Sequence Of Independent Random Variables Research Articles

Non-uniform rates of convergence for double arrays of independent random variables with applications

Abstract)Let be a double array of consteds and be a sequence of independent random variables. DenoteIt is well known thatunder some conditions, where is the standard normal distributionfunction.This paper studies the rate of convergence of Fn(x) toΦ(x) depending on n and x wherex is in a neighborhood of the origin under minimal conditions. Explicitly, we establish thefollowing theorems.Theorem 1. Suppose thatReceived January 20, 1994. Revised February 22, 1995.where 0 1. Then, there exist n) b 0 (not depending on n) such that for all n alld allx E R with lxl 5 "L.'/',Theorem 2. Suppose that (4) holds, 0 L. - 0 and t.n E N is a sequence ofconstants with t. ￣ co such that t. ￣ o(L.'/'). ThenRemark 1. Noting that L. S sup la.kl/A. ￣ L;, therefore, Theorems 1 and 2 alsokhold if L. whenever it appears, is replaced by L;.For the particular caseSa reduces to the partial sum Z:=, uk, which has been studied extensively in the literature(see [1,2]). In this cajse, A: = n, L. = .--i. It is well known that Theorem 2 can't beremoved under (4).In time series analysis, an important class of covariance stationary models is defined bywhere {Ek, --co k co) is a sequence of independent random variables and Cd areconstants. The sequence {Z j, j 2 1} is called lillear process generated by {Ek, --co k co}. Since Z:=, Zj = ZZ=--co ankEk where ark = Z:=, Cj--k, making use of Theorems 1and 2 we obtain.Theorem 3. Let (4) hold, Cd be constants such thatThen, there exist n, b 0 such that for all n all x E R with lxl 5 un'/6,Theorem 4. Assume that the conditions of Theorem 3 hold and t. is a sequence ofconstants with t. ￣ co such that t. ￣ o(n'/'). ThenRemark 2. for the following Ck,where the function l(x) 2 0 is slowly varying at infinity, a 2 1 and in or = t, ZC=, k--'l(k) co, Halll61 obtained the rate of convergence of F.(x) to .(x). Applying the same methodas that of dealing with aam ￣ Z:=, sj--k in [6], we can obtain some results similar to (10),(11) for Cd satisfying (12).TO conclude this paper, we give some classical examples (the so-called summabilitymethods) including the most importallt ones from the paint of view of probability (see [3--5]for details. In [51, the authors have dealt with the rate of convergence for SummabilityMethods "lider an assumption of finite third moments).1. Cesaro Sum:2. Abel Sum:3. Borel Sum:4. Euler Sum: (0 p 1)5. Delayed Sum (or RI Sum): (e 0)6. Valiron Sum:7. Taylor Sum: (0 or 1)8. Meyer-K5nig Sum: (0 a 1)

Limit Theorems of Probability Theory: Sequences of Independent Random Variables.

14. Limit Theorems of Probability Theory: Sequences of Independent Random Variables. By V. V. Petrov. ISBN 0 19 853499 X. Clarendon, Oxford, 1995. x + 292 pp. £50.00.

Changepoint Problems Under Random Censorship

Let X 1, …, X [nθ], X [nθ] + 1, …, X n be a sequence of independent random variables (the “lifetimes”) such that X j ˜ F 1 for 1 ≤ j ≤ [nθ] and X j ˜ F 2 for [nθ] + 1 ≤ j ≤ n, with F 1 ≠ F 2 unknown. In this paper we investigate an estimator θ n for the changepoint θ if the X's are subject to censoring. The rate of almost sure convergence of θ n to θ is established and a test for the hypothesis θ = 0, i.e. “no change”, is proposed.

Records from a power model of independent observations

Records from are analyzed, where {Yj } is an independent sequence of random variables. Each Yj has a continuous distribution function Fj = Fλj for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.

Records from a power model of independent observations

Records from are analyzed, where {Yj} is an independent sequence of random variables. Each Yj has a continuous distribution function Fj = Fλj for some distribution F and some λ j > 0. We study records, record times and related quantities for this sequence. Depending on the sequence of powers , a wide spectrum of behaviour is exhibited.

