Limit Theorems for Randomly Selected Ratios of Order Statistics from a Pareto Distribution

Abstract

Consider independent and identically distributed random variables {X _nk , 1 ≤ k ≤ m, n ≥ 1} from the Pareto distribution. We randomly select a pair of order statistics from each row, X _n(i) and X _n(j), where 1 ≤ i < j ≤ m. Then we test to see whether or not Strong and Weak Laws of Large Numbers with nonzero limits for weighted sums of the random variables X _n(j)/X _n(i) exist where we place a prior distribution on the selection of each of these possible pairs of order statistics.     

On the strong convergence of weighted sums

Let {X _n ,n≥1} be a sequence of independent and identically distributed random variables and {a _ni ,1≤i≤n,n≥1} an array of constants. Some strong convergence results for the weighted sums ∑ _i=1 ⁿ a _ni X _i are obtained.     

A note on asymptotics of linear combinations of iid random variables

Let X₁,X₂, ... be iid random variables, and let a _n = (a ₁,n, ..., a _n,n ) be an arbitrary sequence of weights. We investigate the asymptotic distribution of the linear combination $ S_{a_n } $ S_{a_n } = a _1,n X ₁ + ... + a _n,n X _n under the natural negligibility condition lim _n→∞ max{|a _k,n |: k = 1, ..., n} = 0. We prove that if $ S_{a_n } $ S_{a_n } is asymptotically normal for a weight sequence a _n , in which the components are of the same magnitude, then the common distribution belongs to $ \mathbb{D} $ \mathbb{D} (2).     

Convergence of Point Processes with Weakly Dependent Points

For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process $N_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}}For each n≥1, let {X _j,n }_1≤j≤n be a sequence of strictly stationary random variables. In this article, we give some asymptotic weak dependence conditions for the convergence in distribution of the point process N_n=?_j=1ⁿd_Xj,nN_{n}=\sum_{j=1}^{n}\delta_{X_{j,n}} to an infinitely divisible point process. From the point process convergence we obtain the convergence in distribution of the partial sum sequence S _n =∑ _j=1 ⁿ X _j,n to an infinitely divisible random variable whose Lévy measure is related to the canonical measure of the limiting point process. As examples, we discuss the case of triangular arrays which possess known (row-wise) dependence structures, like the strong mixing property, the association, or the dependence structure of a stochastic volatility model.     

A Refinement of the Kolmogorov–Marcinkiewicz–Zygmund Strong Law of Large Numbers

Let {X _n ; n≥1} be a sequence of independent copies of a real-valued random variable X and set S _n =X ₁+???+X _n , n≥1. This paper is devoted to a refinement of the classical Kolmogorov–Marcinkiewicz–Zygmund strong law of large numbers. We show that for 0<p<2,

$\sum_{n=1}^{\infty}\frac{1}{n}\biggl(\frac{|S_{n}|}{n^{1/p}}\biggr)<\infty\quad \mbox{almost surely}$

if and only if

$\begin{cases}\mathbb{E}|X|^{p}<\infty &; \mbox{if }0 < p < 1,\\\mathbb{E}X=0,\ \sum_{n=1}^{\infty}\frac{|\mathbb{E}XI\{|X|\leq n\}|}{n}<\infty,\mbox{ and }\\\sum_{n=1}^{\infty}\frac{\int_{\min\{u_{n},n\}}^{n}\mathbb{P}(|X|>t)\,dt}{n}<\infty &; \mbox{if }p = 1,\\\mathbb{E}X=0\mbox{ and }\int_{0}^{\infty}\mathbb{P}^{1/p}(|X|>t)\,dt<\infty,&;\mbox{if }1 < p < 2,\end{cases}$

where \(u_{n}=\inf \{t:~\mathbb{P}(|X|>t)<\frac{1}{n}\}\), n≥1. Versions of the above result in a Banach space setting are also presented. To establish these results, we invoke the remarkable Hoffmann-Jørgensen (Stud. Math. 52:159–186, 1974) inequality to obtain some general results for sums of the form \(\sum_{n=1}^{\infty}a_{n}\|\sum_{i=1}^{n}V_{i}\|\) (where {V _n ; n≥1} is a sequence of independent Banach-space-valued random variables, and a _n ≥0, n≥1), which may be of independent interest, but which we apply to \(\sum_{n=1}^{\infty}\frac{1}{n}(\frac{|S_{n}|}{n^{1/p}})\).

     

Local probabilities for random walks conditioned to stay positive

Let S ₀ = 0, {S _n , n ≥ 1} be a random walk generated by a sequence of i.i.d. random variables X ₁, X ₂, . . . and let $\tau ^{-}={\rm min} \{ n \geq 1:S_{n}\leq 0 \}$ and $\tau ^{+}={\rm min}\{n\geq1:S_{n} > 0\} $ . Assuming that the distribution of X ₁ belongs to the domain of attraction of an α-stable law we study the asymptotic behavior, as ${n\rightarrow \infty }$ , of the local probabilities ${\bf P}{(\tau ^{\pm }=n)}$ and prove the Gnedenko and Stone type conditional local limit theorems for the probabilities ${\bf P}{(S_{n} \in [x,x+\Delta )|\tau^{-} > n)}$ with fixed Δ and ${x=x(n)\in (0,\infty )}$ .     

