Identically-distributed Random Variables Research Articles

Variance-Constrained Filtering Fusion for Nonlinear Cyber-Physical Systems With the Denial-of-Service Attacks and Stochastic Communication Protocol

In this paper, a new filtering fusion problem is studied for nonlinear cyber-physical systems under error-variance constraints and denial-of-service attacks. To prevent data collision and reduce communication cost, the stochastic communication protocol is adopted in the sensor-to-filter channels to regulate the transmission order of sensors. Each sensor is allowed to enter the network according to the transmission priority decided by a set of independent and identically-distributed random variables. From the defenders' view, the occurrence of the denial-of-service attack is governed by the randomly Bernoulli-distributed sequence. At the local filtering stage, a set of variance-constrained local filters are designed where the upper bounds (on the filtering error covariances) are first acquired and later minimized by appropriately designing filter parameters. At the fusion stage, all local estimates and error covariances are combined to develop a variance-constrained fusion estimator under the federated fusion rule. Furthermore, the performance of the fusion estimator is examined by studying the boundedness of the fused error covariance. A simulation example is finally presented to demonstrate the effectiveness of the proposed fusion estimator.

Quantum unitary evolution interspersed with repeated non-unitary interactions at random times: the method of stochastic Liouville equation, and two examples of interactions in the context of a tight-binding chain

In the context of unitary evolution of a generic quantum system interrupted at random times with non-unitary evolution due to interactions with either the external environment or a measuring apparatus, we adduce a general theoretical framework to obtain the average density operator of the system at any time during the dynamical evolution. The average is with respect to the classical randomness associated with the random time intervals between successive interactions, which we consider to be independent and identically-distributed random variables. The formalism is very general in that it applies to any quantum system, to any form of non-unitary interaction, and to any probability distribution for the random times. We provide two explicit applications of the formalism in the context of the so-called tight-binding model relevant in various contexts in solid-state physics, e.g. in modelling nano wires. Considering the case of one dimension, the corresponding tight-binding chain models the motion of a charged particle between the sites of a lattice, wherein the particle is for most times localized on the sites, owing to spontaneous quantum fluctuations tunnels between the nearest-neighbour sites. We consider two representative forms of interactions, one that implements a stochastic reset of quantum dynamics in which the density operator is at random times reset to its initial form, and one in which projective measurements are performed on the system at random times. In the former case, we demonstrate with our exact results how the particle is localized on the sites at long times, leading to a time-independent mean-squared displacement (MSD) of the particle about its initial location. This stands in stark contrast to the behavior in the absence of interactions, when the particle has an unbounded growth of the MSD in time, with no signatures of localization. In the case of projective measurements at random times, we show that repeated projection to the initial state of the particle results in an effective suppression of the temporal decay in the probability of the particle to be found on the initial state. The amount of suppression is comparable to the one in conventional Zeno effect scenarios, but which it does not require us to perform measurements at exactly regular intervals that are hallmarks of such scenarios.

Optimization of entropy encoding using random variations

This article describes an optimization method concerning entropy encoding applicable to a source of independent and identically-distributed random variables. The algorithm can be explained with the following example: let us take a source of i.i.d. random variables X with uniform probability density and cardinality 10. With this source, we generate messages of length 1000 which will be encoded in base 10. We call XG the set, of dimension 10^1000, containing all messages that can be generated from the source. According to Shannon's first theorem, if the average entropy of X, calculated on the set XG, is H(X)≈0.9980, the average length of the encoded messages will be 1000∙𝐻(𝑋)=998. Now, we increase the length of the message by one and calculate the average entropy concerning the ten percent of the sequences of length 1001 having less entropy. We call this set, of dimension 10^1000, XG10. The average entropy of X10, calculated on the XG10 set, is H(X10)≈0.9964, consequently, the average length of the encoded messages will be 1001∙𝐻(𝑋10)=997.4. Now, we make the difference between the average length of the encoded sequences belonging to the two sets (XG and XG10) 998.0-997.4 = 0.6. Therefore, if we use the XG10 set, we reduce the average length of the encoded message by 0.6 values in base ten. Consequently, the average information per symbol becomes (1001∙0.9964)/1000=0.9974, which turns out to be less than the average entropy of X H(X)≈0.998. We can use the XG10 set instead of the XG set, because we can create a biunivocal correspondence between all the possible sequences generated by our source, which we know to be 10^1000, and ten percent of the sequences with less entropy of the messages having length 1001, in fact 10^1001∙0.1=10^1000. In this article, we will show that this transformation can be performed by applying random variations on the sequences generated by the source.

