Trimmed Sums Research Articles

Trimmed sums of twists and the area Siegel\u2013Veech constant

We relate trimmed sums of twists in cylinders along a typical Teichmüller geodesic to the area Siegel–Veech constant.

Strong Laws of Large Numbers for Intermediately Trimmed Sums of i.i.d. Random Variables with Infinite Mean

We consider moderately trimmed sums of non-negative i.i.d. random variables. We show that for every distribution function there exists a proper moderate trimming such that for the trimmed sum a non-trivial strong law of large numbers holds. In case that the distribution function has regularly varying tails we give necessary and sufficient conditions on the trimming for a strong law of large numbers to hold.

Tail Probabilities of St. Petersburg Sums, Trimmed Sums, and Their Limit

We provide exact asymptotics for the tail probabilities $${\mathbb {P}}\{ S_{n,r} > x \}$$ as $$x \rightarrow \infty $$ , for fixed 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.

A Limit Theorem for Random Products of Trimmed Sums of i.i.d. Random Variables

Let {X, Xi; i ≥ 1} be a sequence of independent and identically distributed positive random variables with a continuous distribution function F, and F has a medium tail. Denote and , where Mn = max 1≤i≤nXi, , and a > 0 is a fixed constant. Under some suitable conditions, we show that , as n → ∞, where Tk(a) = Sk − Sk(a) is the trimmed sum and {W(t); t ≥ 0} is a standard Wiener process.

ℒ1-Limit of Trimmed Sums of Order Statistics from Location-Scale Distributions with Applications to Type II Censored Data Analysis

We derive the ℒ1-limit of trimmed sums of order statistics from location-scale distributions satisfying certain assumptions. Based on this limit, an approximation to the asymptotic variance of a Best-Asymptotic-Normal (BAN) estimator for the location parameter is developed. Associated formulae are derived for four location-scale distributions commonly used in lifetime data analysis. The approximation is analyzed via the properties of the approximating function and by comparison to the exact values for a special case. Applications are illustrated by applying the approximation to comparing location parameters and to selecting the population with the largest location parameter, using censored samples from location-scale populations.

Trimmed sums of long range dependent moving averages

In this paper we establish asymptotic normality of trimmed sums for long range dependent moving averages. Our results extend those of Ho and Hsing [Ho, H.-C., Hsing, T. 1996. On the asymptotic expansion of the empirical process of long-memory moving averages. Ann. Statist. 24, 992–1024] and Wu [Wu, W.B., 2005. On the Bahadur representation of sample quantiles for dependent sequences. Ann. Statist. 33 1934–1963].

Trimmed sums for non-negative, mixing stationary processes

We consider the effect of “trimming” ergodic sums of their maximal values on the strong law of large numbers for non-negative, non-integrable, mixing stationary processes. The results obtained are used to show the failure of the strong law of large numbers for modified continued fraction coefficients, and to study the “cusp visits” of a certain interval map.

Moments of Limits of Lightly Trimmed Sums of Random Vectors in the Generalized Domain of Normal Attraction of Non-Gaussian Operator-Stable Laws

Existence and nonexistence for moments of limiting random vectors of normalized, lightly trimmed sums of random vectors in the generalized domain of normal attraction of non-Gaussian operator-stable laws are studied. The idea of representing the limiting random vectors by infinite series is essentially used in the proofs.

Precision Calculation of Distributions for Trimmed Sums

Recursive methods are described for computing the frequency and distribution functions of trimmed sums of independent and identically distributed nonnegative integer-valued random variables. Surprisingly, for fixed arguments, these can be evaluated with just a finite number of arithmetic operations (and whatever else it takes to evaluate the common frequency function of the original summands). These methods give rise to very accurate computational algorithms that permit a delicate numerical investigation, herein described, of Feller's weak law of large numbers and its trimmed version for repeated St. Petersburg games. The performance of Stigler's theorem for the asymptotic distribution of trimmed sums is also investigated on the same example.

