Extremes and student’s <Emphasis Type="Italic">t</Emphasis><Subscript>2</Subscript>-distribution

Some new characterizations of Student’s distribution on two degrees of freedom by regressional properties of maximal order statistics are obtained. Bibliography: 5 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 351, 2007, pp. 232–237.     

On families of distributions characterized by certain properties of ordered random variables

In this work, we obtain new characterizations of certain probability distributions by relations with different ordered random variables. Such variables include order statistics, sequential maxima, and records. We consider relations that include not only upper, but also lower record values. The presented ordered objects are based on sequences of independent random variables with a common continuous distribution function. We also investigate equalities in the distribution of sequential maxima exposed by various random shifts. These shifts (one-sided or two-sided) have exponential distributions. Certain theorems and their corollaries present corresponding characterizations of distributions by relations of such a type. In addition, we consider exponentially shifted order statistics such that simple relations among them also characterize certain probability distributions. All of the presented results yield a set of characterizations of various distributions. For particular cases, we present the relations that characterize families of classical exponential and logistic distributions.     

流体力学半径对估算胶体微球聚集速率常数的影响

胶体粒子聚集速率常数实验值远低于理论值一直是被普遍关注的问题.聚集速率常数的理论推导是基于粒子的几何半径来考虑的,但决定粒子扩散速率及聚集速率的应该是粒子的流体力学半径(大于几何半径),因而它是使聚集速率常数实验值低于理论值的因素之一.影响流体力学半径的因素很多,其中,带电粒子在溶液中因表面存在双电层,会明显增大流体力学半径,造成聚集速率减慢.而双电层的厚度又随溶液中离子强度的不同而改变.本工作在聚集速率的公式中引入了修正因子,即几何半径与其流体力学半径之比,以修正由于用几何半径代替流体力学半径带来的误差.其中几何半径和流体力学半径可以分别用扫描电镜(SEM)和动态光散射(DLS)来测定.以两种粒径的聚苯乙烯带电微球为例,考察了在不同离子强度下,该误差的大小.结果发现,对于半径为30 nm的微球,用流体力学半径计算的慢聚集速率常数比理论值偏低约8%.该误差随离子强度增加而减少.对于快聚集情况,流体力学半径对聚集速率基本没有影响.     

On the Problem of the Optimal Choice of Record Values

Let the independent random variables X1, X2, … have the same continuous distribution function. The upper record values X(1) = X₁ < X(2) < … generated by this sequence of variables, as well as the lower record values x(1) = X₁ > x(2) > …, are considered. It is known that in this situation, the mean value c(n) of the total number of the both types of records among the first n variables X is given by the equality c(n)=2(1+1/2+…+1/n), n = 1, 2, …. The problem considered here is following: how, sequentially obtaining the observed values x₁, x₂, … of variables X and selecting one of them as the initial point, to obtain the maximal mean value e(n) of the considered numbers of records among the rest random variables. It is not possible to come back to rejected elements of the sequence. Some procedures of the optimal choice of the initial element X _r are discussed. The corresponding tables for the values e(n) and differences δ(n)= e(n)–c(n) are presented for different values of n. The value of δ= lim_n→∞δ(n)is also given. In some sense, the considered problem and optimization procedure presented in this paper are quite similar to the classical “secretary problem,” in which the probability of selecting the last record value in the set of independent identically distributed X is maximized.     

Joint distribution of order statistics generated by records

Probability distributions of some order statistics connected with the k th records values are discussed. Some useful representations for the exponential records and the corresponding order statistics are given. Bibliography: 8 titles. Translated fromZapiski Nauchnykh Seminarrow POMI, Vol. 228, 1996, pp. 24–30.     

On the maximal value of the expectation of record numbers

There are n independent identically distributed random variables with a continuous distribution function. The problem considered is, getting sequentially the values of these variables and selecting one of them as an initial point, how we can maximize the expected number of records among the rest of this sequence of random variables (without knowledge of the future values).     

Peak effect studies in single crystals CeRu2 and 2H-NbS2

We have studied the peak effect (PE) phenomenon in single crystals of weakly pinned superconductors CeRu₂ and 2H-NbS₂. 2H-NbS₂ is iso-structural and iso-electronic to 2H-NbSe₂, whose similarity with CeRu₂ as regards the PE representing the order-to-disorder transformation of the flux line lattice was claimed some time ago. We report on the step change in equilibrium magnetization across the peak effect in CeRu₂. We also present the vortex phase diagram of 2H-NbS₂ obtained from the magnetization data, and compare the PE phenomenon in 2H-NbS₂ and 2H-NbSe₂.     

Rate of convergence to the normal law of sums of induced order statistics

One obtains a certain estimate of order 1/n of the rate of convergence to the normal law for induced order statistics. The result includes the multidimensional case.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 108, pp. 45–56, 1981.