We show that when $\set{X_j}$ is a sequence of independent (but not necessarily identically distributed) random variables which satisfies a condition similar to the Lindeberg condition, the properly normalized geometric sum $\sum_{j=1}^{ν_p}X_j$ (where $ν_p$ is a geometric random variable with mean $1/p$) converges in distribution to a Laplace distribution as $p\to 0$. The same conclusion holds for the multivariate case. This theorem provides a reason for the ubiquity of the double power law in economic and financial data.
In this paper, joint limit distributions of maxima and minima on independent and non-identically distributed bivariate Gaussian triangular arrays is derived as the correlation coefficient of $i$th vector of given $n$th row is the function of $i/n$. Furthermore, second-order expansions of joint distributions of maxima and minima are established if the correlation function satisfies some regular conditions.
We consider random walks with independent but not necessarily identical distributed increments. Assuming that the increments satisfy the well-known Lindeberg condition, we investigate the asymptotic behaviour of first-passage times over moving boundaries. Furthermore, we prove that a properly rescaled random walk conditioned to stay above the boundary up to time $n$ converges, as $n\to\infty$, towards the Brownian meander.
We consider the problem of finding the optimal upper bound for the tail probability of a sum of $k$ nonnegative, independent and identically distributed random variables with given mean $x$. For $k=1$ the answer is given by Markov's inequality and for $k=2$ the solution was found by Hoeffding and Shrikhande in 1955. We solve the problem for $k=3$ as well as for general $k$ and $x\leq1/(2k-1)$ by showing that it follows from the fractional version of an extremal graph theory problem of Erdős on matchings in hypergraphs.
A notion of conditionally identically distributed (c.i.d.) sequences has been studied as a form of stochastic dependence that is weaker than exchangeability, but is equivalent to exchangeability for stationary sequences. In this article we extend this notion to families of sequences. Paralleling the extension from exchangeability to partial exchangeability in the sense of de Finetti, we propose a notion of partially c.i.d. dependence, that is equivalent to partial exchangeability for stationary processes. Partially c.i.d. families of sequences preserve attractive limit properties of partial ex…
We derive strong laws of large numbers and central limit theorems for Bajraktarević, Gini and exponential- (also called Beta-type) and logarithmic Cauchy quotient means of independent identically distributed (i.i.d.) random variables. The exponential- and logarithmic Cauchy quotient means of a sequence of i.i.d. random variables behave asymptotically normal with the usual square root scaling just like the geometric means of the given random variables. Somewhat surprisingly, the multiplicative Cauchy quotient means of i.i.d. random variables behave asymptotically in a rather different way: in o…
On strong law of large numbers for non identically distributed random variables
M.V. Lomonosov Moscow State University（莫斯科国立罗蒙诺索夫大学）；Institute for Information Transmission Problems (A.A.Kharkevich Institute)（信息传输问题研究所（A.A.哈尔克维奇研究所））；RUDN University（俄罗斯人民友谊大学）；NRU HSE（高等经济国立大学）
Size-biased permutation of a finite sequence with independent and identically distributed terms
本文主要研究了独立同分布（i.i.d.）的$n$的大小偏置置换问题。正随机变量。这是Perman-Pitman-Yor（PPY）[Probab. Theory Related Fields 92（1992）21-39]，以及诱导的顺序统计量的特殊形式[Bull. Inst. Internat.中央集权主义者45（1973）295-300; Ann. Statist. 2（1974）1034-1039]。这个交集赋予我们不同的工具来推导分布特性。他们的比较导致了新的结果，以及现有的更简单的证明。我们的主要贡献，定理25在第6节，描述了最后几项的渐近分布在有限i.i.d.通过Poisson耦合和它的几个最小顺序统计量的大小偏置置换。
On the absolute constants in the Berry-Esseen type inequalities for identically distributed summands
通过对Korolev and Shevtsova，2010中应用的方法进行修改，这里证明了标准正态分布函数与任意数量$n\geq1$的独立同分布随机变量的归一化和的分布函数之间的均匀距离$Δ_n$的不等式$Δ_n\leq0.3328(β_3+0.429)/\sqrt{n}$和$Δ_n\leq0.33554(β_3+0.415)/\sqrt{n}$，这些随机变量具有零均值、单位方差和有限第三绝对矩$β_3$。这两个不等式中的第一个改进了在（Korolev and Shevtsova，2010）中证明的一个，并且也将经典Berry-Esseen不等式中绝对常数PDMATHAMR的最著名的上估计锐化为$C_0<0.4756$，因为凭借条件$β_3\geq1$，$0.3328(β_3+0.429)\leq0.3328\cdot1.429β_3<0.4756β_3$。第二个不等式也是经典Berry-Esseen不等式的一个结构性改进，并将$C_0$的上估计进一步改进为$C_0<0.4748$ .