Infinite Limits and Infinite Limit Points of Random Walks and Trimmed Sums

We consider infinite limit points (in probability) for sums and lightly trimmed sums of i.i.d. random variables normalized by a nonstochastic sequence. More specifically, let $X_1, X_2, \ldots$ be independent random variables with common distribution $F$. Let $M^{(r)}_n$ be the $r$th largest among $X_1, \ldots, X_n$; also let $X^{(r)}_n$ be the observation with the $r$th largest absolute value among $X_1, \ldots, X_n$. Set $S_n = \sum^n_1X_i, ^{(r)}S_n = S_n - M^{(1)}_n - \cdots - M^{(r)}_n$ and $^{(r)}\tilde{S}_n = S_n - X^{(1)}_n - \cdots - X^{(r)}_n (^{(0)}\tilde{S}_n = ^{(0)}\tilde{S}_n = S_n)$. We find simple criteria in terms of $F$ for $^{(r)}S_n/B_n \rightarrow p \pm \infty$ (i.e., $^{(r)}S_n/B_n$ tends to $\infty$ or to $-\infty$ in probability) or $^{(r)}\tilde{S}_n/B_n \rightarrow p \pm \infty$ when $r = 0, 1, \ldots$. Here $B_n \uparrow \infty$ may be given in advance, or its existence may be investigated. In particular, we find a necessary and sufficient condition for $^{(r)}S_n/n \rightarrow p \infty$. Some equivalences for the divergence of $|^{(r)}\tilde{S}_n|/|X^{(r)}_n|$, or of $^{(r)}S_n/(X^-)^{(s)}_n$, where $(X^-)^{(s)}_n$ is the $s$th largest of the negative parts of the $X_i$, and for the convergence $P\{S_n > 0\}\rightarrow 1$, as $n\rightarrow\infty$, are also proven. In some cases we treat divergence along a subsequence as well, and one such result provides an equivalence for a generalized iterated logarithm law due to Pruitt.

A Nonstandard Law of the Iterated Logarithm for Trimmed Sums

Let $X_i, i \geq 1$, be independent random variables with a common distribution in the domain of attraction of a strictly stable law, and for each $n \geq 1$ let $X_{1, n} \leq \cdots \leq X_{n, n}$ denote the order statistics of $X_1, \ldots, X_n$. In 1986, S. Csorgo, Horvath and Mason showed that for each sequence $k_n, n \geq 1$, of nonnegative integers with $k_n \rightarrow \infty$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$, the trimmed sums $S_n(k_n) = X_{k_n + 1, n} + \cdots + X_{n - k_n, n}$ converge in distribution to the standard normal distribution, when properly centered and normalized, despite the fact that the entire sums $X_1 + \cdots + X_n$ have a strictly stable limit, when properly centered and normalized. The asymptotic almost sure behavior of $S_n(k_n)$ strongly depends on the rate at which $k_n$ converges to $\infty$. The sequences $k_n \sim c \log \log n$ as $n \rightarrow \infty$ for $0 < c < \infty$ constitute a borderline case between a classical law of the iterated logarithm and a radically different behavior. This borderline case is investigated in detail for nonnegative summands $X_i$.

Ratios of Trimmed Sums and Order Statistics

Let $X_i$ be independent and identically distributed random variables with distribution $F$. Let $M^{(n)}_n \leq \cdots \leq M^{(1)}_n$ be the sample $X_1, X_2,\ldots, X_n$ arranged in increasing order, with a convention for the breaking of ties, and let $X^{(n)}_n,\ldots, X^{(1)}_n$ be the sample arranged in increasing order of modulus, again with a convention to break ties. Let $S_n = X_1 + \cdots + X_n$ be the sample sum. We consider sums trimmed by large values, $^{(r)}S_n = S_n - M^{(1)}_n - \cdots - M^{(r)}_n, r = 1,2,\ldots, n, ^{(0)}S_n = S_n,$ and sums trimmed by values large in modulus, $^{(r)}\tilde{S}_n = S_n - X^{(1)}_n - \cdots - X^{(r)}_n, r = 1,2,\ldots, n, ^{(0)}\tilde{S}_n = S_n.$ In this paper we give necessary and sufficient conditions for $^{(r)}\tilde{S}_n/|X^{(r)}_n| \rightarrow \infty$ and $^{(r)}S_n/M^{(r)}_n \rightarrow \infty$ to hold almost surely or in probability, when $r = 1,2,\ldots$. These express the dominance of the sum over the large values in the sample in various ways and are of interest in relation to the law of large numbers and to central limit behavior. Our conditions are related to the relative stability almost surely or in probability of the trimmed sum and, hence, to analytic conditions on the tail of the distribution of $X_i$ which give relative stability.

Asymptotic Behavior of Self-Normalized Trimmed Sums: Nonnormal Limits

Let $\{X_j\}$ be independent, identically distributed random variables with continuous nondegenerate distribution $F$ which is symmetric about the origin. Let $\{X_n(1), X_n(2),\ldots, X_n(n)\}$ denote the arrangement of $\{X_1,\ldots, X_n\}$ in decreasing order of magnitude, so that with probability 1, $|X_n(1)| > |X_n(2)| > \cdots > |X_n(n)|$. For integers $r_n \rightarrow \infty$ such that $r_n/n \rightarrow 0$, define the self-normalized trimmed sum $T_n = \sum^n_{i=r_n}X_n(i)/\{\sum^n_{i=r_n}X^2_n(i)\}^{1/2}$. The asymptotic behavior of $T_n$ is studied. Under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion for $T_n$, various interesting nonnormal limit laws for $T_n$ are obtained and represented by means of infinite random series. In general, moreover, criteria for degenerate limits and stochastic compactness for $\{T_n\}$ are also obtained. Finally, more general results and technical difficulties are discussed.

