Let $Y=\sum_{k\ge 1} 1_{A_k}$ be an infinite sum of the indicators of independent events. We investigate a precise (as opposed to logarithmic) first-order asymptotic behavior of the tail probabilities $\mathbb{P}\{Y\ge n\}$ and the point probabilities $\mathbb{P}\{Y=n\}$ as $n\to\infty$. Our analysis provides a reasonably complete classification of the asymptotic behaviors covering most cases of practical interest. These general results are then applied to specific examples where the success probabilities $r_k:=\mathbb{P}(A_k)$ decay polynomially $r_k\sim ck^{-β}$ or (sub-, super-) exponentially $r_k\sim ce^{-k^β}$, yielding the asymptotic tail and point probabilities in explicit forms.
As briefly discussed in the paper, infinite sums of independent indicators arise naturally in numerous settings as diverse as the range of Poissonized samples, the infinite Ginibre point processes and decoupled renewal processes, and records in the $F^α$ scheme. We also explore connections between our results and the theory of Hayman-admissible functions, total positivity, and the Laguerre-Pólya class of type I.
We give exponential upper bounds for $P(S \le k)$, in particular $P(S=0)$, where $S$ is a sum of indicator random variables that are positively associated. These bounds allow, in particular, a comparison with the independent case. We give examples in which we compare with a famous exponential inequality for sums of correlated indicators, the Janson inequality. Here our bound sometimes proves to be superior to Janson's bound.
A survey is given of some Chernoff type bounds for the tail probabilities P(X-EX > a) and P(X-EX < a) when X is a random variable that can be written as a sum of indicator variables that are either independent or negatively related. Most bounds are previously known and some comparisons are made. This paper was written in 1994, but was never published because I had overlooked some existing papers containing some of the inequalities. Because of some recent interest in one of the inequalities, which does not seem to be published anywhere else, it has now been lightly edited and made available her…
We derive the sharp non-asymptotical uniform estimations for tails of distributions for classical normed sums of centered normed independent random vectors having a moderate decreasing individual tails of summands.
We derive in this short report the exponential as well as power decreasing tail estimations for the sums of centered exchangeable random variables, alike ones for the sums of the centered independent ones.
We obtain decay rates of probabilities of tails of polynomials in several independent random variables with heavy tails and derive stable limit theorems for nonconventional sums of such polynomials
We provide exact asymptotics for the tail probabilities $\mathbb{P} \{S_{n,r} > x\}$ as $x \to \infty$, for fix $n$, where $S_{n,r}$ is the $r$-trimmed partial sum of i.i.d. St. Petersburg random variables. In particular, we prove that although the St. Petersburg distribution is only O-subexponential, the subexponential property almost holds. We also determine the exact tail behavior of the $r$-trimmed limits.
We consider sums of $n$ i.i.d. random variables with tails close to $\exp\{-x^β\}$ for some $β>1$. Asymptotics developed by Rootzén (1987) and Balkema, Klüppelberg & Resnick (1993) are discussed from the point of view of tails rather of densities, using a somewhat different angle, and supplemented with bounds, results on a random number $N$ of terms, and simulation algorithms.
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.
A class of stochastic processes strongly related to random sums plays an important role in network and in finance. In this paper we study this kind of stochastic process discuss an overtime unchanged parameter and reveal its asymptotic behavior.
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.
In this paper, we are interested in the limit theorem question for sums of indicator functions. We show that in every aperiodic dynamical system, for every increasing sequence $(a_n)_{n\in\N}\subset\R_+$ such that $a_n\nearrow\infty$ and $\frac{a_n}{n}\to 0$ as $n\to\infty$, there exists a measurable set $A$ such that the sequence of the distributions of the partial sums $\frac{1}{a_n}\sum_{i=0}^{n-1}(\ind_A-μ(A))\circ T^i$ is dense in the set of the probability measures on $\R$. Further, in the ergodic case, we prove that there exists a dense $G_δ$ of such sets.