Average error in the central limit theorem for the cumulative processes

A central limit theorem with a remainder term for a cumulative process W(t) was proved by the author in an earlier paper. Here we show that the average of maximum errors taken over all values of t is actually smaller than what one would expect if a formula for the worst possible case for each t were used. This improvement is in line with what Heyde and Leslie (1972) obtained in a case of central limit theorems for the sequences of independent random variables.

On the law of the iterated logarithm for a sequence of independent random variables

This report discusses the law of the iterated logarithm for a sequence of independent randem variables. Three theorems and a gamma are given.

Decoupling Inequalities for the Tail Probabilities of Multivariate $U$-Statistics

In this paper we present a decoupling inequality that shows that multivariate $U$-statistics can be studied as sums of (conditionally) independent random variables. This result has important implications in several areas of probability and statistics including the study of random graphs and multiple stochastic integration. More precisely, we get the following result: Let $\{X_j\}$ be a sequence of independent random variables on a measurable space $(\mathscr{J}, S)$ and let $\{X^{(j)}_i\}, j = 1,\ldots,k$, be $k$ independent copies of $\{X_i\}$. Let $f_{i_1i_2\ldots i_k}$ be families of functions of $k$ variables taking $(S \times \cdots \times S)$ into a Banach space $(B, \|\cdots\|)$. Then, for all $n \geq k \geq 2, t > 0$, there exist numerical constants $C_k$ depending on $k$ only so that $P\bigg(\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(1)}_{i_2}, \ldots, X^{(1)}_{i_k})\big\|\geq t\bigg)$ $\leq C_kP\bigg(C_k\big\|\sum_{1\leq i_1\neq i_2\neq\cdots\neq i_k\leq n} f_{i_1\cdots i_k}(X^{(1)}_{i_1}, X^{(2)}_{i_2}, \ldots, X^{(k)}_{i_k})\big\|\geq t\bigg).$ The reverse bound holds if, in addition, the following symmetry condition holds almost surely: $f_{i_1i_2\cdots i_k}(X_{i_1}, X_{i_2},\ldots,X_{i_k}) = f_{i_{\pi(1)}i_{\pi(2)}\cdots i_{\pi(k)}} (X_{i_{\pi(1)}, X_{i_{\pi(2)}}, \ldots,X_{i_{\pi(k)}}),$ for all permutations $\pi$ of $(1,\ldots,k)$.

On non-consistency of estimators

Let be a sequence of general filtered statistical parametric models. We describe the limit behaviour of the maximum likelihood estimators and of the Bayesian estimators of θ when the necessary conditions of existence of a consistent estimator of θ are not satisfied. As examples we consider statistical models corresponding to a sequence of independent random variables, Poisson processes, diffusion processes.

Extremes and clustering of nonstationary max-AR(1) sequences

We consider general nonstationary max-autoregressive sequences X i , i ⩾ 1, with X i = Z i max( X i − 1 , Y i ) where Y i , i ⩾ 1 is a sequence of i.i.d. random variables and Z i , i ⩾ 1 is a sequence of independent random variables (0 ⩽ Z i ⩽ 1), independent of Y i . We deal with the limit law of extreme values M n = max X i , i ⩽ n (as n → ∞) and evaluate the extremal index for the case where the marginal distribution of Y i is regularly varying at ∞. The limit of the point process of exceedances of a boundary u n by X i , i ⩽ n, is derived (as n → ∞) by analysing the convergence of the cluster distribution and of the intensity measure.

A class of strong laws for functionals of countable nonhomogeneous Markov chains

In this paper, we use an analytic approach to study the limit properties of { inn(X n)}, the functional of a countable nonhomogeneous Markov chain { X n }. A class of strong laws of large numbers for these processes, which are different from the usual ones, are obtained. In the theorems of this paper, the expectation E( f n ( X n )) in the usual strong laws of large numbers is replaced by the conditional expectation E( f n ( X n ( X n−1 )). Some classical strong laws of large numbers for sequences of independent random variables are implied by the results of this paper.

