Aug 2026· Scientia Sinica Mathematica· 0 citations· 16 references
Mathematics
Abstract
Let $\{Y_n; n\ge 1\}$ be a sequence of independent and identically distributed random variables with mean zero in Peng's framework of the sub-linear expectation space $(\Omega,\mathscr{H},\widehat{\mathbb E})$, and $S_n=\sum_{i=1}^nY_i$. In this paper, we establish a limit law of \begin{align*}\lim_{n\to \infty}\max_{k\le n}\frac{S_k}{\sqrt{2k \log\log n}}. \end{align*} Different from the result obtained by Chen (2015) in which the limit is a constant, it is shown that under the upper capacity the limit may be prescribed as a given function of $Y_1,Y_2,\ldots$, taking values in the standard deviation interval. As a result, it is also shown that the set of limit points in the compact law of the iterated logarithm can be a symmetric random interval. This paper (Chinese version) has been submitted to Special Issue of Science in China-Mathematics in Celebration of Professor Peng Shige's 80th Birthday. In Theorem 2.2 of the original paper, an additional condition (2.6) is needed.
Let $A=(A_n)_{n\ge2}$ be a triangular array of random matrices, where $A_n=(a_{ij})_{1\le i,j\le n}$ is an $n\times n$ random matrix with independent real entries satisfying $\mathbb E a_{ij}=0$ and $\mathbb Ea_{ij}^2=1$, and put $\mathcal L_n=\log|\det A_n|$ and \[ W_n^{\mathrm d}(A_n):=\frac{\mathcal L_n - \frac12\lo...
Let $X$ be a centered isotropic log-concave random vector in $\mathbb{R}^n$. For $\theta\in S^{n-1}$, let $\mu_\theta$ be the law of $\langle X,\theta\rangle$, and let $\Theta$ be uniformly distributed on $S^{n-1}$, independently of $X$. We prove the sharp estimate \[ \textsf{E} W_1(\mu_\Theta,\gamma_1) \le \frac{C}{n}...
For each $s>0$, we consider a random Dirichlet series $X(s)=\sum_{k\geq 1}k^{-1/2-s}a_k\eta_k$, where $\eta_1$, $\eta_2,\ldots$ are independent and identically distributed random variables with mean zero and finite positive variance, and $(a_k)_{k\geq 1}$ is a deterministic sequence of real numbers satisfying $\sum_{k\...
Consider $n$ independent, non-negative, mean at most one random variables, $X_1,X_2,\ldots$. We show the following bound on the probability of their sum exceeding a threshold $t$: \[ \mathbb{P}\left[\sum_{i=1}^n X_i\ge t\right] \leq 1-\left(1-\frac{1}{t}\right)^n \text{ for all } t\ge 2n+1 \,. \] To prove this, we cons...
Let $W=n^{-1/2}\sum_{i=1}^n X_i$, where the $X_i$ are independent centered random vectors in ${\mathbb R}^p$ with $|X_{ij}|\le B$ almost surely. Suppose that $\text{Cov}(W)$ has unit diagonal and smallest eigenvalue at least $b^2>0$. We prove that the distance between $W$ and a Gaussian vector with the same covariance,...
For a positive integer $n$, let $$S_0(n)=\{1,2,\ldots,\lfloor n/2\rfloor\},\qquad S_{j+1}(n)=\{n\bmod k:k\in S_j(n)\setminus\{0\}\},$$ and put $s_j(n) := |S_j(n)|$. The sets defined above arise naturally in the study of the length of the Pierce series expansion of a rational number. In \cite{Ba-Vu}, Baraskar and Vukusi...
Omkar Baraskar, Prashant Gokhale, Sarvagya Jain et al.· 1 citation
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