Limit theorems for trimmed sums

This paper studies the heavily trimmed sums (*) _{[ns] + 1} ^[nt] X _j ⁽ⁿ⁾ , where {X _j ⁽ⁿ⁾ } _{j = 1} ⁿ are the order statistics from independent random variables {X ₁,...,X _n } having a common distributionF. The main theorem gives the limiting process of (*) as a process oft. More smoothly trimmed sums like _{j = 1} ^[nt] J(j/n)X _j ⁽ⁿ⁾ are also discussed.     

Universal asymptotic normality for conditionally trimmed sums

The conditionally trimmed sums formed from an arbitrary i.i.d. sample are shown to satisfy both a probabilistic and empirical central limit theorem with a normal limit law. The specific method of trimming attempts to retain as many summands as possible and deletes only terms of sufficient magnitude. The behavior of the deleted terms is also studied for random variables which generate affinely stochastically compact partial sums.     

Non-classical law of the iterated logarithm behaviour for trimmed sums

Summary We study the law of the iterated logarithm for the partial sum of i.i.d. random variables when the r _n largest summands are excluded, where r _n=o(log logn). This complements earlier work in which the case log log_n=O(r_n) was considered. A law of the iterated logarithm is again seen to prevail for a wide class of distributions, but for reasons quite different from previously.Research supported in part by NSF Grant DMS-8501732     

Asymptotic behavior of self-normalized trimmed sums: Nonnormal limits II

Let {X _j} be independent, identically distributed random variables which are symmetric about the origin and have a continuous nondegenerate distributionF. Let {X _n(1),...,X _n(n)} denote the arrangement of {X ₁,...,X _n} in decreasing order of magnitude, so that with probability one, |X _n(1)|>|X _n(2)|>...> |X _n(n)|. For initegersr _n such thatr _n/n0, define the self-normalized trimmed sumT _n= _i=rn ⁿ X _n(i)/{ _i=rn ⁿ X _n ² (i)}^1/2. Hahn and Weiner⁽⁶⁾ showed that under a probabilistically meaningful analytic condition generalizing the asymptotic normality criterion forT _n, various nonnormal limit laws forT _n arise which are represented by means of infinite random series. The analytic condition is now extended and the previous approach is refined to obtain limits which are mixtures of a normal, a Rademacher, and a law represented by a more general random series. Each such limit law actually arises as can be seen from the construction of a single distribution whose correspondingL(T _n ) generates all of the law along different subsequences, at least if {r _n} grows sufficiency fast. Another example clarifies the limitations of the basic approach.     

On the central limit theorem for modulus trimmed sums

We prove a functional central limit theorem for modulus trimmed i.i.d. variables in the domain of attraction of a nonnormal stable law. In contrast to the corresponding result under ordinary trimming, our CLT contains a random centering factor which is inevitable in the nonsymmetric case. The proof is based on the weak convergence of a two-parameter process where one of the parameters is time and the second one is the fraction of truncation.     

An almost sure invariance principle for trimmed sums of random vectors

Let {X _n ; n ≥ 1} be a sequence of independent and identically distributed random vectors in ℜ ^p with Euclidean norm |·|, and let X _n ^(r) = X _m if |X _m | is the r-th maximum of {|X _k |; k ≤ n}. Define S _n = Σ _k≤n X _k and ^(r) S _n − (X _n ⁽¹⁾ + ... + X _n ^(r)). In this paper a generalized strong invariance principle for the trimmed sums ^(r) S _n is derived.     

On the functional law of the iterated logarithm for trimmed sums

New conditions for the functional law of the iterated logarithm for trimmed sums of symmetric independent identically distributed random variables are obtained. Bibliography: 7 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 244, 1997, pp. 119–125. Translated by N. B. Lebedinskaya.     

Relative norms of Gauss sums

The explicit determination of the values of Gauss sums is a very classical problem and has some rather deep arithmetic consequences in classfield theory. Here we study the simpler problem of finding their relative norms. We give a complete determination of the relative norms of Gauss sums for norms whose values are known to be in an imaginary quadratic extension of the rational field.     

A general LIL for trimmed sums of random fields in banach spaces

Given a field of independent identically distributed (i.i.d.) random variables $ \left\{ {X_{\bar n} ;\bar n \in \aleph ^d } \right\} $ indexed by d-tuples of positive integers and taking values in a separable Banach space B, let $ X_{\bar n}^{(r)} = X_{\bar m} $ is the r-th maximum of $ \left\{ {\left\| {X_{\bar k} } \right\|;\bar k \leqq \bar n} \right\} $ and let $ ^{(r)} S_{\bar n} = S_{\bar n} - \left( {X_{\bar n}^{(1)} + \cdots + X_{\bar n}^{(r)} } \right) $ be the trimmed sums, where $ S_{\bar n} = \sum\nolimits_{\bar k \leqq \bar n} {X_{\bar k} } $ . This paper aims to obtain a general law of the iterated logarithm (LIL) for the trimmed sums which improves previous works.     

