Paweł Hitczenko, Stephen Montgomery-Smith
In this note a two sided bound on the tail probability of sums of independent, and either symmetric or nonnegative, random variables is obtained. We utilize a recent result by Latała on bounds on moments of such sums. We also give a new proof of Latała's result for nonnegative random variables, and improve one of the constants in his inequality.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
In literature, the central limit theorems for the product of sums of various random variables have studied. The purpose of this note is to show that this kind of results are corollary of the invariance principle.
This note provides the Stein equation for weighted sums of independent $χ^{2}$ distributions.
We present two short proofs giving the best known asymptotic lower bound for the maximum element in a set of $n$ positive integers with distinct subset sums.
Let $\bX=\{X_n\}_{n\geq 1}$ and $\bY=\{Y_n\}_{n\geq 1}$ be two independent random sequences. We obtain rates of convergence to the normal law of randomly weighted self-normalized sums $$ ψ_n(\bX,\bY)=\sum_{i=1}^nX_iY_i/V_n,\quad V_n=\sqrt{Y_1^2+...+Y_n^2}. $$ These rates are seen to hold for the convergence of a number of important statistics, such as for instance Student's $t$-statistic or the empirical correlation coefficient.
Kwapien and Woyczynski asked in their monograph (1992) whether their notion of superstrong domination is inherited when taking sums of independent symmetric random vectors (one vector dominates another if, essentially, tail probabilities of any norm of the two vectors compare up to some scaling constants). We answer this question positively. As a by-product of our methods, we establish that a certain notion of weak concentration is also preserved by taking sums of independent symmetric random vectors.
This paper considers how to measure the magnitude of the sum of independent random variables in several ways. We give a formula for the tail distribution for sequences that satisfy the so called Levy property. We then give a connection between the tail distribution and the pth moment, and between the pth moment and the rearrangement invariant norms.
This note studies the asymptotic properties of the variable $$Z_d:=\frac{X_1}{d}|\{X_1+X_2=d\},$$ as $d\to \infty$. Here $X_1$ and $X_2$ are non-negative i.i.d. variables with a common twice differentiable density function $f$. General results concerning the distributional limits of $Z_d$ are discussed with various examples. Eventual log-convexity or log-concavity of $f$ turns out to be the key ingredient that determines how the variable $Z_d$ behaves. As a consequence, two surprising discoveries are presented: Firstly, it is noted that the distributional limit is not strictly determined by th…
We derive a central limit theorem for sums of a function of independent sums of independent and identically distributed random variables. In particular we show that previously known result from Rempa\la and Weso\lowski (Statist. Probab. Lett. 74 (2005) 129--138), which can be obtained by applying the logarithm as the function, holds true under weaker assumptions.
Our starting point is an improved version of a result of D. Hajela related to a question of Komlós: we show that if $f(n)$ is a function such that $\lim\limits_{n\to\infty }f(n)=\infty $ and $f(n)=o(n)$, there exists $n_0=n_0(f)$ such that for every $n\geqslant n_0$ and any $S\subseteq \{-1,1\}^n$ with cardinality $|S|\leqslant 2^{n/f(n)}$ one can find orthonormal vectors $x_1,\ldots ,x_n\in {\mathbb R}^n$ that satisfy $$\|ε_1x_1+\cdots +ε_nx_n\|_{\infty }\geqslant c\sqrt{\log f(n)}$$ for all $(ε_1,\ldots ,ε_n)\in S$. We obtain analogous results in the case where $x_1,\ldots ,x_n$ are independ…
This paper re-examines the density for sums of independent exponential, Erlang and gamma random variables. By using a divided difference perspective, the paper provides a unified approach to finding closed-form formulae for such convolutions. In particular, the divided difference perspective for sums of Erlang variates suggests a new approach to finding the density for sums of independent gamma variates using fractional calculus.
We investigate the sums $(1/\sqrt{H}) \sum_{X < n \leq X+H} χ(n)$, where $χ$ is a fixed non-principal Dirichlet character modulo a prime $q$, and $0 \leq X \leq q-1$ is uniformly random. Davenport and Erdős, and more recently Lamzouri, proved central limit theorems for these sums provided $H \rightarrow \infty$ and $(\log H)/\log q \rightarrow 0$ as $q \rightarrow \infty$, and Lamzouri conjectured these should hold subject to the much weaker upper bound $H=o(q/\log q)$. We prove this is false for some $χ$, even when $H = q/\log^{A}q$ for any fixed $A > 0$. On the other hand, we show it is true…
Known Bernstein-type upper bounds on the tail probabilities for sums of independent zero-mean sub-exponential random variables are improved in several ways at once. The new upper bounds have a certain optimality property.
