Real Random Variables Research Articles

Asymptotic Behaviour of Stable Measures Near the Origin

We investigate the lower tail of $q_r = (\sum^\infty_{i=1} | \alpha_i\theta_i|^r)^{1/r}$ seminorms on $R^\infty$, where $r \geq 1$ and $\theta_i$ are standard $p$-stable real random variables. We prove that for $p < r \leq 2$ we have $P\{q_r \leq t\} \geq \exp\{-ct^{-pr/(r-p)}\}$ in some neighbourhood of 0, where $c$ is a nonnegative constant. If $r \leq p$, then for any positive, increasing function $f$, we can find $q_r$ such that $P\{q_r \leq t\} \leq f(t)$ for $t \leq 1$. We also give a new characterization of Banach spaces of stable type $p$ in terms of the behaviour of $\mu\{\| \cdot \| \leq t\}$ near 0, where $\mu$ is a symmetric and $p$-stable measure.

Studies in the history of probability and statistics XLI Waring and Sylvester on random algebraic equations

Modern research on random algebraic equations was initiated by Offord & Littlewood (1938), and has been concerned mainly with the probability distribution of the number of real roots of the equation xaixi = 0, where the ai are independent identically distributed real random variables. It is known that if they are normal with mean P * 0, the expectation of the number of real roots is asymptotically i'log n, while if pu = 0 it is 27-1 log n. Much more is known but there is unfortunately no monograph on the subject. Earlier than this, both Waring and Sylvester used probability in their discussion of algebraic equations. Their writings are of interest as early examples of the use of probabilitistic and statistical ideas in looking at pure mathematical problems. They involve studies of the Rule of Signs, stated without proof by Newton in his Lectures on Algebra, published as the Arithmetica Universalis (1707). Both writers present problems of interpretation. The rule was expressed by Sylvester (1864) as follows: we associate with the algebraic equation

A note on a theorem of Berkes and Philipp for dependent sequences

We improve an almost sure invariance principle for f-mixing sequences of real random variables with finite (2 + δ)th moment (0 < δ ⩽ 2) due to Berkes and Philipp (1979). The exponent of the slow logarithmic rate of mixing in that theorem is relaxed from 160/δ to (1 + ϵ)(1 + 2/δ) and the undetermined quantities a N are replaced by Nσ 2 for some σ > 0.

A Further Problem of Interval Estimation of an Unknown Regressor in Model I of Linear Regression

AbstractFor the model y = α + βx + ϵ (model I) of linear regression we dealt with in KUHNERT and HORN (1980) the determination of a confidence interval for that x0 where the expectation Ey reaches a given value y0. Here we start with realizations of random variables y (i = 1,…, m) being independent of x which are given in addition to the realizations of‐y. Now y0 denotes the unknown value of \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop \sum \limits_{i = 1}^m $\end{document} ciEy and x0 the x‐value where the expectation Ey reaches that value y0. For this x0 we give a confidence interval. Applications stem from dose response assays.

On Random Sequences with Centred Spherical Symmetry

SUMMARY An infinite sequence of real random variables, any finite sample from which has a centred spherically symmetric joint distribution, is shown to have the property that, conditional on two further random variables, M and V, the variables are independent with the same normal distribution N(M, V).

Characterization of exponential and Weibull distributions through truncated statistics

Abstract Let Y 1(a 1) be the result of truncating at a 1 the ±th power of each of n real random variables X 1 .... , X n. Suppose for some 1 ≤ s ≤ n − 1. (Y 1, ..., Y 1) and (Y a+l ..., Y n) are independent. Then all X 1 are mutually independent and follow the Weibull distribution it and only if EΠn j-1 Y(aj) depends on the aj only through their sum.

On the ordered partial sums of real random variables

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

On the ordered partial sums of real random variables

In 1952 Pollaczek discovered a remarkable formula for the Laplace-Stieltjes transforms of the distributions of the ordered partial sums for a sequence of independent and identically distributed real random variables. In this paper Pollaczek's result is proved in a simple way and is extended for a semi-Markov sequence of real random variables.

