Sequence Of Independent Random Variables Research Articles

Density and Correlation in a Random Sequential Adsorption Model

We consider the random sequential adsorption process on the one- dimensional lattice with nearest-neighbor exclusion. In this model, each site s 2 Z starts empty and a particle will be deposited in it at time ts, where (ts)s2Z is a sequence of independent random variables uniformly distributed on the interval [0; 1]. The site will be occupied if both of its neighbors are vacant. Analytical expressions for the density of occupied sites and the pair correlation function, for all time t, are well-established and have been obtained through methods such as generating functions and differential equations. In this study, we present a method based on probabilistic arguments for the calculation of these expressions.

Asymptotic Duration for Optimal Multiple Stopping Problems

We study the asymptotic duration of optimal stopping problems involving a sequence of independent random variables that are drawn from a known continuous distribution. These variables are observed as a sequence, where no recall of previous observations is permitted, and the objective is to form an optimal strategy to maximise the expected reward. In our previous work, we presented a methodology, borrowing techniques from applied mathematics, for obtaining asymptotic expressions for the expectation duration of the optimal stopping time where one stop is permitted. In this study, we generalise further to the case where more than one stop is permitted, with an updated objective function of maximising the expected sum of the variables chosen. We formulate a complete generalisation for an exponential family as well as the uniform distribution by utilising an inductive approach in the formulation of the stopping rule. Explicit examples are shown for common probability functions as well as simulations to verify the asymptotic calculations.

Strong convergence of peaks over a threshold

AbstractExtreme value theory plays an important role in providing approximation results for the extremes of a sequence of independent random variables when their distribution is unknown. An important one is given by the generalised Pareto distribution $H_\gamma(x)$ as an approximation of the distribution $F_t(s(t)x)$ of the excesses over a threshold t, where s(t) is a suitable norming function. We study the rate of convergence of $F_t(s(t)\cdot)$ to $H_\gamma$ in variational and Hellinger distances and translate it into that regarding the Kullback–Leibler divergence between the respective densities.

The limit joint distributions of statistics of three tests of the NIST package

Abstract For sequences of independent random variables having a Bernoulli distribution the joint distribution and the limit joint distribution of statistics of three tests of the NIST statistical package («Monobit Test», «Frequency Test within a Block», and «Cumulative Sums Test») are obtained. In the case when two blocks are used in «Frequency Test within a Block», pairwise covariance of these statistics is given.

Almost sure central limit theorems for the maxima of randomly chosen random variables

In this paper, we give an almost sure central limit theorem (ASCLT) version of a maximum limit theorem (MLT) with an arbitrary sequence {dn, n ≥ 1} of weighted means of max{Xk, k ∈ An}, where {Xn, n ≥ 1} is a sequence of independent random variables, and {An, n ≥ 1} is a sequence of almost surely finite random subsets of positive integers independent of {Xn, n ≥ 1}. Thus we generalize the cases considered in the literature: (i) the nonrandom version of ASCLT for the MLT; (ii) the version of ASCLT for randomly indexed MLT; and (iii) the version of maximum schema of observed and unobserved random variables. We complete the paper with illustrative examples.

Higher-order expansions of sample range from general error distribution

Let be a sequence of independent random variables with common general error distribution with shape parameter , and denote and the partial maximum and minimum of . With different normalizing constants, the distributional expansions of normalized sample range are established in this article. A byproduct is to deduce the convergence rates of distributions of normalized sample range to their limits, which shows that the optimal convergence rate is proportional to as contrary to the case of , which is proportional to . Furthermore, numerical analysis is provided to illustrate the theoretical findings.

The limit joint distributions of statistics of four tests of the NIST package

Abstract For sequences of independent random variables having a Bernoulli distribution with parameter p the limit joint distribution of statistics of four tests of the NIST statistical package (« Monobit Test », « Frequency Test within a Block », « Runs Test » and a generalization of « Non-overlapping Template Matching Test ») is obtained. Conditions of asymptotic uncorrelatedness and/or asymptotic independence of these statistics are given.