We deduce in this short report the non-asymptotic for exponential tail of distribution for sums of independent centered random variables.
We study how well moments of sums of independent symmetric random variables with logarithmically concave tails may be approximated by moments of Gaussian random variables.
We study the exact constants in the moment inequalities for sums of centered independent random variables: improve their asymptotics, low and upper bounds, calculate more exact asymptotics, elaborate the numerical algorithm for their calculation, study the class of smoothing etc.
We modify the classical Bernstein's inequality for the sums of independent centered random variables (r.v.) in the terms of relative tails or moments. We built also some examples in order to show the exactness of offered results.
We derive in this article the exact non-asymptotical exponential and power estimates for self-normalized sums of centered independent random variables (r.v.) under natural norming. We will use also the theory of the so-called Grand Lebesgue Spaces (GLS) of random variables.
We continue the research of Latała on improving estimates of $p$-th moments of sums of independent random variables. We generalize some of his results in the case when $2 \leq p \leq 4$ and present a combinatorial approach for even moments.
This paper develops Rio's method [C. R. Acad. Sci. Paris Sér. I Math., 1995] to prove the weak law of large numbers for maximal partial sums of pairwise independent random variables. The method allows us to avoid using the Kolmogorov maximal inequality. As an application, a weak law of large numbers for maximal partial sums of pairwise independent random variables under a uniform integrability condition is also established. The sharpness of the result is illustrated by an example.
Tail bounds for sums of independent two-sided exponential random variables
本文为双边指数随机变量加权和建立匹配主项的尾部上下界，推广了Janson的单边指数结果。
In this article, we consider Poisson and Poisson convoluted geometric approximation to the sums of $n$ independent random variables under moment conditions. We use Stein's method to derive the approximation results in total variation distance. The error bounds obtained are either comparable to or improvement over the existing bounds available in the literature. Also, we give an application to the waiting time distribution of 2-runs.
Variance gamma approximation to sums of triplewise independent random variables
探讨三元独立随机变量与完全二分图的联系并构造其序列，用斯坦因方法对其和进行方差伽马近似并推导边界，还讨论了相关极限定理。
We obtain some optimal inequalities on tail probabilities for sums of independent bounded random variables. Our main result completes an upper bound on tail probabilities due to Talagrand by giving a one-term asymptotic expansion for large deviations. This result can also be regarded as sharp large deviations of types of Cramér and Bahadur-Ranga Rao.
Moderate Deviations of Hitting Times for Trajectories of Sums of Independent and Identically Distributed Random Variables
本文研究独立同分布随机变量和的轨迹首达时的中偏差问题，通过区间转单点的思路结合已有大、中偏差结果，建立了对应的中偏差原理并证明转化不改变速率函数。
We study randomly stopped sums via their asymptotic scales. First, finiteness of moments is considered. To generalise this study, asymptotic scales applicable to the class of all heavy-tailed random variables are used. The stopping is assumed to be independent of the underlying process, which is a random walk. The main result enables one to identify whether the asymptotic behaviour of a stopped sum is dominated by the increment, or the stopping variable. As a consequence of this result, new sufficient conditions for the moment determinacy of compounded sums are obtained.
We give a comparison inequality that allows one to estimate the tail probabilities of sums of independent Banach space valued random variables in terms of those of independent identically distributed random variables. More precisely, let X_1,...,X_n be independent Banach-valued random variables. Let I be a random variable independent of X_1,...,X_n and uniformly distributed over {1,...,n}. Put Z_1 = X_I, and let Z_2,...,Z_n be independent identically distributed copies of Z_1. Then, P(||X_1+...+X_n|| > t) < c P(||Z_1+...+Z_n|| > t/c), for all t>0, where c is an absolute constant.
Classical and non-classical central limit theorems for random sums of independent random variables of a double sequence
本文研究了二重序列独立随机变量随机和的经典与非经典中心极限定理，提出随机化方法并给出有效性条件，是对已知结果的扩展与推广。
The approach of Kleitman (1970) and Kanter (1976) to multivariate concentration function inequalities is generalized in order to obtain for deviation probabilities of sums of independent symmetric random variables a lower bound depending only on deviation probabilities of the terms of the sum. This bound is optimal up to discretization effects, improves on a result of Nagaev (2001), and complements the comparison theorems of Birnbaum (1948) and Pruss (1997). Birnbaum's theorem for unimodal random variables is extended to the lattice case.