Bounds for CDFs of Order Statistics Arising from INID Random Variables

In recent decades, studying order statistics arising from independent and not necessary identically distributed (INID) random variables has been a main concern for researchers. A cumulative distribution function (CDF) of these random variables (Fi:n) is a complex manipulating, long time consuming and a software-intensive tool that takes more and more times. Therefore, obtaining approximations and boundaries for Fi:n and other theoretical properties of these variables, such as moments, quantiles, characteristic function, and some related probabilities, has always been a main chal- lenge. Recently, Bayramoglu (2018) provided a new definition of ordering, by point to point ordering Fi’s (D-order) and showed that these new functions are CDFs and also, the corresponding random variables are independent. Thus, he suggested new CDFs (F[i]) that can be used as an alternative of Fi:n. Now with using, just F[1], and F[n], we have found the upper and lower bounds of Fi:n. Furthermore, specially a precisely approximation for F1:n and Fn:n (F1;n:n). Also in many cases approximations for other CDFs are derived. In addition, we compare approximated function with those oered by Bayramoglu and it is shown that our results of these proposed functions are far better than D-order functions.

Граничні випадки критерію мінімуму протяжності

Представлені граничні випадки критерію мінімуму протяжності, які відбивають його зв'язок з вiдомими критерiями обробки даних. Установленi вiдповiдностi мiж критерієм мiнiмуму протяжностi й критерiями найменших квадратiв, найменших модулiв, максимуму правдоподiбностi в задачi оцiнювання параметра зсуву за умови фiксованого параметра масштабу для незалежних однаково розподiлених випадкових величин iз законами розподiлу Гаусса, Лапласа, Кошi, "гостровершинним" меридiанним законом розподiлу, законами узагальнених розподiлiв Гаусса й Кошi, критерієм узагальненої максимальної правдоподібності з вартісними функціями Мішалкіна й Демиденка, а також критеріїм максимуму гістограми. Крім цього, у рамках концепції функціонала протяжності представлено кілька варіантів граничного переходу від критерію мінімуму квазіпротяжності до критерію мінімуму строгої протяжності, де останній критерій для дискретного випадку дає постановку NP-складної задачі мінімізації квазінорми простору l0. Підкреслено, що більшість із зазначених критеріїв приводить до постановки задач оптимізації з неопуклою та неунімодальною цільовою функцією.

Type-A Worst-Case Uncertainty for Gaussian noise instruments

An analytical type-A approach is proposed for predicting the Worst-Case Uncertainty of a measurement system. In a set of independent observations of the same measurand, modelled as independent- and identically-distributed random variables, the upcoming extreme values (e.g. peaks) can be forecast by only characterizing the measurement system noise level, assumed to be white and Gaussian. Simulation and experimental results are presented to validate the model for a case study on the worst-case repeatability of a pulsed power supply for the klystron modulators of the Compact LInear Collider at CERN. The experimental validation highlights satisfying results for an acquisition system repeatable in the order of ±25 ppm over a bandwidth of 5 MHz.