This paper studies the tail probability of weighted sums of the form $\sum_{i=1}^n c_i X_i$, where random variables $X_i$'s are either independent or pairwise quasi-asymptotical independent with heavy tails. Using $h$-insensitive function, the uniform asymptotic equivalence of the tail probabilities of $\sum_{i=1}^n c_iX_i$, $\max_{1\le k\le n}\sum_{i=1}^k c_iX_i$ and $\sum_{i=1}^n c_iX_i^+$ is established, where $X_i$'s are independent and follow the long-tailed distribution, and $c_i$'s take value in a broad interval. Some further uniform asymptotic results for the weighted sums of $X_i$'s w…
Laws of the iterated and single logarithm for sums of independent indicators, with applications to the Ginibre point process and Karlin's occupancy scheme
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.
Correcting for skewness can result in more accurate tail probability approximations in the central limit theorem for sums of independent random variables. In this paper, we extend the theory to sums of local statistics of independent random variables and apply the result to $k$-runs, U-statistics, and subgraph counts in the Erdös-Rényi random graph. To prove our main result, we develop exponential concentration inequalities and higher-order Cramér-type moderate deviations via Stein's method.
The approach used by Kalashnikov and Tsitsiashvili for constructing upper bounds for the tail distribution of a geometric sum with subexponential summands is reconsidered. By expressing the problem in a more probabilistic light, several improvements and one correction are made, which enables the constructed bound to be significantly tighter. Several examples are given, showing how to implement the theoretical result.
This work prepares new probability bounds for sums of random, independent, Hermitian tensors. These probability bounds characterize large-deviation behavior of the extreme eigenvalue of the sums of random tensors. We extend Lapalace transform method and Lieb's concavity theorem from matrices to tensors, and apply these tools to generalize the classical bounds associated with the names Chernoff, Bennett, and Bernstein from the scalar to the tensor setting. Tail bounds for the norm of a sum of random rectangular tensors are also derived from corollaries of random Hermitian tensors cases. The pro…
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 $\{X_1, X_2, ... \}$ be a sequence of dependent heavy-tailed random variables with distributions $F_1, F_2,...$ on $(-\infty,\infty)$, and let $τ$ be a nonnegative integer-valued random variable independent of the sequence $\{X_k, k \ge 1\}$. In this framework, we study the asymptotic behavior of the tail probabilities of the quantities $X_{(n)} = \max_{1\le k \le n} X_k$, $S_n =\sum_{k=1}^n X_k$ and $S_{(n)}=\max_{1\le k\le n} S_k$ for $n>1$, and for those of their randomized versions $X_{(τ)}$, $S_τ$ and $S_{(τ)}$. We also consider applications of the results obtained to some commonly-us…
We obtain first decay rates of probabilities of tails of multivariate polynomials built on independent random variables with heavy tails. Then we derive stable limit theorems for nonconventional sums of the form $\sum_{Nt\geq n\geq 1}F(X(q_1(n)),...,X(q_\ell(n)))$ where $F$ is a polynomial, $1\leq q_1(n)<\cdots <q_\ell(n)$ are integer valued increasing functions satisfying certain conditions and $X(n),\, n\geq 0$ is a sequence of independent random variables with heavy tails.
We derive in this preprint the moment and exponential tail estimates, sufficient conditions for the Non-Central Limit Theorem (NCLT) in the ordinary one-dimensional space as well as in the space of continuous functions for the properly (natural) normalized multi-indexed sums of function of random variables, processes or fields (r.f.), on the other words V-statistics, parametric, in general case. We construct also some examples in order to show the exactness of obtained estimates. We will use the theory of the so-called degenerate approximation of the functions of several variables as well as t…
This paper is the Part II of a serious work about T product tensors focusing at establishing new probability bounds for sums of random, independent, T product tensors. These probability bounds characterize large deviation behavior of the extreme eigenvalue of the sums of random T product tensors. We apply Lapalace transform method and Lieb concavity theorem for T product tensors obtained from our Part I paper, and apply these tools to generalize the classical bounds associated with the names Chernoff, and Bernstein from the scalar to the T product tensor setting. Tail bounds for the norm of a …
We study the large deviation probabilities of infinite weighted sums of independent random variables that have stretched exponential tails. This generalizes Kiesel and Stadtmüller (2000), who study the same objects under the assumption of finite exponential moments, and Gantert et al.\ (2014), who study finite weighted sums with stretched exponential tails.