Let $X_1,X_2,\ldots,X_n$ be independent random variables and $S_k=\sum_{i=1}^k X_i$. We show that for any constants $a_k$, \[ \Pr(\max_{1\leq k\leq n}||S_{k}|-a_{k}|>11t)\leq 30 \max_{1\leq k\leq n}\Pr(||S_{k}|-a_{k}|>t). \] We also discuss similar inequalities for sums of Hilbert and Banach space valued random vectors.
A Perron-Frobenius comonotonic approximation for sums of lognormals
本文基于Perron-Frobenius定理，提出一种新的对数正态随机变量和的共单调逼近方法。
We obtain an uniform tail estimates for natural normed sums of independent random variables (r.v.) with regular varying tails of distributions. We give also many examples on order to show the exactness of offered estimates and discuss some applications in the method Monte-Carlo and statistics, and obtain the sufficient conditions for Central and stable limit theorem in the Banach space of continuous function. There are considered a slight generalization on a random variables with super-heavy tails and martingale difference scheme.
We formulate and prove in this report some sufficient conditions for exponential tightness (ET) of a family of independent identical distributed (i.i.d.) random fields (r.f.) (processes) in the space of continuous functions defined on certain metric compact set. We will use the entropic method as well as too modern approach through the majorizing measures (generic chaining).
We prove that if a rectangular matrix with uniformly small entries and approximately orthogonal rows is applied to the independent standardized random variables with uniformly bounded third moments, then the empirical CDF of the resulting partial sums converges to the normal CDF with probability one. This implies almost sure convergence of empirical periodograms, almost sure convergence of spectra of circulant and reverse circulant matrices, and almost sure convergence of the CDF's generated from independent random variables by independent random orthogonal matrices. For special trigonometric …
A well-known longstanding conjecture on the supremum of the tails of normalized sums of independent Rademacher random variables is disproved. A related conjecture, also recently disproved, is discussed.
The asymptotic tail behaviour of sums of independent subexponential random variables is well understood, one of the main characteristics being the principle of the single big jump. We study the case of dependent subexponential random variables, for both deterministic and random sums, using a fresh approach, by considering conditional independence structures on the random variables. We seek sufficient conditions for the results of the theory with independent random variables still to hold. For a subexponential distribution, we introduce the concept of a boundary class of functions, which we hop…
In a recent author's work the cyclic behavior of maxima in a hierarchical summation scheme was discovered. In the present note we show how the same phenomenon appears in the scheme of conventional summation: the distribution of maximum of $2^n$ independent copies of a sum of $n$ i.i.d. random variables approaches, as $n$ grows, some helix in the space of distributions.
A clarification of Fermi's formula for the maximum of partial sums
本研究澄清费米1926年笔记中部分和最大值概率公式的不一致，从扩散方程得到正确概率，明确独立增量极限的充分假设，补充反例说明小方差不足以支撑该极限。
This note investigates invariance principles for sums of N(nt) iid radom variables, where n is an integer, t is a positive real number and N(u) is a stochastic process with nonnegative integer values. We show that the sequence of sums of these random variables denoted S(n,t), when appropriately centered and normalized, weakly converges to a Gaussian process. We give sufficient conditions depending on the expectation of N(nt) which allows to rescale S(n,t) into a stochastic S(n,a(t)) weakly converging to a Brownian motion.
We show sharp bounds for probabilities of large deviations for sums of independent random variables satisfying Bernstein's condition. One such bound is very close to the tail of the standard Gaussian law in certain case; other bounds improve the inequalities of Bennett and Hoeffding by adding missing factors in the spirit of Talagrand (1995). We also complete Talagrand's inequality by giving a lower bound of the same form, leading to an equality. As a consequence, we obtain large deviation expansions similar to those of Cramér (1938), Bahadur-Rao (1960) and Sakhanenko (1991). We also show that…
The aim of the present paper is to establish the multidimensional counterpart of the \textit{fourth moment criterion} for homogeneous sums in independent leptokurtic and mesokurtic random variables (that is, having positive and zero fourth cumulant, respectively), recently established in \cite{NPPS} in both the classical and in the free setting. As a consequence, the transfer principle for the Central limit Theorem between Wiener and Wigner chaos can be extended to a multidimensional transfer principle between vectors of homogeneous sums in independent commutative random variables with zero th…
On the Rotar central limit theorem for sums of a random number of independent random variables
本文将Rotar中心极限定理推广到随机数量独立但不必同分布的随机变量之和情形，给出其成立条件并研究逼近阶。
We consider weighted sums of independent random variables regulated by an increment sequence. We provide operative conditions that ensure strong law of large numbers for such sums to hold in both the centered and non-centered case. The existing criteria for the strong law are either implicit or assume some sufficient decay for the sequence of coefficients. In our set up we allow for arbitrary sequence of coefficients, possibly random, provided the random variables regulated by such increments satisfy some mild concentration conditions. In the non-centered case, convergence can be translated in…
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.