On the asymptotic efficiency of estimators in an autoregressive process

Let {Xt} be defined recursively byXt=θXt−1+Ut(t=1,2, ...), whereX0=0 and {Ut} is a sequence of independent identically distributed real random variables having a density functionf with mean 0 and varianceσ2. We assume that |θ|<1. In the present paper we obtain the bound of the asymptotic distributions of asymptotically median unbiased (AMU) estimators of θ and the sufficient condition that an AMU estimator be asymptotically efficient in the sense that its distribution attains the above bound. It is also shown that the least squares estimator of θ is asymptotically efficient if and only iff is a normal density function.

An extension of Tukey's method of multiple comparisons to experimental designs with random concomitant variables

SUMMARY A procedure is provided for the construction, of simultaneous confidence intervals on all contrasts of a set of parametric functions, or tests of hypotheses concerning them, in an analysis of covariance wherein the dependent and concomitant variates have a joint (p + 1)dimensional normal distribution. This new procedure is analogous to Tukey's method of multiple comparisons in the analysis of variance, both in application and in the conditions which are necessary for its use. The probability distribution of the range of adjusted parametric functions, suitably studentized, is derived and its critical values tabulated. The procedure is demonstrated using a two-way cross classification model. 1. ITTRODUCTION In many practical design and analysis of experiments situations, especially in psychometric settings, it is possible to obtain cheaply concomitant information in addition to the main response variable. Frequently, this information represents the realization of random variables. It is this situation which we shall address in the sequel. More precisely, we consider the model

A Simple proof of a Known Result in Random Walk Theory

Let $\{X_n, n \geqq 1\}$ be a stationary independent sequence of real random variables, $S_n = X_1 + \cdots + X_n$, and $\alpha_A$ the hitting time of the set $A$ by the process $\{S_n, n \geqq 1\}$, where $A$ is one of the half-lines $(0, \infty), \lbrack 0, \infty), (-\infty, 0 \rbrack$ or $(-\infty, 0)$. This note provides a simple proof of a known result in random walk theory on necessary and sufficient conditions for $E\{\alpha_A\}$ to be finite. The method requires neither generating functions nor moment conditions on $X_1$.

The oscillating random walk

{ Y n ; n=0, 1, …} denotes a stationary Markov chain taking values in R d . As long as the process stays on the same side of a fixed hyperplane E 0, it behaves as an ordinary random walk with jump measure μ or ν, respectively. Thus ordinary random walk would be the special case μ = ν. Also the process Y′ n = | Y′ n−1 − Z n | (with the Z n as i.i.d. real random varia bles) may be regarded as a special case. The general process is studied by a Wiener–Hopf type method. Exact formulae are obtained for many quantities of interest. For the special case that the Y n are integral-valued, renewal type conditions are established which are necessary and sufficient for recurrence.

Improving Subjective Probability Assessment for Planning and Control in Team-Like Organizations

Improving Subjective Probability Assessment for Planning and Control in Team-Like Organizations

Integration of direct limits of real random variables

In our preceding paper [1], we have studied the fundamental properties of the direct limits of measure spaces. The present article is devoted to the study of the integration of direct limits of direct systems of real measurable functions, and in particular, of direct limits of systems of real random variables. These results will be useful in the stochastic calculus whenever the direct limit theory is essentially employed.

Characterizing measurability, distribution and weak convergence of random variables in a Banach space by total subsets of linear functionals

Consider a generalized random variable X assuming values in a Banach space X with conjugate space X ∗. For separable or reflexive X the measurability, probability distribution, and other properties of X are characterized in terms of a collection of real random variables { a ∗(X) : a ∗ ∈ A } and their linear combinations, where A is a total subset of X ∗, i.e., A distinguishes points of X . Convergence in distribution of a sequence { X n } is characterized in terms of uniform convergence of finite-dimensional distributions formed from { x ∗(X n) : x ∗ ∈ X ∗} (for X separable) or from { a ∗(X n) : a ∗ ∈ A } (for X separable and reflexive). These results extend earlier ones known for the special cases A = X ∗ or X = C[0, 1]. The proofs are based on theorems of Banach, Krein, and Šmulian characterizing the weak ∗ closure of a convex set in X ∗. An application to random Fourier series is given.

A Counterexample on Translation Invariant Estimators

It seems to be generally known that the proof of the continuity part of Theorem 1 in Hodges' and Lehmann's paper (1963) is incorrect. The fact that the theorem is incorrect is--perhaps--not so well known. We show this by constructing independent real random variables $X_1,\cdots, X_n$, each having the same non-atomic symmetric distribution, and an odd translation invariant estimator $h(X_1,\cdots, X_n)$ such that $P(h(X_1,\cdots, X_n) = 0) > 0. h$ may be chosen symmetric provided $n \geqq 3$.