Let $ξ_1, ξ_2, \dots$ be i.i.d. non-negative random variables whose tail varies regularly with index $-1$, let $S_n$ be the sum and $M_n$ the largest of the first $n$ values. We clarify for which sequences $x_n\to\infty$ we have $\mathbb P(S_n \ge x_n) \sim \mathbb P(M_n \ge x_n)$ as $n\to\infty$. Outside this regime, the typical size of $S_n$ conditioned on exceeding $x_n$ is not completely determined by the largest summand and we provide an appropriate correction term which involves the integrated tail of $ξ_1$.
In this paper, we study the fluctuations of sums of random variables with distribution defined as a mixture of light-tail and truncated heavy-tail distributions. We focus on the case when both the mixing coefficient and the truncation level depend on the number of summands. The aim of this research is to characterize the limiting distributions of the sums due to various relations between these parameters.
In this work we present concentration inequalities for the sum $S_n$ of independent integer-valued not necessary indentically distributed random variables, where each variable has tail function that can be bounded by some power function with exponent $-α$. We show that when $0<α\leq 1$, then the sum does not have finite expectation, however, with high probability we have that $|S_n|=O\left(n^{1/α}\right)$. When $α>1$, then the sum $S_n$ is concentrated around its mean. Since the r.vs. that constitute the sum has tails, which can be bounded by some power function, it follows that results of thi…
In this paper we show under weak assumptions that for $R\stackrel{d}{=}1+M_1+M_1M_2+\ldots$, where $P(M\in[0,1])=1$ and $M_i$ are independent copies of $M$, we have $\ln P(R>x)\sim C\, x\ln P(M>1-\frac1x)$ as $x\to\infty$. The constant $C$ is given explicitly and its value depends on the rate of convergence of $\ln P(M>1-\frac1x)$. Random variable $R$ satisfies the stochastic equation $R\stackrel{d}{=}1+MR$ with $M$ and $R$ independent, thus this result fits into the study of tails of iterated random equations, or more specifically, of perpetuities.
In this paper, we obtain some results on precise large deviations for non-random and random sums of widely dependent random variables with common dominatedly varying tail distribution or consistently varying tail distribution on $(-\infty,\infty)$. Then we apply the results to reinsurance and insurance and give some asymptotic estimates on proportional reinsurance, random-time ruin probability and the finite-time ruin probability.
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.
This paper presents new probability inequalities for sums of independent, random, self-adjoint matrices. These results place simple and easily verifiable hypotheses on the summands, and they deliver strong conclusions about the large-deviation behavior of the maximum eigenvalue of the sum. Tail bounds for the norm of a sum of random rectangular matrices follow as an immediate corollary. The proof techniques also yield some information about matrix-valued martingales. In other words, this paper provides noncommutative generalizations of the classical bounds associated with the names Azuma, Benn…
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 given. These conditions extend and strengthen a series of previous results. Additionally, a connection with subexponential distributions is demonstrated. That is, $F$ is subexponential if and only if the condition above holds for some $t_n$ and $$ \lim_{t\to\infty}{1-F(t+x)\over 1-F(t)} = 1 \quad\text{for each real $x$.}$$
In two recent works, Kuba and Mahmoud (arXiv:1503.090691 and arXiv:1509.09053) introduced the family of two-color affine balanced Polya urn schemes with multiple drawings. We show that, in large-index urns (urn index between $1/2$ and $1$) and triangular urns, the martingale tail sum for the number of balls of a given color admits both a Gaussian central limit theorem as well as a law of the iterated logarithm. The laws of the iterated logarithm are new even in the standard model when only one ball is drawn from the urn in each step (except for the classical Polya urn model). Finally, we prove…
For a distribution $F^{*τ}$ of a random sum $S_τ=ξ_1+...+ξ_τ$ of i.i.d. random variables with a common distribution $F$ on the half-line $[0,\infty)$, we study the limits of the ratios of tails $\bar{F^{*τ}}(x)/\bar{F}(x)$ as $x\to\infty$ (here, $τ$ is a counting random variable which does not depend on $\{ξ_n\}_{n\ge1}$). We also consider applications of the results obtained to random walks, compound Poisson distributions, infinitely divisible laws, and subcritical branching processes.