Nonstandard local empirical processes indexed by sets

Abstract Let μn denote the empirical measure based on the first n observations from a sequence of independent random variables with common distribution μ in . Let be a class of Borel subsets of . For any and sequence k(n) with (log log n)−1k(n) → c ϵ (0, ∞), let , , be the empirical process at indexed by . Under suitable assumptions on μ and , we obtain functional laws of the iterated logarithm for .

Asymptotic distributions of records in nonstationary schemes

Abstract We investigate an asymptotic behaviour of record values X ( n ) for nonstationary sequences of independent random variables. The so-called F α -scheme is considered. It is shown that the set of all possible asymptotic distributions for X ( n ) under some conditions on parameters of the F α -scheme coincides with the corresponding set of asymptotic distributions in the stationary scheme.

Best constants in Kahane-Khintchine inequalities for complex Steinhaus functions

Let { φ k } k ≥ 1 {\{ {\varphi _k}\} _{k \geq 1}} be a sequence of independent random variables uniformly distributed on [ 0 , 2 π [ [0,2\pi [ , and let ‖ ∙ ‖ ψ {\left \| \bullet \right \|_\psi } denote the Orlicz norm induced by the function ψ ( x ) = exp ⁡ ( | x | 2 ) − 1 \psi (x) = \exp (|x{|^2}) - 1 . Then \[ ‖ ∑ k = 1 n z k e i φ k ‖ ψ ≤ 2 ( ∑ k = 1 n | z k | 2 ) 1 / 2 {\left \| {\sum \limits _{k = 1}^n {{z_k}{e^{i{\varphi _k}}}} } \right \|_\psi } \leq \sqrt 2 {\left ( {\sum \limits _{k = 1}^n {|{z_k}{|^2}} } \right )^{1/2}} \] for all z 1 , … , z n ∈ C {z_1}, \ldots ,{z_n} \in {\mathbf {C}} and all n ≥ 1 n \geq 1 . The constant 2 \sqrt 2 is shown to be the best possible. The method of proof relies upon a combinatorial argument, Taylor expansion, and the central limit theorem. The result is additionally strengthened by showing that the underlying functions are Schur-concave. The proof of this fact uses a result on the multinomial distribution of Rinott, and Schur’s proposition on the sum of convex functions. The estimates obtained throughout are shown to be the best possible. The result extends and generalizes to provide similar inequalities and estimates for other Orlicz norms.

Formula for the joint distribution of stack filters

A formula for the joint distribution function of two stack filters with common arguments is derived. Using this formula, the joint distribution function of any two samples in the output of a stack filter can be obtained. Here, it is presumed that the input signal is modelled as a sequence of independent random variables. This also allows analytic determination of the autocorrelation function of the output of an arbitrary stack filter.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Measure change estimates for hidden Markov models

Using a change of measure, the finite state observation process of a Markov chain is transformed into a sequence of independent random variables. By computing unnormalized conditional estimates under the new measure, simple, recursive formulate are obtained.

A Central Limit Theorem for Cumulative Processes

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

A Central Limit Theorem for Cumulative Processes

A central limit theorem for cumulative processes was first derived by Smith (1955). No remainder term was given. We use a different approach to obtain such a term here. The rate of convergence is the same as that in the central limit theorems for sequences of independent random variables.

Estimation of density functionals

Given a sequence of independent random variables with densityf we estimate quantities θ of the form θ=∫ φ(f(x))dx, φ a known function, by inserting histograms and kernel density estimators for the unknownf. We obtain conditions for consistency and asymptotic normality and discuss the choice of cell size and bandwidth.

On Probabilities of Linear Sets

If ( X k ) ∞ k = 1 is a sequence of independent random variables with probabilities of atoms bounded away from one and E is a Borel linear subspace of R ∞, then the event {( X k ) ∞ k = 1 ∈ E} is a.s. equivalent to the event {( X k ) n k = 1 ∈ F}, where F is an affine subspace of R n for some n ≥ 1.