scholarly journals Asymptotics for randomly reinforced urns with random barriers

2016 ◽  
Vol 53 (4) ◽  
pp. 1206-1220
Author(s):  
Patrizia Berti ◽  
Irene Crimaldi ◽  
Luca Pratelli ◽  
Pietro Rigo
Keyword(s):  
Conditional Distribution ◽  
Random Number ◽  
Random Variable ◽  
Conditional Convergence ◽  
Mild Conditions ◽  
The Past ◽  
Random Variance

Abstract An urn contains black and red balls. Let Zn be the proportion of black balls at time n and 0≤L<U≤1 random barriers. At each time n, a ball bn is drawn. If bn is black and Zn-1<U, then bn is replaced together with a random number Bn of black balls. If bn is red and Zn-1>L, then bn is replaced together with a random number Rn of red balls. Otherwise, no additional balls are added, and bn alone is replaced. In this paper we assume that Rn=Bn. Then, under mild conditions, it is shown that Zn→a.s.Z for some random variable Z, and Dn≔√n(Zn-Z)→𝒩(0,σ2) conditionally almost surely (a.s.), where σ2 is a certain random variance. Almost sure conditional convergence means that ℙ(Dn∈⋅|𝒢n)→w 𝒩(0,σ2) a.s., where ℙ(Dn∈⋅|𝒢n) is a regular version of the conditional distribution of Dn given the past 𝒢n. Thus, in particular, one obtains Dn→𝒩(0,σ2) stably. It is also shown that L<Z<U a.s. and Z has nonatomic distribution.

Two characterizations of the geometric distribution

1980 ◽  
Vol 17 (02) ◽  
pp. 570-573 ◽  
Author(s):  
Barry C. Arnold
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Conditional Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Mild Conditions ◽  
Independent Identically Distributed

Let X 1, X 2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X 1:n , X 2:n , …, Xn :n. If the Xi 's have a geometric distribution then the conditional distribution of Xk +1:n – Xk :n given Xk+ 1:n – Xk :n &gt; 0 is the same as the distribution of X 1:n–k . Also the random variable X 2:n – X 1:n is independent of the event [X 1:n = 1]. Under mild conditions each of these two properties characterizes the geometric distribution.

scholarly journals Central limit theorems for a hypergeometric randomly reinforced urn

2016 ◽  
Vol 53 (3) ◽  
pp. 899-913
Author(s):  
Irene Crimaldi
Keyword(s):  
Central Limit ◽  
Conditional Distribution ◽  
Limit Theorems ◽  
Common Factors ◽  
Central Limit Theorems ◽  
Stable Convergence ◽  
Conditional Convergence ◽  
Time Step ◽  
The Past ◽  
Asymptotic Confidence Intervals

AbstractWe consider a variant of the randomly reinforced urn where more balls can be simultaneously drawn out and balls of different colors can be simultaneously added. More precisely, at each time-step, the conditional distribution of the number of extracted balls of a certain color, given the past, is assumed to be hypergeometric. We prove some central limit theorems in the sense of stable convergence and of almost sure conditional convergence, which are stronger than convergence in distribution. The proven results provide asymptotic confidence intervals for the limit proportion, whose distribution is generally unknown. Moreover, we also consider the case of more urns subjected to some random common factors.

Two characterizations of the geometric distribution

1980 ◽  
Vol 17 (2) ◽  
pp. 570-573 ◽  
Author(s):  
Barry C. Arnold
Keyword(s):  
Positive Integer ◽  
Order Statistics ◽  
Conditional Distribution ◽  
Geometric Distribution ◽  
Random Variables ◽  
Random Variable ◽  
Mild Conditions ◽  
Independent Identically Distributed

Let X1, X2, …, Xn be independent identically distributed positive integer-valued random variables with order statistics X1:n, X2:n, …, Xn:n. If the Xi's have a geometric distribution then the conditional distribution of Xk+1:n – Xk:n given Xk+1:n – Xk:n > 0 is the same as the distribution of X1:n–k. Also the random variable X2:n – X1:n is independent of the event [X1:n = 1]. Under mild conditions each of these two properties characterizes the geometric distribution.