Algebraic structures in the set of sequences of independent random variables

The space whose subsets we analyse with respect to lineability is L_0(Omega ,P)^{mathbb {N}} consisting of random variables sequences on probability space (Omega ,P) with atomless probability measure P. We study lineability and algebrability of L_0(Omega ,P)^{mathbb {N}}-subsets of independent random variables with additional properties connected with various types of convergence, laws of large numbers, and Markov and Kolgomorov conditions.

Rates of convergence of powered order statistics from general error distribution

ABSTRACT Let be a sequence of independent random variables with common general error distribution with shape parameter v>0, and let denote the rth largest order statistics of . With different normalizing constants the distributional expansions and the uniform convergence rates of normalized powered order statistics are established. An alternative method is presented to estimate the probability of the rth extremes. Numerical analyses are provided to support the main results.

On the (p,q)$(p,q)$‐type strong law of large numbers for sequences of independent random variables

AbstractLi, Qi, and Rosalsky (Trans. Amer. Math. Soc., 368 (2016), no. 1, 539–561) introduced a refinement of the Marcinkiewicz–Zygmund strong law of large numbers (SLLN), the so‐called ‐type SLLN, where and . They obtained sets of necessary and sufficient conditions for this new type SLLN for two cases: , , and . Results for the case where and remain open problems. This paper gives a complete solution to these problems. We consider random variables taking values in a real separable Banach space , but the results are new even when is the real line. Furthermore, the conditions for a sequence of random variables satisfying the ‐type SLLN are shown to provide an exact characterization of stable type p Banach spaces.

Regression methods for the appearances of extremes in climate data

AbstractFor any given city, on any calendar day, there will be record high and low temperatures. Which record occurred earlier? If there is a trend towards warming then, intuitively, there should be a preponderance of record highs occurring more recently than the record lows for each of the 365 calendar days. We are interested in modeling the joint distribution of appearances of the extremes but not these values themselves. We develop a bivariate discrete distribution modeling the joint indices of maximum and minimum in a sequence of independent random variables sampled from different distributions. We assume these distributions share a proportional hazard rate and develop regression methods for these paired values. This approach has reasonable power to detect a small mean change over a decade. Using readily available public data, we examine the daily calendar extreme values of five US cities for the decade 2011–2020. We develop linear regression models for these data, describe models to account for calendar‐date dependence, and use diagnostic measures to detect remarkable observations.

On Asymptotics of Optimal Stopping Times

We consider optimal stopping problems, in which a sequence of independent random variables is drawn from a known continuous density. The objective of such problems is to find a procedure which maximizes the expected reward. In this analysis, we obtained asymptotic expressions for the expectation and variance of the optimal stopping time as the number of drawn variables became large. In the case of distributions with infinite upper bound, the asymptotic behaviour of these statistics depends solely on the algebraic power of the probability distribution decay rate in the upper limit. In the case of densities with finite upper bound, the asymptotic behaviour of these statistics depends on the algebraic form of the distribution near the finite upper bound. Explicit calculations are provided for several common probability density functions.

Berry-Esseen bounds for functionals of independent random variables

We derive Berry-Esseen approximation bounds for general functionals of independent random variables, based on a continuous-time integration by parts setting and discrete chaos expansions methods. Our approach improves on related results obtained in discrete-time integration by parts settings and applies to U-statistics satisfying the weak assumption of decomposability in the Hoeffding sense, and yield Kolmogorov distance bounds instead of the Wasserstein bounds previously derived in the special case of degenerate U-statistics. Linear and quadratic functionals of arbitrary sequences of independent random variables are included as particular cases, with new fourth moment bounds, and applications are given to Hoeffding decompositions, weighted U-statistics, quadratic forms, and random subgraph weighing. In the case of quadratic forms, our results recover and improve the bounds available in the literature, and apply to matrices with non-empty diagonals.

Simultaneous Joint Lower and Upper record values Probability Laws for Absolutely Continuous or Discrete Data

This paper investigates the probability density function (pdf) of the \((2n-1)\)-vector \((n\geq 1)\) of both lower and upper record values for a sequence of independent random variables with common \textit{pdf} \(f\) defined on the same probability space, provided that the lower and upper record times are finite up to \(n\). A lot is known about the lower or the upper record values when they are studied separately. When put together, the challenges are far complicated. The rare results in the literature still present some flaws. This paper begins a new and complete investigation with a few number of records: (n=2\) and \(n=3\). Lessons from these simple cases will allow addressing the general formulation of simultaneous joint lower-upper records.

