Ergodic theorems for subadditive superstationary families of convex compact random sets

1983 ◽  
Vol 62 (1) ◽  
pp. 125-135 ◽  
Author(s):  
Klaus Sch�rger
Keyword(s):  
Random Sets ◽  
Convex Compact ◽  
Ergodic Theorems

scholarly journals On Laslett's transform for the Boolean model

2001 ◽  
Vol 33 (1) ◽  
pp. 1-5 ◽  
Author(s):  
A. D. Barbour ◽  
V. Schmidt
Keyword(s):  
Poisson Process ◽  
Point Process ◽  
Simple Proof ◽  
Poisson Point Process ◽  
Boolean Model ◽  
Random Sets ◽  
Convex Compact ◽  
Homogeneous Poisson Process ◽  
Vertical Strip ◽  
Cumulative Process

Consider the Boolean model in ℝ2, where the germs form a homogeneous Poisson point process with intensity λ and the grains are convex compact random sets. It is known (see, e.g., Cressie (1993, Section 9.5.3)) that Laslett's rule transforms the exposed tangent points of the Boolean model into a homogeneous Poisson process with the same intensity. In the present paper, we give a simple proof of this result, which is based on a martingale argument. We also consider the cumulative process of uncovered area in a vertical strip and show that a (linear) Poisson process with intensity λ can be embedded in it.

Ergodic theorems for random sets with density zero

1994 ◽  
Vol 14 (1) ◽  
pp. 141-149
Author(s):  
Yenkun Huang
Keyword(s):  
Positive Solution ◽  
Ergodic Theorem ◽  
The Other ◽  
Random Sets ◽  
Asymptotic Density ◽  
Necessary Condition ◽  
Ergodic Theorems ◽  
Mean Ergodic Theorem ◽  
Density Zero ◽  
Sufficient And Necessary Condition

AbstractWe generalize a result of Bourgain and a result of Huang. We also give a positive solution to A. Bellow's question: the a.e. convergence of the averages for σn = 1/n. On the other hand, we establish a sufficient and necessary condition for random sets in Z+ with asymptotic density zero which almost surely satisfy a mean ergodic theorem.

scholarly journals ASSESSING DISSIMILARITY OF RANDOM SETS THROUGH CONVEX COMPACT APPROXIMATIONS, SUPPORT FUNCTIONS AND ENVELOPE TESTS

2016 ◽  
Vol 35 (3) ◽  
pp. 181 ◽  
Author(s):  
Vesna Gotovac ◽  
Kateřina Helisová ◽  
Ivo Ugrina
Keyword(s):  
Random Sets ◽  
Convex Compact ◽  
Full Potential ◽  
Simulation Studies ◽  
Support Functions ◽  
Interaction Processes ◽  
Compact Approximations ◽  
Material Sciences ◽  
Underlying Processes ◽  
Isotropic Random

In recent years random sets were recognized as a valuable tool in modelling different processes from fields like biology, biomedicine or material sciences. Nevertheless, the full potential of applications has not still been reached and one of the main problems in advancement is the usual inability to correctly differentiate between underlying processes generating real world realisations. This paper presents a measure of dissimilarity of stationary and isotropic random sets through a heuristic based on convex compact approximations, support functions and envelope tests. The choice is justified through simulation studies of common random models like Boolean and Quermass-interaction processes. 

scholarly journals Multiple Alu exonization in 3’UTR of a primate specific isoform of CYP20A1 creates a potential miRNA sponge

2020 ◽  
Author(s):  
Aniket Bhattacharya ◽  
Vineet Jha ◽  
Khushboo Singhal ◽  
Mahar Fatima ◽  
Dayanidhi Singh ◽  
...  
Keyword(s):  
Heat Shock ◽  
Cortical Neurons ◽  
Regulatory Networks ◽  
Large Scale ◽  
Neuronal Development ◽  
Random Sets ◽  
Rna Seq ◽  
Orphan Gene ◽  
Mirna Sponge ◽  
Human Neurons

Abstract Alu repeats contribute to phylogenetic novelties in conserved regulatory networks in primates. Our study highlights how exonized Alus could nucleate large-scale mRNA-miRNA interactions. Using a functional genomics approach, we characterize a transcript isoform of an orphan gene, CYP20A1 (CYP20A1_Alu-LT) that has exonization of 23 Alus in its 3’UTR. CYP20A1_Alu-LT, confirmed by 3’RACE, is an outlier in length (9 kb 3’UTR) and widely expressed. Using publically available datasets, we demonstrate its expression in higher primates and presence in single nucleus RNA-seq of 15928 human cortical neurons. miRanda predicts ∼4700 miRNA recognition elements (MREs) for ∼1000 miRNAs, primarily originated within these 3’UTR-Alus. CYP20A1_Alu-LT could be a potential multi-miRNA sponge as it harbors ≥10 MREs for 140 miRNAs and has cytosolic localization. We further tested whether expression of CYP20A1_Alu-LT correlates with mRNAs harboring similar MRE targets. RNA-seq with conjoint miRNA-seq analysis was done in primary human neurons where we observed CYP20A1_Alu-LT to be downregulated during heat shock response and upregulated in HIV1-Tat treatment. 380 genes were positively correlated with its expression (significantly downregulated in heat shock and upregulated in Tat) and they harbored MREs for nine expressed miRNAs which were also enriched in CYP20A1_Alu-LT. MREs were significantly enriched in these 380 genes compared to random sets of differentially expressed genes (p = 8.134e-12). Gene ontology suggested involvement of these genes in neuronal development and hemostasis pathways thus proposing a novel component of Alu-miRNA mediated transcriptional modulation that could govern specific physiological outcomes in higher primates.

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