相关论文: Some results on two-sided LIL behavior

Let X_1,X_2, . . . be a sequence of i.i.d. mean zero random variables and let S_n the sum of the first n random variables. We show that whenever lim sup_n |S_n|/c_n is finite with probability one and the normalizing sequence {c_n} is…

Let $\{X,X_n,n\ge 1\}$ be a sequence of identically distributed, negatively dependent (NA) random variables under sub-linear expectations, and denote $S_n=\sum_{i=1}^{n}X_i$, $n\ge 1$. Assume that $h(\cdot)$ is a positive non-decreasing…

A milestone in Probability Theory is the law of the iterated logarithm (LIL), proved by Khinchin and independently by Kolmogorov in the 1920s, which asserts that for iid random variables $\{t_i\}_{i=1}^{\infty}$ with mean $0$ and variance…

Let X,X_1,X_2,... be independent identically distributed random variables and let h(x,y)=h(y,x) be a measurable function of two variables. It is shown that the bounded law of the iterated logarithm, $\limsup_n (n\log\log n)^{-1}|\sum_{1<=…

Let denote $S_n(p) = k_n^{-1} \sum_{i=1}^{k_n} \left( \log (X_{n+1-i,n} / X_{n-k_n, n}) \right)^p$, where $p > 0$, $k_n \leq n$ is a sequence of integers such that $k_n \to \infty$ and $k_n / n \to 0$, and $X_{1,n} \leq \ldots \leq X_{n,n}$…

We study the almost sure behaviour of suitably normalised multivariate Levy processes as t goes to zero. Among other results we find necessary and sufficient conditions for a law of a very slowly varying function which includes a general…

Let $X_p, p\in\cP$ be a sequence of independent random variables s.t. $\bbP(X_p=\pm 1)=1/2$. Let $\te_j=\prod_{p|j}X_p$ if $j$ is square free and $\te_j=0$ otherwise. Denote $S_n=\sum_{\ell=1}^n\te_\ell$. The from this point of view proving…

Let $(X_n)_{n\in \mathbb{N}}$ be a sequence of i.i.d. random variables with distribution $\mathbb P(X_1=1)=\mathbb P(X_1=-1)=1/2$. Let $F(\sigma)=\sum_{n=1}^\infty X_nn^{-\sigma}$. We prove that the following holds almost surely…

There has been recent interest in the conditional central limit question for (strictly) stationary, ergodic processes $...,X_{-1},X_0,X_1,...$ whose partial sums $S_n=X_1+...+X_n$ are of the form $S_n=M_n+R_n$, where $M_n$ is a square…

Let $S_n$ be the sum of independent random variables with distribution $F$. Under the assumption that $-\log(1-F(x))$ is slowly varying, conditions for $$ \lim_{n\to\infty}\sup_{s\ge t_n}\left|{P[S_n>s]\over n(1-F(s))}-1\right| =0 $$ are…

Let $S=(S_n)$ be an oscillatory random walk on the integer lattice $\mathbb{Z}$ with i.i.d. increments. Let $V_{{\rm d}}(x)$ be the renewal function of the strictly descending ladder height process for $S$. We obtain several sufficient…

Take a centered random walk S_n and consider the sequence of its partial sums A_n = S_1 + ... + S_n. Suppose S_1 is in the domain of normal attraction of an \alpha-stable law with 1 < \alpha <= 2. Assuming that S_1 is either…

Let $X$, $X_1$, $X_2$, $...$ be i.i.d. random variables, and let $S_n=X_1+... + X_n$ be the partial sums and $M_n=\max_{k\le n}|S_k|$ be the maximum partial sums. We give the sufficient and necessary conditions for a kind of limit theorems…

Let $X=\{X_n: n\in\mathbb{N}\}$ be a long memory linear process in which the coefficients are regularly varying and innovations are independent and identically distributed and belong to the domain of attraction of an $\alpha$-stable law…

Let $\{X_i,i=1,2,...\}$ be i.i.d. standard gaussian variables. Let $S_n=X_1+...+X_n$ be the sequence of partial sums and $$ L_n=\max_{0\leq i<j\leq n}\frac{S_j-S_i}{\sqrt{j-i}}. $$ We show that the distribution of $L_n$, appropriately…

We prove a central limit theorem for random sums of the form $\sum_{i=1}^{N_n} X_i$, where $\{X_i\}_{i \geq 1}$ is a stationary $m-$dependent process and $N_n$ is a random index independent of $\{X_i\}_{i\geq 1}$. Our proof is a…

Let $X_1$, $X_2$,... be a sequence of independent random variables with common distribution function $F$ in the domain of attraction of a Gumbel extreme value distribution and for each integer $n\geq 1$, let $X_{1,n} \leq ... X_{n,n}$…

The article studies the almost surely asymptotics of extreme values $\bar{\xi}_n = \max_{1\leq i \leq n} \xi_i$, where $ \xi , \xi_1 , \xi_2 , \ldots$ are discrete identically distributed random variables. One of the main results on this…

Let $X=\{X_n: n\in\mathbb{N}\}$ be a linear process in which the coefficients are of the form $a_i=i^{-1}\ell(i)$ with $\ell$ being a slowly varying function at the infinity and the innovations are independent and identically distributed…

Consider a random walk $S_n=\sum_{i=1}^n X_i$ with independent and identically distributed real-valued increments $X_i$ of zero mean and finite variance. Assume that $X_i$ is non-lattice and has a moment of order $2+\delta$. For any $x\geq…