Stochastic properties of INID progressive Type-II censored order statistics

Let X 1 : n ≤ X 2 : n ≤ ⋯ ≤ X n : n denote the order statistics of random variables X 1 , X 2 , … , X n which are independent but not necessarily identically distributed (INID), and let K 1 , K 2 be two integer-valued random variables, independent of { X 1 , … , X n } , such that 1 ≤ K 1 ≤ K 2 ≤ n . It is shown that if K 1 has a log-concave probability function and SI ( K 2 | K 1 ) then RTI ( X K 2 : n | X K 1 : n ) , and if K 2 has a log-concave probability function and SI ( K 1 | K 2 ) then LTD ( X K 1 : n | X K 2 : n ) , where SI, RTI and LTD are three notions of bivariate positive dependence. Based on these, we obtain that RTI ( X j : m , n R | X i : m , n R ) and LTD ( X i : m , n R | X j : m , n R ) whenever 1 ≤ i < j ≤ m , where { X 1 : m , n R , … , X m : m , n R } are progressive Type-II censored order statistics from INID random variables { X 1 , … , X n } . Furthermore, one result concerning the likelihood ratio ordering of the progressive Type-II censored order statistics is also given.

The norm of products of free random variables

Let Xi denote free identically-distributed random variables. This paper investigates how the norm of products \({\Pi_{n} = X_{1}X_{2}\cdots X_{n}}\) behaves as n approaches infinity. In addition, for positive Xi it studies the asymptotic behavior of the norm of \({Y_{n} = X_{1}\circ X_{2}\circ \cdots \circ X_{n},}\) where \({\circ}\) denotes the symmetric product of two positive operators: \({A\circ B=:A^{1/2}BA^{1/2}}\) . It is proved that if EXi = 1, then \({\left\Vert Y_{n}\right\Vert }\) is between \({c_{1}\sqrt{n}}\) and c2n for certain constant c1 and c2. For \({\left\Vert \Pi_{n}\right\Vert ,}\) it is proved that the limit of \({n^{-1}\log \left\Vert \Pi _{n}\right\Vert }\) exists and equals \({\log \sqrt{E\left( X_{i}^{\ast }X_{i}\right) }.}\) Finally, if π is a cyclic representation of the algebra generated by Xi, and if ξ is a cyclic vector, then \({n^{-1}\log \left\Vert \pi \left( \Pi _{n}\right) \xi \right\Vert = \log \sqrt{E\left( X_{i}^{\ast }X_{i}\right) }}\) for all n. These results are significantly different from analogous results for commuting random variables.

The Joint Effect of Leadtime Variance and Lot Size in a Parallel Processing Environment

We study a basic (r, q) system, in which the demand is a Poisson process and the leadtimes are independent, identically-distributed random variables. The key issue is the joint effect of the leadtime variance and the lot size q on performance. We know that, under a simple base-stock policy (with q = 1), the leadtime variance has no effect at all. We find here that, for larger q, the leadtime variance can have a significant adverse impact on performance. To explore this effect, we test two simple approximations. The simplest ignores the leadtime variance. The second approach is only a bit more complex; it captures the variance effect through a hybrid of two limiting approximations. Both methods provide useful information, but the second is more robust.

On confidence limits and statistical convergence of Monte Carlo point-flux estimators with unbounded variance

The problem of analyzing the statistical error of Monte Carlo point-flux estimators with unbounded variance is considered. The results are based on the generalization of the classical limit theorem for summands of identically-distributed random variables on the unbounded-variance case. The Lévy representation of the characteristic functions of stable attracting distributions is implemented. The explicit expression of the stable attracting probability density function for the uncollided point-flux estimates is obtained and the corresponding cumulative probability function is tabulated. It is shown that a 1 x singularity of an unbounded-variance point-flux estimator in the vicinity of the detector point leads to 1 N 1 2 convergence to the normal law. The procedure of estimation of confidence limits and statistical accuracy of unbounded-variance point-flux estimators for intermediate sample size is discussed. The results may be utilized for the calculation of statistical error of Monte Carlo calculations of flux at a point.