Related papers: Average first-passage times for character sums

A simple lemma bounds $\mathrm{s.d.}(T)/\mathbb{E} T$ for hitting times $T$ in Markov chains with a certain strong monotonicity property. We show how this lemma may be applied to several increasing set-valued processes. Our main result…

A sequence $\textbf{p}=(p_{n})$ of real numbers is called Abel convergent to $\ell$ if the series $\Sigma_{k=0}^{\infty}p_{k}x^{k}$ is convergent for $0\leq x<1$ and \[\lim_{x \to 1^{-}}(1-x) \sum_{k=0}^{\infty}p_{k}x^{k}=\ell.\] We…

Let $s$ be a string of length $n$ over an alphabet of constant size $\sigma$ and let $c$ and $\epsilon$ be constants with (1 \geq c \geq 0) and (\epsilon > 0). Using (O (n)) time, (O (n^c)) bits of memory and one pass we can always encode…

We consider first-passage percolation on $\mathbb{Z}^2$ with i.i.d. weights, whose distribution function satisfies $F(0) = p_c = 1/2$. This is sometimes known as the "critical case" because large clusters of zero-weight edges force passage…

Let $\Omega(n)$ denote the number of prime factors of $n$. We show that for any bounded $f\colon\mathbb{N}\to\mathbb{C}$ one has \[ \frac{1}{N}\sum_{n=1}^N\, f(\Omega(n)+1)=\frac{1}{N}\sum_{n=1}^N\, f(\Omega(n))+\mathrm{o}_{N\to\infty}(1).…

We study the rate of convergence in the Shape Theorem of first-passage percolation, obtaining the precise asymptotic rate of decay for the probability of linear order deviations under a moment condition. Our results are stated for a given…

We observe an infinite sequence of independent identically distributed random variables $X_1,X_2,\ldots$ drawn from an unknown distribution $p$ over $[n]$, and our goal is to estimate the entropy $H(p)=-\mathbb{E}[\log p(X)]$ within an…

Let $X$ be an isotropic random vector in $R^d$ that satisfies that for every $v \in S^{d-1}$, $\|<X,v>\|_{L_q} \leq L \|<X,v>\|_{L_p}$ for some $q \geq 2p$. We show that for $0<\varepsilon<1$, a set of $N = c(p,q,\varepsilon) d$ random…

We consider a run-and-tumble particle on a finite interval $[a,b]$ with two absorbing end points. The particle has an internal velocity state that switches between three values $v,0,-v$ at exponential times, thus incorporating positive…

We prove two lower bounds for stopping times of sequential tests between general composite nulls and alternatives. The first lower bound is for the setting where the type-1 error level $\alpha$ approaches zero, and equals $\log(1/\alpha)$…

The main result is the following Theorem: Let p=p(n) be such that p(n) in [0,1] for all n and either p(n)<< n^{-1} or for some positive integer k, n^{-1/k}<< p(n)<< n^{-1/(k+1)} or for all epsilon >0, n^{- epsilon}<< p(n) and n^{-…

For a one-dimensional simple random walk, let $g_1(n)$ denote the number of sites visited exactly once at time $n$. Major (1988) proved that \begin{equation*} \limsup_{n\to\infty}\frac{g_1(n)}{\log^2 n}=C\qquad a.s. \end{equation*} where…

We investigate the first moment of primes in progressions $$ \sum_{\substack{q\leq x/N \\ (q,a)=1}} \Big(\psi(x; q, a) - \frac x{\varphi(q)}\Big) $$ as $x, N \to \infty$. We show unconditionally that, when $a=1$, there is a significant bias…

For arbitrary $c_0>0$, if $A$ is a subset of the primes less than $x$ with cardinality $\delta x (\log x)^{-1}$ with $\delta\geq (\log x)^{-c_0}$, then there exists a positive constant $c$ such that the cardinality of $A+A$ is larger than…

Consider (independent) first-passage percolation on the sites of the triangular lattice $\mathbb{T}$. Denote the passage time of the site $v$ in $\mathbb{T}$ by $t(v)$, and assume that $P(t(v)=0)=P(t(v)=1)=1/2$. Denote by $b_{0,n}$ the…

In [2], it was claimed that the time constant $\mu_{d}(e_{1})$ for the first-passage percolation model on $\mathbb Z^{d}$ is $\mu_{d}(e_{1}) \sim \log d/(2ad)$ as $d\to \infty$, if the passage times $(\tau_{e})_{e\in \mathbb E^{d}}$ are…

Let $\xi_0,\xi_1,\ldots$ be i.i.d. random variables with zero mean and unit variance. Consider a random Taylor series of the form $f(z)=\sum_{k=0}^\infty \xi_k c_k z^k$, where $c_0,c_1,\ldots$ is a real sequence such that $c_n^2$ is…

Let $p$ be a prime number, $\mathbb{F}_{p^n}$ be the finite field of order $p^n$, and $\{\omega_1,\ldots\omega_n\}$ be a basis of $\mathbb{F}_{p^n}$ over $\mathbb{F}_p$. Let, further, $N_i,H_i$ be integers such that $1\leq H_i\leq p$,…

Let $\pi$ be a $SL(3,\mathbb{Z})$ Hecke Maass-cusp form, $f$ be a $SL(2,\mathbb{Z})$ holomorphic cusp form or Maass-cusp form with normalized Fourier coefficients $\lambda_{\pi}(r,n) \text{ and }\lambda_{f}(n)$ respectively and $\chi$ be…

For a given Dirichlet character $\chi (n) = e^{i \theta_n}$, we prove central limit theorems for the series $\sum_{p'} \cos \theta_{p'}$ for non-principal characters, and $\sum_{p' } \cos (t \log p')$ for principal characters, where $p'$…