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.     

Convergence rates of the LIL for random fields in Hilbert spaces

Considering the positive d-dimensional lattice point Z ₊ ^d (d ≥ 2) with partial ordering ≤, let {X _k: k ∈ Z ₊ ^d } be i.i.d. random variables taking values in a real separable Hilbert space (H, ‖ · ‖) with mean zero and covariance operator Σ, and set $ S_n = \sum\limits_{k \leqslant n} {X_k } $ S_n = \sum\limits_{k \leqslant n} {X_k } , n ∈ Z ₊ ^d . Let σ _i ², i ≥ 1, be the eigenvalues of Σ arranged in the non-increasing order and taking into account the multiplicities. Let l be the dimension of the corresponding eigenspace, and denote the largest eigenvalue of Σ by σ ². Let logx = ln(x ∨ e), x ≥ 0. This paper studies the convergence rates for $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) $ \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}} P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt {2\left| n \right|\log \log \left| n \right|} } \right) . We show that when l ≥ 2 and b > −l/2, E[‖X‖²(log ‖X‖) ^d−2(log log ‖X‖) ^b+4] < ∞ implies $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} $ \begin{gathered} \mathop {\lim }\limits_{\varepsilon \searrow \sqrt {d - 1} } (\varepsilon ^2 - d + 1)^{b + l/2} \sum\limits_n {\frac{{\left( {\log \log \left| n \right|} \right)^b }} {{\left| n \right|\log \left| n \right|}}P\left( {\left\| {S_n } \right\| \geqslant \sigma \varepsilon \sqrt 2 \left| n \right|\log \log \left| n \right|} \right)} \hfill \\ = \frac{{K(\Sigma )(d - 1)^{\frac{{l - 2}} {2}} \Gamma (b + l/2)}} {{\Gamma (l/2)(d - 1)!}} \hfill \\ \end{gathered} , where Γ(·) is the Gamma function and $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } $ \prod\limits_{i = l + 1}^\infty {((\sigma ^2 - \sigma _i^2 )/\sigma ^2 )^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}} } .     

A weak law for normed weighted sums of random elements in rademacher type p banach spaces

For weighted sums Σ_{j = 1}ⁿa_jV_j of independent random elements {V_n, n ≥ 1} in real separable, Rademacher type p (1 ≤ p ≤ 2) Banach spaces, a general weak law of large numbers of the form (Σ_{j = 1}ⁿa_jV_j − v_n)/b_n →^p 0 is established, where {v_n, n ≥ 1} and b_n → ∞ are suitable sequences. It is assumed that {V_n, n ≥ 1} is stochastically dominated by a random element V, and the hypotheses involve both the behavior of the tail of the distribution of |V| and the growth behaviors of the constants {a_n, n ≥ 1} and {b_n, n ≥ 1}. No assumption is made concerning the existence of expected values or absolute moments of the {V_n, n >- 1}.     

A general metric regularity in asplund banach spaces

This paper establishes a simple and easily-applied criterion for determining whether a multivalued mapping is metrically regular relatively to a subset in the range space.     

Generalized LIL for geometrically weighted random series in Banach spaces

Let $\{X,X_n;n?0\}$ be a sequence of independent and identically distributed random variables, taking values in a separable Banach space B with topological dual $B^{?}$. Considering the geometrically weighted series $\xi (\beta )={\sum }_{n=0}^{\infty }{\beta }^n{X}_n$ for $0<\beta <1$, motivated by Einmahl and Li (2005, 2008), a general law of the iterated logarithm for $\xi (\beta )$ is established.     

LIL behavior for weakly dependent random variables in Banach spaces

Given a sequence of identically distributed ψ-mixing random variables {X _n ; n ≧ 1} with values in a type 2 Banach space B, under certain conditions, the law of the iterated logarithm for this sequence is obtained without second moment.     

Complemented subspaces of sums and products of banach spaces

Summary We prove that, when X is one of the Banach spaces l^p (1p ) or c₀, then every infinite-dimensional complemented subspace of X^N (resp. X^(N)) is isomorphic to one of the following spaces: (, X, × X, X^N (resp. , X, X, X^(N)). Therefore, X^N and X^(N) are primary. We also give some consequences and related results.The second author acknowledges partial support from the Italian Ministero della Pubblica Istruzione.     

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.     

A nonclassical LIL for sums of B-valued random variables when extreme terms are excluded

Let {X,X _n ; n≧1} be a sequence of B-valued i.i.d. random variables. Denote $X_{{n}}^{(r)}=X_{{m}}$ if ∥X _m ∥ is the r-th maximum of {∥X _k ∥; k≦n}, and let ${}^{(r)}S_{{n}}=S_{{n}}-(X_{{n}}^{(1)}+\cdots+X_{{n}}^{(r)})$ be the trimmed sums, where $S_{{n}}=\sum_{ k=1}^{n}X_{{k}}$ . Given a sequence of positive constants {h(n), n≧1}, which is monotonically approaching infinity and not asymptotically equivalent to loglogn, a limit result for $^{(r)}S_{{n}}/\sqrt{2nh(n)}$ is derived.     

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.     

Limsup results and LIL for finite dimensional Gaussian random fields

We establish strong limsup theorems related to the law of the iterated logarithm (LIL) for finite dimensional Gaussian random fields by using the second Borel-Cantelli lemma. Supported by KRF-2003-C00098.     

Bounds for some general sums of random variables

Rogers and Shi (1995) have used the technique of conditional expectations to derive approximations for the distribution of a sum of lognormals. In this paper we extend their results to more general sums of random variables. In particular we study sums of functions of dependent random variables that are multivariate normally distributed and also derive results for sums of functions of dependent random variables from the additive exponential dispersion family. The usefulness of our results for practical applications is also discussed.     

Relative stability of trimmed sums

Summary This paper gives extensions of Mori's strong law for ^(r) S _n =S _n –X _n ^(1)} ...–X _n ^(r) , where S _n =X₁+X₂+...+X _n ,X _i are iidrv's and (X _i ⁿ ()) is (X _i ) arranged in decreasing order of absolute magnitude. The methods differ from Mori's. Continuity of the distribution of the X _i is assumed throughout. Necessary and sufficient conditions for relative stability (^(r) S _n /B _n ±1 a.s. for some B _n ), including a generalised condition of Spitzer's and a dominated ergodic theorem, are proved. A one-sided version of the relative stability results is also given. A theorem of Kesten's is generalised to show that if (^(r) S _n –A _n )/B _n is bounded almost surely for constants A _n ,B _n + then for some _n . A corollary to this is that if ¦ ^(r) S _n ¦/B _n is bounded away from 0 and + a.s. then ^(r) S _n is relatively stable. This generalises a result of Chow and Robbins, apart from the continuity assumption.     

Fourier sums for the banach indicatrix

We prove the existence of a functionf(t), which is continuous on the interval [0, 1], is of bounded variation, minf(t)=0, maxf(t)=1, for which the integral $$I(x) = \frac{1}{\pi }\int_0^\infty {\left[ {\int_0^1 {cos} y(f(l) - x)\left| {df(l)} \right|} \right]} dy$$ diverges for almost all X ∈ [0, 1]. This result gives a negative answer to a question posed by Z. Ciesielski.     

Nonlinear perturbations and abstract evolution equations in general banach spaces

Summary Part I deals with the problem of determining sufficient conditions under which the sum of two m-accretive operators on a closed convex set Q₁ is m-accretive on Q₁. Part II is concerned with the initial value problem: u′+Au+g(u)=v, u(0)=u₀. Applications are given to the Boltzmann equation. Entrata in Redazione il 2 luglio 1975.