Bivariate change point detection: Joint detection of changes in expectation and variance

AbstractA method for change point detection is proposed. We consider a univariate sequence of independent random variables with piecewise constant expectation and variance, apart from which the distribution may vary periodically. We aim to detect change points in both expectation and variance. For that, we propose a statistical test for the null hypothesis of no change points and an algorithm for change point detection. Both are based on a bivariate moving sum approach that jointly evaluates the mean and the empirical variance. The joint consideration helps improve inference compared with separate univariate approaches. We infer on the strength and the type of changes with confidence. Nonparametric methodology supports the analysis of diverse data. Additionally, a multiscale approach addresses complex patterns in change points and effects. We demonstrate the performance through theoretical results and simulation studies. A companion R‐package jcp (available on CRAN) is discussed.

Convergence for weighted sums of Lüroth type random variables

In this work we prove an asymptotic result, that under some conditions on the involved distribution functions, is valid for any Oppenheim expansion, extending a classical result proven by W. Vervaat in 1972 for denominators of the Lüroth case. Furthermore, we study the convergence in distribution of weighted sums of a sequence of independent random variables. Although the result is of its own interest, in the present setting it is used to prove convergence in distribution of specific sequences of random variables generalizing known results obtained for Lüroth random variables.

Counterexamples to the classical central limit theorem for triplewise independent random variables having a common arbitrary margin

Abstract We present a general methodology to construct triplewise independent sequences of random variables having a common but arbitrary marginal distribution F (satisfying very mild conditions). For two specific sequences, we obtain in closed form the asymptotic distribution of the sample mean. It is non-Gaussian (and depends on the specific choice of F). This allows us to illustrate the extent of the ‘failure’ of the classical central limit theorem (CLT) under triplewise independence. Our methodology is simple and can also be used to create, for any integer K, new K-tuplewise independent sequences that are not mutually independent. For K [four.tf], it appears that the sequences created using our methodology do verify a CLT, and we explain heuristically why this is the case.

Scanning for Clusters of Large Values in Time Series: Application of the Stein-Chen Method

The purpose of this application paper is to apply the Stein-Chen (SC) method to provide a Poisson-based approximation and corresponding total variation distance bounds in a time series context. The SC method that is used approximates the probability density function (PDF) defined on how many times a pattern such as It,It+1,It+2 = {1 0 1} occurs starting at position t in a time series of length N that has been converted to binary values using a threshold. The original time series that is converted to binary is assumed to consist of a sequence of independent random variables, and could, for example, be a series of residuals that result from fitting any type of time series model. Note that if {1 0 1} is known to not occur, for example, starting at position t = 1, then this information impacts the probability that {1 0 1} occurs starting at position t = 2 or t = 3, because the trials to obtain {1 0 1} are overlapping and thus not independent, so the Poisson distribution assumptions are not met. Nevertheless, the results shown in four examples demonstrate that Poisson-based approximation (that is strictly correct only for independent trials) can be remarkably accurate, and the SC method provides a bound on the total variation distance between the true and approximate PDF.

On multidimensional record patterns

Multidimensional record patterns are random sets of lattice points defined by means of a recursive stochastic construction. The patterns thus generated owe their richness to the fact that the construction is not based on a total order, except in one dimension, where usual records in sequences of independent random variables are recovered. We derive many exact results on the statistics of multidimensional record patterns on finite samples drawn on hypercubic lattices in any dimension D. The most detailed analysis concerns the two-dimensional situation, where we also investigate the distribution of the landing position of the record point which is closest to the origin. Asymptotic expressions for the full distribution and the moments of the number of records on large hypercubic samples are also obtained. The latter distribution is related to that of the largest of D standard Gaussian variables.

Segmentation and estimation of change-point models: False positive control and confidence regions

To segment a sequence of independent random variables at an unknown number of change-points, we introduce new procedures that are based on thresholding the likelihood ratio statistic, and give approximations for the probability of a false positive error when there are no change-points. We also study confidence regions based on the likelihood ratio statistic for the change-points and joint confidence regions for the change-points and the parameter values. Applications to segment array CGH data are discussed.