Related papers: The Reverse of The Law of Large Numbers
The success of today's large language models (LLMs) depends on the observation that larger models perform better. However, the origin of this neural scaling law, that loss decreases as a power law with model size, remains unclear. We…
Let $(d_n)$ be a sequence of positive numbers and let $(X_n)$ be a sequence of positive independent random variables. We provide an upper bound for the deviation between the distribution of the mantissaes of $(X_n^{d_n})$ and the Benford's…
We consider a family of positive operator valued measures associated with representations of compact connected Lie groups. For many independent copies of a single state and a tensor power representation we show that the observed probability…
We derive a necessary and sufficient condition for the sum of M independent continuous random variables modulo 1 to converge to the uniform distribution in L^1([0,1]), and discuss generalizations to discrete random variables. A consequence…
Given p independent normal populations, we consider the problem of estimating the mean of those populations, that based on the observed data, give the strongest signals. We explicitly condition on the ranking of the sample means, and…
We show that the left (right) sample quantile tends to the left (right) distribution quantile at p in [0,1], if the left and right quantiles are identical at p. We show that the sample quantiles diverge almost surely otherwise. The latter…
Standard neutral population genetics theory with a strictly fixed population size has important limitations. An alternative model that allows independently fluctuating population sizes and reproduces the standard neutral evolution is…
Consider Ginibre's ensemble of $N \times N$ non-Hermitian random matrices in which all entries are independent complex Gaussians of mean zero and variance $\frac{1}{N}$. As $N \uparrow \infty$ the normalized counting measure of the…
We consider the problem of finding, for a given quadratic measure of non-uniformity of a set of $N$ points (such as $L_2$ star-discrepancy or diaphony), the asymptotic distribution of this discrepancy for truly random points in the limit…
We show an extension of Sanov's theorem on large deviations, controlling the tail probabilities of i.i.d. random variables with matching concentration and anti-concentration bounds. This result has a general scope, applies to samples of any…
We prove a strong law of large numbers for simultaneously testing parameters of a large number of dependent, Lancaster bivariate random variables with infinite supports, and discuss its implications.
The literature on "mechanism design from samples," which has flourished in recent years at the interface of economics and computer science, offers a bridge between the classic computer-science approach of worst-case analysis (corresponding…
It is known that increasing powers of a continuous random variable converge in distribution to Benford's law as the exponent approaches infinity. The rate of convergence has been estimated using Fourier analysis, but we present an…
Equip the edges of the lattice $\mathbb{Z}^2$ with i.i.d. random capacities. We prove a law of large numbers for the maximal flow crossing a rectangle in $\mathbb{R}^2$ when the side lengths of the rectangle go to infinity. The value of the…
Large Language Models (LLMs) are increasingly utilized in autonomous decision-making, where they sample options from vast action spaces. However, the heuristics that guide this sampling process remain under explored. We study this sampling…
The strong law of large numbers for linear combinations of functions of order statistics ($L$-statistics) based on weakly dependent random variables is proven. We also establish the Glivenko--Cantelli theorem for $\phi$-mixing sequences of…
We analyze the asymptotic behavior of random variables $x(n,x\_0)$ defined by $x(0,x\_0)=x\_0$ and $x(n+1,x\_0)=A(n)x(n,x\_0)$, where $\sAn$ is a stationary and ergodic sequence of random matrices with entries in the semi-ring…
The probability that the sum of independent, centered, identically distributed, heavy-tailed random variables achieves a very large value is asymptotically equal to the probability that there exists a single summand equalling that value. We…
Rejection sampling is a popular method used to generate numbers that follow some given distribution. We study the use of this method to generate random numbers in the unit interval from increasing probability density functions. We focus on…
Machine learning models that are developed with invariance to certain types of data transformations have demonstrated superior generalization performance in practice. However, the underlying mechanism that explains why invariance leads to…