Organophotoredox-Catalyzed C–H Alkylation of Imidazoheterocycles with Malonates: Total Synthesis of Zolpidem

Synthesis ◽  
2020 ◽  
Author(s):  
Narendra R. Chaubey ◽  
Anant R. Kapdi ◽  
Biswanath Maity
Keyword(s):  
Total Synthesis ◽  
Room Temperature ◽  
Drug Molecule ◽  
Bond Functionalization ◽  
First Report ◽  
Mild Conditions ◽  
The Past ◽  
Hypnotic Drug ◽  
Free Environment

AbstractOrganophotocatalytic C–H bond functionalization has attracted a lot of attention in the past several years due to the possibility of catalyzing reactions in a metal- and peroxide-free environment. Continuing on these lines, an organophotoredox-catalyzed C–H functionalization of imidazo[1,2-a]pyridines and related heterocycles with bromomalonates under mild conditions is reported, providing excellent yields of the products at room temperature. This is the first report involving malonates as coupling partners leading to the synthesis of a range of functionalized products including total synthesis of zolpidem, a sedative­-hypnotic drug molecule.

scholarly journals Analysis of the Past Lifetime in a Replacement Model through Stochastic Comparisons and Differential Entropy

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1203 ◽  
Author(s):  
Antonio Di Crescenzo ◽  
Patrizia Di Gironimo ◽  
Suchandan Kayal
Keyword(s):  
Random Variable ◽  
Fixed Time ◽  
Neuronal Model ◽  
Lifetime Distribution ◽  
Differential Entropy ◽  
Firing Activity ◽  
Stochastic Comparisons ◽  
The Past ◽  
Replacement Procedure ◽  
Replacement Model

A suitable replacement model for random lifetimes is extended to the context of past lifetimes. At a fixed time u an item is planned to be replaced by another one having the same age but a different lifetime distribution. We investigate the past lifetime of this system, given that at a larger time t the system is found to be failed. Subsequently, we perform some stochastic comparisons between the random lifetimes of the single items and the doubly truncated random variable that describes the system lifetime. Moreover, we consider the relative ratio of improvement evaluated at x ∈ ( u , t ) , which is finalized to measure the goodness of the replacement procedure. The characterization and the properties of the differential entropy of the system lifetime are also discussed. Finally, an example of application to the firing activity of a stochastic neuronal model is provided.

scholarly journals Natural Test for Random Numbers Generator Based on Exponential Distribution

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 920 ◽  
Author(s):  
Tanackov ◽  
Sinani ◽  
Stanković ◽  
Bogdanović ◽  
Stević ◽  
...  
Keyword(s):  
Standard Deviation ◽  
Mathematical Expectation ◽  
Random Number ◽  
Random Variable ◽  
Random Numbers ◽  
The Third ◽  
Atmospheric Noise ◽  
Constant E ◽  
Fourth Series ◽  
Natural Test

We will prove that when uniformly distributed random numbers are sorted by value, their successive differences are a exponentially distributed random variable Ex(λ). For a set of n random numbers, the parameters of mathematical expectation and standard deviation is λ =n−1. The theorem was verified on four series of 200 sets of 101 random numbers each. The first series was obtained on the basis of decimals of the constant e=2.718281…, the second on the decimals of the constant π =3.141592…, the third on a Pseudo Random Number generated from Excel function RAND, and the fourth series of True Random Number generated from atmospheric noise. The obtained results confirm the application of the derived theorem in practice.

Remarks on a model of competitive bidding for employment

1983 ◽  
Vol 20 (02) ◽  
pp. 349-357
Author(s):  
Anthony G. Pakes
Keyword(s):  
Random Number ◽  
Random Variable ◽  
Competitive Bidding ◽  
Complete Analysis ◽  
Income Distributions ◽  
Present Generation

Arnold and Laguna introduced a model for income distributions in which the income of the present generation of individuals has the same distribution as the minimum of a random number Nn of independent copies of some random variable and {Nn } is independent. The present paper gives a fairly complete analysis of this model and a number of extensions of it.

Supercritical age-dependent branching processes with immigration

1974 ◽  
Vol 11 (4) ◽  
pp. 695-702 ◽  
Author(s):  
K. B. Athreya ◽  
P. R. Parthasarathy ◽  
G. Sankaranarayanan
Keyword(s):  
Branching Process ◽  
Random Number ◽  
Branching Processes ◽  
Life Time ◽  
Random Variable ◽  
One Dimensional ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Branching Process With Immigration ◽  
Mutually Independent

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

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