In this work, conditions are provided under which a normed double sum of independent random elements in a real separable Rademacher type $p$ Banach space converges completely to $0$ in mean of order $p$. These conditions for the complete convergence in mean of order $p$ are shown to provide an exact characterization of Rademacher type $p$ Banach spaces. In case the Banach space is not of Rademacher type $p$, it is proved that the complete convergence in mean of order $p$ of a normed double sum implies a strong law of large numbers.
Convergence in $χ^2$ Distance to the Normal Distribution for Sums of Independent Random Variables
本文证明在独立同方差零均值随机变量的 $\chi^2$ 距离平均值有界时，其标准化和的 $\chi^2$ 距离以 $O(1/n)$ 速率收敛于正态分布。
Sharp bounds on $p$-norms for sums of independent uniform random variables, $0 < p < 1$
本文针对 $0<p<1$ 的独立均匀随机变量和，建立了以其方差表示的 $p$-范数尖锐下界，并推广至 $p$-Rényi 熵的类似问题。
This note establishes convergence in mean of order $p$, $0<p\le 1$ for $d$-dimensional arrays of random vectors in Hilbert spaces under the Cesàro uniform integrability conditions. In the case where $0<p<1$, our $L_p$ convergence is valid irrespective of any dependence structure. In the case where $p=1$, the underlying random vectors are supposed to be pairwise independent. The mean convergence results are established for maximal partial sums while previous contributions were so far considered partial sums only. Some results in the literature are extended. Various properties of the Cesàro unif…
A well-known discovery of Feige's is the following: Let $X_1, \ldots, X_n$ be nonnegative independent random variables, with $\mathbb{E}[X_i] \leq 1 \;\forall i$, and let $X = \sum_{i=1}^n X_i$. Then for any $n$, \[\Pr[X < \mathbb{E}[X] + 1] \geq α> 0,\] for some $α\geq 1/13$. This bound was later improved to $1/8$ by He, Zhang, and Zhang. By a finer consideration of the first four moments, we further improve the bound to approximately $.14$. The conjectured true bound is $1/e \simeq .368$, so there is still (possibly) quite a gap left to fill.
We give explicit bounds for the tail probabilities for sums of independent geometric or exponential variables, possibly with different parameters.
Entropies of sums of independent gamma random variables
本文在固定方差下研究独立伽马随机变量加权和的Rényi熵，建立Schur-凸性结果并给出非渐近界，简化并扩展了已有结论。
Let $X,X_1,\ldots,X_n$ be independent identically distributed random variables. The paper deals with the question about the behavior of the concentration function of the random variable $\sum\limits_{k=1}^{n}X_k a_k$ according to the arithmetic structure of vectors $a_k$. Recently, the interest to this question has increased significantly due to the study of distributions of eigenvalues of random matrices. In this paper we formulate and prove multidimensional generalizations of the results Eliseeva and Zaitsev (2012). They are also the refinements of the results of Friedland and Sodin (2007) a…
Exact exponential tail estimation for sums of independent centered random variables, under natural norming, with applications to the theory of U-statistics
本文利用 GLS 理论推导自然规范化独立中心化随机变量之和的精确指数尾部，并给出 U 统计量中的精细尾部估计。
We study some SDEs derived from the $q\to 1$ limit of a 2D surface growth model called the $q$-Whittaker process. The fluctuations are proven to exhibit Gaussian characteristics that "come down from infinity": After rescaling and re-centering, convergence to the time-inverted stationary additive stochastic heat equation holds. The point of view in this paper is a probabilistic representation of the SDEs by independent sums. By this connection, the normal and Poisson approximations and the in-between slow decorrelation, all in particular integrated forms, explain the convergence of the re-cente…
We calculate the p-the moment of the sum of n independent random variables with respect to symmetric norm in R^n. The order of growth for upper bound p/ln p obtained in ths estimate is optimal. The result extends to generalized Lorentz spaces l_{f,w} under mild assumptions on f. Indeed, the key combinatorial estimate is obtained for the weak l_1 (l_{1,infinity})-norm. Similar results have been obtained independently by Gordon, Litvak, Schuett and Werner for Orlicz norms and by Montgomery-Smith using different techniques and avoiding the combinatorial estimate.
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.
Concentration and moment inequalities for sums of independent heavy-tailed random matrices
本文针对仅存在低阶有限矩的独立重尾随机矩阵和，证明了依赖有效秩等内在维度的Fuk-Nagaev型与Rosenthal型不等式，并通过多个应用验证了结果优势。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。