Record values and the exponential distribution

A sequence {X _n,n≧1} of independent and identically distributed random variables with continuous cumulative distribution functionF(x) is considered.X _j is a record value of this sequence ifX _j>max (X ₁, …,X _j−1). Let {X _L(n) n≧0} be the sequence of such record values. Some properties ofX _L(n) andX _L(n)−X_L(n−1) are studied when {X _n,n≧1} has the exponential distribution. Characterizations of the exponential distribution are given in terms of the sequence {X _L(n),n≧0} The work was partly completed when the author was at the Department of Statistics, University of Brasilia, Brazil.     

Strassen-Type Law of the Iterated Logarithm for Self-normalized Sums

Let {X,X _n ,n≥1} be a sequence of independent identically distributed random variables with EX=0 and assume that EX ² I(|X|≤x) is slowly varying as x→∞. In this paper it is shown that a Strassen-type law of the iterated logarithm holds for self-normalized sums of such random variables, i.e., when X is in the domain of attraction of the normal law.     

On the almost sure central limit theorem for randomly indexed sums

Let {S_n, n ≥ 1} be partial sums of independent identically distributed random variables. The almost sure version of CLT is generalized on the case of randomly indexed sums {S_Nn, n ≥ 1}, where {N_n, n ≥ 1} is a sequence of positive integer‐valued random variables independent of {S_n, n ≥ 1}. The affects of nonrandom centering and norming are considered too (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The multiplier ideal is a universal test ideal

Abstract

For a canonical threefold X, we know that h ⁰(X, 𝒪 _X (nK _X )) ≥ 1 for a sufficiently large n. When χ(𝒪 _X ) > 0, it is not easy to get such an integer n. Fletcher showed that h ⁰(X, 𝒪 _X (12K _X )) ≥ 1 and h ⁰(X, 𝒪 _X (24K _X )) ≥ 2 when χ(𝒪 _X ) = 1. He inquired about existence of a canonical threefold with given conditions which shows the result sharp. We show that such an example does not exist. Using a different technique, we prove h ⁰(X, 𝒪 _X (12K _X )) ≥ 2.     

Precise rates in the law of the logarithm for the moment convergence in Hilbert spaces

Let （X, Xn; n ≥1） be a sequence of i.i.d, random variables taking values in a real separable Hilbert space （H, ｜｜·｜｜） with covariance operator ∑. Set Sn = X1 ＋ X2 ＋ ... ＋ Xn, n≥ 1. We prove that, for b 〉 -1,
lim ε→0 ε^2（b＋1） ∞ ∑n=1 （logn）^b/n^3/2 E{｜｜Sn｜｜-σε√nlogn}=σ^-2（b＋1）/（2b＋3）（b＋1） B｜｜Y｜^2b＋3
holds if EX=0,and E｜｜X｜｜^2（log｜｜x｜｜）^3bv（b＋4）〈∞ where Y is a Gaussian random variable taking value in a real separable Hilbert space with mean zero and covariance operator ∑, and σ^2 denotes the largest eigenvalue of ∑.     

Uncorrelatedness and correlatedness of powers of random variables

Let x₁, ?, x_n \xi_1, \ldots, \xi_n be random variables and U be a subset of the Cartesian product \mathbbZ₊ⁿ, \mathbbZ₊ \mathbb{Z}_+^n, \mathbb{Z}_+ being the set of all non-negative integers. The random variables are said to be strictly U-uncorrelated if¶¶E(x₁^j₁ ?x_n^j_n) = E(x₁^j₁) ?E(x_n^j_n) ? (j₁, ... ,j_n) ? U. \textbf {E}\big(\xi_1^{j_1} \cdots \xi_n^{j_n}\big) = \textbf {E}\big(\xi_1^{j_1}\big) \cdots \textbf {E}\big(\xi_n^{j_n}\big) \iff (j_1, \dots ,j_n) \in U. ¶It is proved that for an arbitrary subset U \subseteqq \mathbbZ₊ⁿ U \subseteqq \mathbb{Z}_+^n containing all points with 0 or 1 non-zero coordinates there exists a collection of n strictly U-uncorrelated random variables.     

A general LIL for B-valued geometrically weighted series under dependent assumption

Let {X,X _n ; n?R0} be a sequence of identically distributed ??-mixing dependent random variables taking values in a type 2 Banach space B with topological dual B ^?. Considering the geometrically weighted series $\xi(\beta)= \sum_{n=0}^{\infty}\beta^{n}X_{n}$ for 0<??<1, a general law of the iterated logarithm for ??(??) is obtained without second moment.