A completion of counterexamples to the classical central limit theorem for pair- and triplewise independent and identically distributed random variables
本文完善了经典中心极限定理的反例研究，证明对于任意具有有限方差且不集中于单点的概率分布，均存在三重独立同分布随机变量序列使其标准化部分和依分布收敛于非正态分布，并可推广至 $ k $ 重独立性。
Conditionally identically distributed species sampling sequences
分布中的条件恒等式（Berti et al.（2004））是随机变量的一种新的依赖关系，它推广了众所周知的交换变量的概念。本文定义了一类随机序列，称为广义种群抽样序列，并给出了分布具有条件恒等式的条件。此外，还引入并研究了一类条件同分布的广义种群抽样序列：广义Ottawa序列（GOS）。这个类包含了波利亚瓮和布莱克威尔-麦奎因瓮方案的“随机复制”版本。对于GOS的经验平均值和预测平均值，我们证明了两个收敛结果向合适的混合高斯分布。第一个是稳定收敛意义下的，第二个是几乎必然条件收敛意义下的。在本文的最后一部分，我们研究了在时间$n$，即直到时间$n$的GOS的不同值的随机数的GOS诱导的分区的长度。在适当的条件下，证明了一个稳定收敛意义下的强大数律和一个中心极限定理。文中的所有结果都附有一些例子。
The complete convergence for weighted sums of sequences of independent, identically distributed random variables under sublinear expectations space was studied. By moment inequality and truncation methods, we establish the equivalent conditions of complete convergence for weighted sums of sequences of independent, identically distributed random variables under sublinear expectations space. The results extend the corresponding results obtained by Guo (2012) to those for sequences of independent, identically distributed random variables under sublinear expectations space.
We establish new sufficient conditions for the applicability of the strong law of large numbers (SLLN) for sequences of pairwise independent non-identically distributed random variables. These results generalize Etemadi's extension of Kolmogorov's SLLN for identically distributed random variables. Some of the obtained results hold with an arbitrary norming sequence in place of the classical normalization.
As an extension of a central limit theorem established by Svante Janson, we prove a Berry-Esseen inequality for a sum of independent and identically distributed random variables conditioned by a sum of independent and identically distributed integer-valued random variables.
It is known that limit theorems for triangular arrays with identically distributed rows yields convergence of densities rather than just convergence in distribution. We show that this superconvergence result holds -- at least at points at which the limit density is nonzero -- even if the rows of the array are not identically distributed.
We investigate the complete $p$-th moment convergence for weighted sums of independent, identically distributed random variables under sublinear expectations space. Using moment inequality and truncation methods, we prove the equivalent conditions of complete $p$-th moment convergence of weighted sums of independent, identically distributed random variables under sublinear expectations space, which complement the corresponding results obtained in Guo and Shan (2020).
The law of large numbers is one of the most fundamental results in Probability Theory. In the case of independent sequences, there are some known characterizations; for instance, in the independent and identically distributed setting it is known that the law of large numbers is equivalent to integrability. In the case of dependent sequences, there are no known general characterizations -- to the best of our knowledge. We provide such a characterization for identically distributed Bernoulli sequences in terms of a product disintegration.
