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A simple random walk and a Brownian motion are considered on a spider that is a collection of half lines (we call them legs) joined in the origin. We give a strong approximation of these two objects and their local times. For fixed number of legs we establish limit theorems on local and occupation times in n steps., Comment: 22 pages,contains small corrections,and some detailed explanations than the previous version

In this paper we introduce randomized $t$-type statistics that will be referred to as randomized pivots. We show that these randomized pivots yield central limit theorems with a significantly smaller magnitude of error as compared to that of their classical counterparts under the same conditions. This constitutes a desirable result when a relatively small number of data is available. When a data set is too big to be processed, we use our randomized pivots to make inference about the mean based on significantly smaller sub-samples. The approach taken is shown to relate naturally to estimating distributions of both small and big data sets., Comment: 37 pages and three tables. arXiv admin note: text overlap with arXiv:1307.5476

In this paper we introduce randomized pivots for the means of short and long memory linear processes. We show that, under the same conditions, these pivots converge in distribution to the same limit as that of their classical non-randomized counterparts. We also present numerical results that indicate that these randomized pivots significantly outperform their classical counterparts and as a result they lead to a more accurate inference about the population mean., Comment: 35 pages, 12 figures

In this paper we introduce the concept of bootstrapped pivots for the sample and the population means. This is in contrast to the classical method of constructing bootstrapped confidence intervals for the population mean via estimating the cutoff points via drawing a number of bootstrap sub-samples. We show that this new method leads to constructing asymptotic confidence intervals with significantly smaller error in comparison to both of the traditional t-intervals and the classical bootstrapped confidence intervals. The approach taken in this paper relates naturally to super-population modeling, as well as to estimating empirical and theoretical distributions., Comment: 50 pages

This paper studies the problem of testing the null assumption of no-change in the mean of chronologically ordered independent observations on a random variable $X$ {\it versus} the at most one change in the mean alternative hypothesis. The approach taken is via a Darling-Erd\H{o}s type self-normalized maximal deviation between sample means before and sample means after possible times of a change in the expected values of the observations of a random sample. Asymptotically, the thus formulated maximal deviations are shown to have a standard Gumbel distribution under the null assumption of no change in the mean. A first such result is proved under the condition that $EX^2\log\log (|X|+1)<\infty$, while in the case of a second one, $X$ is assumed to be in a specific class of the domain of attraction of the normal law, possibly with infinite variance.

This paper is mainly concerned with asymptotic studies of weighted bootstrap for u- and v-statistics. We derive the consistency of the weighted bootstrap u- and v-statistics, based on i.i.d. and non i.i.d. observations, from some more general results which we first establish for sums of randomly weighted arrays of random variables. Some of the results in this paper significantly extend some well-known results on consistency of u-statistics and also consistency of sums of arrays of random variables. We also employ a new approach to conditioning to derive a conditional CLT for weighted bootstrap u- and v-statistics, assuming the same conditions as the classical central limit theorems for regular u- and v-statistics.

Let X, X_1,X_2,... be a sequence of i.i.d. random variables with mean $\mu=E X$. Let ${v_1^{(n)},...,v_n^{(n)}}_{n=1}^\infty$ be vectors of non-negative random variables (weights), independent of the data sequence ${X_1,...,X_n}_{n=1}^\infty$, and put $m_n=\sumn v_i^{(n)}$. Consider $ X^{*}_1,..., X^{*}_{m_n}$, $m_n\geq 1$, a bootstrap sample, resulting from re-sampling or stochastically re-weighing a random sample $X_1,...,X_n$, $n\geq 1$. Put $\bar{X}_n= \sumn X_i/n$, the original sample mean, and define $\bar{X^*}_{m_n}=\sumn v_i^{(n)} X_i/m_n$, the bootstrap sample mean. Thus, $\bar{X^*}_{m_n}- \bar{X}_n=\sumn ({v_i^{(n)}}/{m_n}-{1}/{n}) X_i$. Put $V_n^{2}=\sumn ({v_i^{(n)}}/{m_n}-{1}/{n})^2$ and let $S_n^{2}$, $S_{m_{n}}^{*^{2}}$ respectively be the the original sample variance and the bootstrap sample variance. The main aim of this exposition is to study the asymptotic behavior of the bootstrapped $t$-statistics $T_{m_n}^{*}:= (\bar{X^*}_{m_n}- \bar{X}_n)/(S_n V_n)$ and $T_{m_n}^{**}:= \sqrt{m_n}(\bar{X^*}_{m_n}- \bar{X}_n)/ S_{m_{n}}^{*} $ in terms of conditioning on the weights via assuming that, as $n,m_n\to \infty$, $\max_{1\leq i \leq n}({v_i^{(n)}}/{m_n}-{1}/{n})^2\big/ V_n^{2}=o(1)$ almost surely or in probability on the probability space of the weights. This view of justifying the validity of the bootstrap is believed to be new. The need for it arises naturally in practice when exploring the nature of information contained in a random sample via re-sampling, for example. Conditioning on the data is also revisited for Efron's bootstrap weights under conditions on $n,m_n$ as $n\to \infty $ that differ from requiring $m_n /n$ to be in the interval $(\lambda_1,\lambda_2)$ with 0< \lambda_1 < \lambda_2 < \infty as in Mason and Shao. Also, the validity of the bootstrapped $t$-intervals for both approaches to conditioning is established., Comment: 33 pages