On the minimax estimator of an unknown mean value

where H is some set of estimators “by observation” of the random process 5 = KU), u E u. We consider the functional a(8), 6 E 0, which is linear in the sense that @) = 4&) + X2@,>, f i a function 19 = B(u) is O(u) = +9,(u) + A,&&), u E U, where A, , A, are constants. We assume that H is the Hilbert space of all (real) random variables 3, generated by the values I(G), u E U, with the usual inner product

The behavior of analog communication systems

We consider the problem of transmission of analog data over a noisy channel. It is assumed that the channel input is of the form \surd S f(t, X) , where X is an n -dimensional source vector, and S is the allowable transmitted power. The performance of any given modulation scheme f(t, \cdot ) as a function of the transmitted power S is studied. Lower bounds on the average distortion produced by noise for a class of distortion functions are derived. These bounds relate the smoothness of modulation techniques to the minimum error that can be achieved with them. It is shown that when the analog source emits a sequence of mutually independent real random variables at a rate of R per second, the mean-square error that is associated with any practical modulation scheme f(t, \cdot) decays no faster than S^{-2} as the signal power S \rightarrow \infty . It follows that in the case of a band-limited additive white Gaussian channel no single modulation scheme f(t, \cdot ) can achieve the ideal rate-distortion bound on the mean-square error for all values of S , if the channel bandwidth is larger than the source rate R .

Estimation for Monotone Parameter Sequences: The Discrete Case

Let $x_1, x_2, \cdots$ be a sequence of independent, observable, discrete, real random variables with respective density functions $f(x, \theta_1), f(x, \theta_2), \cdots$, where $\theta_i\in\Omega$ (a real interval), $i = 1,2, \cdots$. At any stage $s$, having observed $x_1, \cdots, x_s$, we wish to estimate $\theta_s$, relative to the loss function $L(d, \theta) \geqq 0$, under the restraint that $\theta_1 \leqq \cdots \leqq \theta_s$. Most of the literature concerned with estimating ordered parameters deals with maximum likelihood estimators (for example, Robertson and Waltman [3] and references contained therein). Recently, Blumenthal and Cohen [1] considered questions of minimaxity and admissibility of certain estimators of ordered translation parameters of continuous distributions. In Section 3 we obtain results concerning admissible and minimax estimators of $\theta_s$ for a large class of discrete distributions. Theorem 3.2 states that the minimax value $M_k$ of the problem of estimating $\theta_k$ at stage $k$ is the same for all $k = 1,2,\cdots$. Furthermore, the Corollary to Theorem 3.1 states that if $t(x_1, \cdots, x_s)$ is admissible for estimating $\theta_s$ at stage $s$, then $t(x_{k+1}, \cdots, x_{k+s})$ is admissible for estimating $\theta_{k+s}$ at stage $k + s$. This is not true, for example, when $x_k$ is normally distributed with mean $\theta_k$ and unit variance [4]. Hence in situations satisfying the conditions of Theorems 3.1 and 3.2, if there exists an admissible estimator $t(x_1)$ of $\theta_1$ having constant risk, then $t(x_s)$ will be the unique admissible minimax estimator of $\theta_s$, at stage $s$. It is this undesirable property (i.e. being based only on the last observation) which prompts us to look for other estimators which may be better in some sense. In Section 4 we use the early observations $(x_1, \cdots, x_{s-1}$, at stage $s$) to construct a sequence of estimators which is asymptotically subminimax $(s \rightarrow \infty)$ as well as having desirable properties for finite $s$. The results in Section 4 are obtained for general cumulative distribution functions as the methods are not peculiar to the discrete situation. However, it should be noted that the main results of Section 4 (Theorem 4.2 and Theorem 4.3) do not yield a worthwhile sequence of estimators unless we are in a situation similar to that of Section 3. That is, unless the existing admissible minimax estimators are undesirable as they are in the discrete case of Section 3.

The characteristic functions of functions

The characteristic functions of various functions of a real or vector random variable are expressed in terms of the characteristic function of that variable. In the examples there is special emphasis on the stable distributions that have real characteristic functions. Some of the results suggest the practicability of generalizing traditional multivariate analysis beyond the multi-Gaussian model.