We prove tail estimates for variables $\sum_i f(X_i)$, where $(X_i)_i$ is the trajectory of a random walk on an undirected graph (or, equivalently, a reversible Markov chain). The estimates are in terms of the maximum of the function $f$, its variance, and the spectrum of the graph. Our proofs are more elementary than other proofs in the literature, and our results are sharper. We obtain Bernstein and Bennett-type inequalities, as well as an inequality for subgaussian variables.
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.
We study functional convergence of sums of moving averages with random coefficients and heavy-tailed innovations. Under some standard moment conditions and the assumption that all partial sums of the series of coefficients are a.s. bounded between zero and the sum of the series we obtain functional convergence of the corresponding partial sum stochastic process in the space $D[0,1]$ of càdlàg functions with the Skorohod $M_{2}$ topology.
We study conditions under which $P(S_τ>x)\sim P(M_τ>x)\sim EτP(ξ_1>x)$ as $x\to\infty$, where $S_τ$ is a sum $ξ_1+...+ξ_τ$ of random size $τ$ and $M_τ$ is a maximum of partial sums $M_τ=\max_{n\leτ}S_n$. Here $ξ_n$, $n=1$, 2, ..., are independent identically distributed random variables whose common distribution is assumed to be subexponential. We consider mostly the case where $τ$ is independent of the summands; also, in a particular situation, we deal with a stopping time. Also we consider the case where $Eξ>0$ and where the tail of $τ$ is comparable with or heavier than that of $ξ$, and obt…
We take an $L_1$-dense class of functions $\Cal F$ on a measurable space $(X,\Cal X)$ and a sequence of i.i.d. $X$-valued random variables $ξ_1,\dots,ξ_n$, and give a good estimate on the tail behaviour of $\sup\limits_{f\in\Cal F}\sum\limits_{j=1}^nf(ξ_j)$ if the conditions $\sup\limits_{x\in X}|f(x)|\le1$, $Ef(ξ_1)=0$ and $Ef(ξ_1)^2<σ^2$ with some $0\leσ\le1$ hold for all $f\in\Cal F$. Roughly speaking this estimate states that under some natural conditions the above considered supremum is not much larger than the worst element taking part in it. The proof heavily depends on the main result …
Two old conjectures from problem sections, one of which from SIAM Review, concern the question of finding distributions that maximize P(Sn <= t), where Sn is the sum of i.i.d. random variables X1, ..., Xn on the interval [0,1], satisfying E[X1]=m. In this paper a Lagrange multiplier technique is applied to this problem, yielding necessary conditions for distributions to be extremal, for arbitrary n. For n=2, a complete solution is derived from them: extremal distributions are discrete and have one of the following supports, depending on m and t: {0,t}, {t-1,1}, {t/2,1}, or {0,t,1}. These resul…
The extremal tail probabilities of moving sums in a marked Poisson random field is examined here. These sums are computed by adding up the weighted occurrences of events lying within a scanning set of fixed shape and size. Change of measure and analysis of local random fields are used to provide tail probabilities. The asymptotic constants are initially expressed in a form that seems hard to evaluate and do not seem to provide any additional information on the properties of the constants. A more sophisticated approach is then undertaken giving rise to an expression that is not only neater but …
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.