The central limit theorem for sums of trimmed variables with heavy tails

Trimming is a standard method to decrease the effect of large sample elements in statistical procedures, used, e.g., for constructing robust estimators and tests. Trimming also provides a profound insight into the partial sum behavior of i.i.d. sequences. There is a wide and nearly complete asymptotic theory of trimming, with one remarkable gap: no satisfactory criteria for the central limit theorem for modulus trimmed sums have been found, except for symmetric random variables. In this paper we investigate this problem in the case when the variables are in the domain of attraction of a stable law. Our results show that for modulus trimmed sums the validity of the central limit theorem depends sensitively on the behavior of the tail ratio P(X>t)/P(|X|>t) of the underlying variable X as t→∞ and paradoxically, increasing the number of trimmed elements does not generally improve partial sum behavior.     

The asymptotic distribution of magnitude trimmed sums for distributions in the Feller class

LetX, X ₁,X ₂,... be i.i.d. with common distribution functionF. Rather than study limit behavior of the sum,S _n =X ₁++X _n , under constant normalizations, we consider the sum with ther _n summands largest in magnitude removed from the sumS _n , wherer _n andr _n n ^–10. This is known as an intermediate magnitude trimmed sum. LetF be such that lim inf_t lim inf _t EX ² I(|X|t/)t ² P((|X|>t)>0. The collection of suchF is known as the Feller class, a large class of distributions which includes all domains of attraction (in particular the stable laws). Pruitt(¹³) showed that asymptotic normality always holds for the trimmed sums ifF is in the Feller class and ifF is symmetric. Here, for anyF in the Feller class, we obtain complete results including the form of the possible limit laws and their convergence criteria, thus generalizing Pruitt's result to the asymmetric setting.This paper forms a portion of the author's Ph.D. dissertation under the supervision of Daniel C. Weiner.     

Strong laws of large numbers for intermediately trimmed Birkhoff sums of observables with infinite mean

We consider dynamical systems on a finite measure space fulfilling a spectral gap property and Birkhoff sums of a non-negative, non-integrable observable. For such systems we generalize strong laws of large numbers for intermediately trimmed sums only known for independent random variables. The results split up in trimming statements for general distribution functions and for regularly varying tail distributions. In both cases the trimming rate can be chosen in the same or almost the same way as in the i.i.d. case. As an example we show that piecewise expanding interval maps fulfill the necessary conditions for our limit laws. As a side result we obtain strong laws of large numbers for truncated Birkhoff sums.     

A LIL and limit distributions for trimmed sums of random vectors attracted to operator semi-stable laws

Let θ∈ Rdbe a unit vector and let X,X1,X2,...be a sequence of i.i.d.Rd-valued random vectors attracted to operator semi-stable laws.For each integer n ≥ 1,let X1,n ≤···≤ Xn,n denote the order statistics of X1,X2,...,Xn according to priority of index,namely | X1,n,θ | ≥···≥ | Xn,n,θ |,where ·,· is an inner product on Rd.For all integers r ≥ 0,define by(r)Sn = n-ri=1Xi,n the trimmed sum.In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums(r)Sn.Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded.A stochastically compactness of(r)Sn is obtained.     

On stability of sums of nonnegative random variables

We present new sufficient conditions for stability of sums of nonnegative random variables having finite moments of second order. We demonstrate that these conditions are nonimprovable in some sense. Bibliography: 4 titles. Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 361, 2008, pp. 78—82.     

Integral trimmed regions

We define a new family of central regions with respect to a probability measure. They are induced by a set or a family of sets of functions and we name them integral trimmed regions. The halfspace trimming and the zonoid trimming are particular cases of integral trimmed regions. We focus our work on the derivation of properties of such integral trimmed regions from conditions satisfied by the generating classes of functions. Further we show that, under mild conditions, the population integral trimmed region of a given depth can be characterized in terms of certain regions based on empirical distributions.     

Relative length of long paths and cycles in graphs with large degree sums

For a graph G, p(G) denotes the order of a longest path in G and c(G) the order of a longest cycle. We show that if G is a connected graph n ≥ 3 vertices such that d(u) + d(v) + d(w) ≧ n for all triples u, v, w of independent vertices, then G satisfies c(G) ≥ p(G) – 1, or G is in one of six families of exceptional graphs. This generalizes results of Bondy and of Bauer, Morgana, Schmeichel, and Veldman. © 1995, John Wiley & Sons, Inc.     

Relative stability in strictly stationary random sequences

Relative stability results for weakly dependent and strongly mixing strictly stationary sequences are established. As a consequence, some infinite memory models, including ARCH(1) processes, are relatively stable.