimal averaging procedures in almost sure central limit theory

Let \(X_1, X_2, \ldots\) be i.i.d. random variables with \(\mathbb{E}[X_1] = 0\), \(\mathbb{E}[X_1^2] = 1\), \(S_n = X_1 + \cdots + X_n\) and let \((d_k)\) be a positive numerical sequence. We investigate the a.s. convergence of the averages \[\frac{1}{D_N} \sum_{k = 1}^{N} d_k I \{S_k / \sqrt{k} \leq x\},\]where \(D_N = \sum_{k = 1}^{N} d_k\). In the case of \(d_k = 1/k\) we have logarithmic means and by the almost sure central limit theorem the above averages converge a.s. to \(\Phi(x)\), the standard normal distribution function. It is also known that the analogous convergence relation fails for \(d_k = 1\) (ordinary averages). In this paper we give a fairly complete solution of the problem for which weight sequences the above convergence relation and its refinements hold.