相关论文: Explicit Universal Bounds for Cumulants via Moment…

In this article, we show that a linear combination $X$ of $n$ independent, unbiased Bernoulli random variables $\{X_k\}$ can match the first $2n$ moments of a random variable $Y$ which is uniform on an interval. More generally, for each $p…

We provide some upper bounds for the Mertens function ($M(n)$: the cumulative sum of the M$\ddot{\mathrm{o}}$bius function) by an approach of statistical mechanics, in which the M$\ddot{\mathrm{o}}$bius function is taken as a particular…

We combine the fixed-order evaluation of the $b\bar{b}$ sum rules with a non-relativistic effective-theory approach. The combined result for the $n$-th moment includes all terms suppressed with respect to the leading-order result by ${\cal…

Let $M_n$ be the number of steps of the loop-erasure of a simple random walk on $\mathbb{Z}^2$ from the origin to the circle of radius $n$. We relate the moments of $M_n$ to $Es(n)$, the probability that a random walk and an independent…

Let $X_1, X_2, \dots, X_n$ be independent and uniformly distributed random variables in the unit square $[0,1]^2$ and let $L(X_1, \dots, X_n)$ be the length of the shortest traveling salesman path through these points. In 1959, Beardwood,…

This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{eqnarray*} \begin{array}{llc} u_t=\Delta u-\chi\nabla\cdot (u\nabla v)+\kappa u-\mu u^2,\\ v_t=\Delta v-uv, \end{array}…

A runtime analysis of the Univariate Marginal Distribution Algorithm (UMDA) is presented on the OneMax function for wide ranges of its parameters $\mu$ and $\lambda$. If $\mu\ge c\log n$ for some constant $c>0$ and…

In the paper we develop an approach to asymptotic normality through factorial cumulants. Factorial cumulants arise in the same manner from factorial moments, as do (ordinary) cumulants from (ordinary) moments. Another tool we exploit is a…

For an array $\left\{X_{n,j}, \, 1 \leqslant j \leqslant k_{n}, n \geqslant 1 \right\}$ of random variables and a sequence $\{c_{n} \}$ of positive numbers, sufficient conditions are given under which, for all $\varepsilon > 0$,…

For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $$ P_N(\theta) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(n\theta), $$ and show that the $2k$-th moments on the unit circle $$ \int_0^1…

This note displays an interesting phenomenon for percentiles of independent but non-identical random variables. Let $X_1,\cdots,X_n$ be independent random variables obeying non-identical continuous distributions and $X^{(1)}\geq \cdots\geq…

The dynamics of quantum fields become nonperturbative when their interactions are probed by a large number of particles. To explore this regime we study correlation functions which involve a large number of fields, focussing on massive…

Spectral properties of Gram matrices are central to high dimensional asymptotic analyses of statistical estimators in regression and covariance estimation. These properties, in turn, depend critically on the extreme singular values and…

Chen [Ann. Appl. Probab. {\bf 11} (2001), 1242--1262] derived exact convergence rates in a central limit theorem and a local limit theorem for a supercritical branching Wiener process.We extend Chen's results to a branching random walk…

Let $\phi(n)$ be the Euler totient function and $\phi_k(n)$ its $k$-fold iterate. In this note, we improve the upper bound for the number of positive $n\leqslant x$ such that $\phi_{k+1}(n)\geqslant cn$. Comparing with the upper bound which…

We present the alternative derivation of the excluded volume equation. The resulting equation is mathematically identical to the one proposed in the preceding paper. As a result, the theory reproduces well the observed points by SANS (small…

We establish completely log-free bounds for exponential sums over the primes and the M\"{o}bius function. Let $0<\eta \leq 1/10$, and suppose $\alpha = a/q + \delta/x$, with $(a,q)=1$ and $|\delta| \leq x^{1/5 + \eta}/q$, and set $\delta_0…

We present an analytic method for computing the moments of a sum of independent and identically distributed random variables. The limiting behavior of these sums is very important to statistical theory, and the moment expressions that we…

We derive sharper probabilistic concentration bounds for the Monte Carlo Empirical Rademacher Averages (MCERA), which are proved through recent results on the concentration of self-bounding functions. Our novel bounds are characterized by…

We address the question of the asymptotic description of random tensors that are local-unitary invariant, that is, invariant by conjugation by tensor products of independent unitary matrices. We consider both the mixed case of a tensor with…