Let $X_{1},\ldots ,X_{n}$ be $n$ real-valued dependent random variables. With motivation from Mitra and Resnick (2009), we derive the tail asymptotic expansion for the weighted sum of order statistics $X_{1:n}\leq \cdots \leq X_{n:n}$ of $X_{1},\ldots ,X_{n}$ under the general case in which the distribution function of $X_{n:n}$ is long-tailed or rapidly varying and $% X_{1},\ldots ,X_{n}$ may not be comparable in terms of their tail probability. We also present two examples and an application of our results in risk theory.
Let X_1,X_2,... be a sequence of independent and identically distributed random variables, and put S_n=X_1+...+X_n. Under some conditions on the positive sequence tau_n and the positive increasing sequence a_n, we give necessary and sufficient conditions for the convergence of sum_{n=1}^infty tau_n P(|S_n|>t a_n) for all t>0, generalizing Baum and Katz's (1965) generalization of the Hsu-Robbins-Erdos (1947, 1949) law of large numbers, also allowing us to characterize the convergence of the above series in the case where tau_n=1/n and a_n=(n log n)^{1/2} for n>1, thereby answering a question of…
Let $X_1,X_2,...$ be a sequence of independent and identically distributed random variables, and put $S_n=X_1+...+X_n$. Under some conditions on the positive sequence $τ_n$ and the positive increasing sequence $a_n$, we give necessary and sufficient conditions for the convergence of $\sum_{n=1}^\infty τ_n P(|S_n|\ge εa_n)$ for all $ε>0$, generalizing Baum and Katz's (1965) generalization of the Hsu-Robbins-Erdos (1947, 1949) law of large numbers, also allowing us to characterize the convergence of the above series in the case where $τ_n=n^{-1}$ and $a_n=(n\log n)^{1/2}$ for $n\ge 2$, thereby a…
We investigate asymptotic behaviour of probabilities of large deviations for normalized combinatorial sums. We find a zone in which these probabilities are equivalent to the tail of the standard normal law. Our conditions are similar to the classical Bernstein condition. The range of the zone of the normal convergence can be of power order.
The sums and maxima of weighted non-stationary random length sequences of regularly varying random variables may have the same tail and extremal indices, Markovich and Rodionov (2020). The main constraints are that there exists a unique series in a scheme of series with the minimum tail index, the tail of the term number is lighter than the tail of the terms and the weights are positive constants. These assumptions are changed here: a bounded random number of series is allowed to have the minimum tail index, the tail of the term number may be heavier than the tail of the terms and the weights …
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.
The tail of the distribution of a sum of a random number of independent and identically distributed nonnegative random variables depends on the tails of the number of terms and of the terms themselves. This situation is of interest in the collective risk model, where the total claim size in a portfolio is the sum of a random number of claims. If the tail of the claim number is heavier than the tail of the claim sizes, then under certain conditions the tail of the total claim size does not change asymptotically if the individual claim sizes are replaced by their expectations. The conditions all…
Rutgers University（罗格斯大学）；University of Pennsylvania（宾夕法尼亚大学）
We derive exponential tail inequalities for sums of random matrices with no dependence on the explicit matrix dimensions. These are similar to the matrix versions of the Chernoff bound and Bernstein inequality except with the explicit matrix dimensions replaced by a trace quantity that can be small even when the dimension is large or infinite. Some applications to principal component analysis and approximate matrix multiplication are given to illustrate the utility of the new bounds.
Consider the classical Gaussian unitary ensemble of size $N$ and the real Wishart ensemble $W_N(n,I)$. In the limits as $N \to \infty$ and $N/n \to γ> 0$, the expected number of eigenvalues that exit the upper bulk edge is less than one, 0.031 and 0.170 respectively, the latter number being independent of $γ$. These statements are consequences of quantitative bounds on tail sums of eigenvalues outside the bulk which are established here for applications in high dimensional covariance matrix estimation.