Let $\{X_n;n\ge 1\}$ be a sequence of independent and identically distributed random variables on a sub-linear expectation space $(Ω,\mathscr{H},\widehat{\mathbb E})$, $S_n=X_1+\ldots+X_n$. We consider the moments of $\max_{n\ge 1}|S_n|/\sqrt{2n\log\log n}$. The sufficient and necessary conditions for the moments to be finite are given. As an application, we obtain the law of the iterated logarithm for moving average processes of independent and identically distributed random variables.
In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng, we give a theorem about the convergence of a random series and establish a three series theorem of independent random variables under the sub-linear expectations. As an application, we obtain the Marcinkiewicz's strong law of large numbers for independent and identically distributed random variables under the sub-linear expectations. The technical details are different from those for classical theorems because the sub-linear expectation and its re…
The aim of this paper is to study a dimorphic property associated with two different sums of identically independent Bernoulli random variables having two different families of probability mass functions. In addition, we give two expressions on sums of products of degenerate Stirling numbers of the second kind and Stirling numbers of the first kind connected with those two different sums of identically independent Bernoulli random variables.
In this paper, we establish some general forms of the law of the iterated logarithm for independent random variables in a sub-linear expectation space, where the random variables are not necessarily identically distributed. Exponential inequalities for the maximum sum of independent random variables and Kolmogorov's converse exponential inequalities are established as tools for showing the law of the iterated logarithm. As an application, the sufficient and necessary conditions of the law of iterated logarithm for independent and identically distributed random variables under the sub-linear ex…
Birkner et al. obtained necessary and sufficient conditions for the frequency between two independent and identically distributed continuous-state branching processes time-changed by a functional of the total mass process to be a Markov process. Foucart et al. extended this result to continuous-state branching processes with immigration. We generalize these results by dropping the independent and identically distributed assumption. Our result clarifies under which conditions a multi-type $Λ$-coalescent can be constructed from a multi-type branching process by a time change using the total mass…
Let $\{Y_i,-\infty<i<\infty\}$ be a doubly infinite sequence of identically distributed, negatively dependent random variables under sub-linear expectations, $\{a_i,-\infty<i<\infty\}$ be an absolutely summable sequence of real numbers. In this article, we study complete convergence and Marcinkiewicz-Zygmund strog law of large numbers for the partial sums of moving average processes $\{X_n=\sum_{i=-\infty}^{\infty}a_{i}Y_{i+n},n\ge 1\}$ based on the sequence $\{Y_i,-\infty<i<\infty\}$ of identically distributed, negatively dependent random variables under sub-linear expectations, complementing…
The validity of the strong law of large numbers for multiple sums $S_n$ of independent identically distributed random variables $Z_k$, $k\leq n$, with $r$-dimensional indices is equivalent to the integrability of $|Z|(\log^+|Z|)^{r-1}$, where $Z$ is the typical summand. We consider the strong law of large numbers for more general normalisations, without assuming that the summands $Z_k$ are identically distributed, and prove a multiple sum generalisation of the Brunk--Prohorov strong law of large numbers. In the case of identical finite moments of irder $2q$ with integer $q\geq1$, we show that …
Classical Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of random variables are basic tools for studying the strong laws of large numbers. In this paper, motived by the notion of independent and identically distributed random variables under the sub-linear expectation initiated by Peng (2006, 2008b), we introduce the concept of negative dependence of random variables and establish Kolmogorov's and Rosenthal's inequalities for the maximum partial sums of negatively dependent random variables under the sub-linear expectations. As an application, we show that Kolmogorov's…
We present a formalization of the well-known thesis that, in the case of independent identically distributed random variables $X_1,\dots,X_n$ with power-like tails of index $α\in(0,2)$, large deviations of the sum $X_1+\dots+X_n$ are primarily due to just one of the summands.