Asymptotic Normality and Subsequential Limits of Trimmed Sums

Let $\{X_i\}$ be i.i.d. and $S_n(s_n, r_n)$ the sum of the first $n X_i$ with the $r_n$ largest and $s_n$ smallest excluded. Assume $r_n \rightarrow \infty, s_n \rightarrow \infty, n^{-1}r_n \rightarrow 0, n^{-1}s_n \rightarrow 0.$ Necessary and sufficient conditions are obtained for the existence of $\{\delta_n\}, \{\gamma_n\}$ such that $\gamma^{-1}_n(S_n(s_n, r_n) - \delta_n)$ converges weakly to a standard normal. The set of all subsequential limit laws for these sequences is characterized and sufficient conditions are given for $X_i$ to be in the domain of partial attraction of a given law in the class. These conditions are also necessary if a unique factorization result for characteristic functions is true.

Asymptotic Behavior of a Class of Infinitely Divisible Distributions

Previous article Next article Asymptotic Behavior of a Class of Infinitely Divisible DistributionsA. L. YakymivA. L. Yakymivhttps://doi.org/10.1137/1132095PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] I. S. Gradshtein and , I. M. Ryzhik, Tables of Integrals, Sums, Series and Products, Fizmatgiz, Moscow, 1963, (In Russian.) Google Scholar[2] V. M. Zolotarev, On the asymptotic behavior of a class of infinitely divisible laws, Theory Probab. Appl., 6 (1961), 304–307 10.1137/1106039 0124.34501 LinkGoogle Scholar[3] E. Seneta, Regularly Varying Functions, Nauka, Moscow, 1985, (In Russian.) Google Scholar[4] William Feller, An introduction to probability theory and its applications. Vol. II. , Second edition, John Wiley & Sons Inc., New York, 1971xxiv+669 42:5292 0219.60003 Google Scholar[5] A. L. Yakymiv, Asymptotic behavior of the probability of continuing critical Bellman-Harris branching processes, Trudy Mat. Inst. Steklov, 177 (1986), 177–205, (In Russian.) 0607.60075 Google Scholar[6] A. L. Yakymiv, Asymptotic properties of the times the states change in a random record process, Theory Probab. Appl., 31 (1986), 508–512 10.1137/1131068 0623.60062 LinkGoogle Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Tail behaviors of semi-stable distributionsJournal of Mathematical Analysis and Applications, Vol. 393, No. 1 | 1 Sep 2012 Cross Ref On the Asymptotics of the Density of an Infinitely Divisible Distribution at InfinityA. L. YakymivTheory of Probability & Its Applications, Vol. 47, No. 1 | 25 July 2006AbstractPDF (122 KB)Remarks on the absolute maximum of a Lévy processJournal of Applied Probability, Vol. 39, No. 2 | 14 July 2016 Cross Ref A Probabilistic Approach to the Tails of Infinitely Divisible LawsSums, Trimmed Sums and Extremes | 1 Jan 1991 Cross Ref Volume 32, Issue 4| 1988Theory of Probability & Its Applications573-758 History Submitted:23 December 1985Published online:17 July 2006 InformationCopyright © 1987 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/1132095Article page range:pp. 628-639ISSN (print):0040-585XISSN (online):1095-7219Publisher:Society for Industrial and Applied Mathematics

The Asymptotic Distribution of Trimmed Sums

Let $X_{1,n} \leq \cdots \leq X_{n,n}$ be the order statistics of $n$ independent and identically distributed random variables and let $m_n$ and $k_n$ be positive integers such that $m_n \rightarrow \infty, k_n \rightarrow \infty, m_n/n \rightarrow 0$ and $k_n/n \rightarrow 0$ as $n \rightarrow \infty$. We find a necessary and sufficient condition for the existence of normalizing and centering constants $A_n > 0$ and $B_n$ such that the sequence $T_n = A^{-1}_n \big\{\sum_{i=m_n+1}^{n-k_n} X_{i,n} - B_n\big\}$ is stochastically compact and completely describe the possible subsequential limiting distributions. We also give a necessary and sufficient condition for the existence of $A_n$ and $B_n$ such that $T_n$ is asymptotically normal. A variant of Stigler's theorem when $m_n/n \rightarrow \alpha$ and $k_n/n \rightarrow 1 - \beta$, where $0 < \alpha < \beta < 1$, is also obtained as a by-product.

Asymptotic Normality of Trimmed Sums of $\Phi$-Mixing Random Variables

A Gaussian central limit theorem for trimmed sums of $\Phi$-mixing Hilbert-space-valued random variables is obtained, and implications regarding Ibragimov's conjecture are examined.