caa a22 4500 445391553 CHVBK 20180317143048.0 cr unu---uuuuu 170323e20110101xx s 000 0 eng 10.1007/s10955-010-0087-5 doi (NATIONALLICENCE)springer-10.1007/s10955-010-0087-5 Trajectories in Random Monads [Elektronische Daten] [A. Ramos, A. Toom] Let us have a non-empty finite set S with n>1 elements which we call points and a map M:S→S. After V.I. Arnold, we call such pairs (S, M) monads, but we consider random monads in which all the values of M(⋅) are random, independent and uniformly distributed in S. We fix some ⊙∈S and consider the infinite sequence M t(⊙), t=0,1,2,  . Apoint is called visited if it coincides with at least one term of this sequence. A visited point is called recurrent if it appears in this sequence at least twice; if a visited point appears in this sequence only once, it is called transient. We denote by Vis, Rec, Tra the numbers of visited, recurrent and transient points respectively and study their distributions. The distributions of Vis, Rec, Tra are unimodal. The modes of Rec and Tra equal their minimal values, that is 1 and 0 respectively. The mode of Vis is approximated by $\sqrt{n}$, plus-minus a constant. The mathematical expectations: $\mathbb{E}(\mathit{Vis})$ is approximated by $2 \sqrt{\pi\, n/8}$ plus-minus a constant; $\mathbb{E}(\mathit{Rec})$ and $\mathbb{E}(\mathit{Tra})$ are approximated by $\sqrt{\pi\, n/8}$ plus-minus a constant. For the standard deviations σ(Vis) and σ(Rec)=σ(Tra) respectively we present the approximations $$\sqrt{\frac{4-\pi}{2} \cdot n} \quad\mbox{and}\quad \sqrt{\frac{16-3\pi}{24} \cdot n},$$ from which they also deviate at most by a constant. We prove that when n tends to infinity, the correlations Corr(Rec,Tra) and Corr(Rec,Vis)=Corr(Tra,Vis) converge to $$\frac{8-3\pi}{16-3\pi}\quad \mbox{and}\quad \sqrt{\frac{12-3\pi}{16-3\pi}}.$$ Springer Science+Business Media, LLC, 2010 Arnold's monads nationallicence Random dynamical systems nationallicence Incomplete gamma function nationallicence Recurrent nationallicence Transient nationallicence Ramos A. Department of Statistics, Federal University of Pernambuco, 50740-540, Recife, PE, Brazil aut Toom A. Department of Statistics, Federal University of Pernambuco, 50740-540, Recife, PE, Brazil aut Journal of Statistical Physics Springer US; http://www.springer-ny.com 142/1(2011-01-01), 201-219 0022-4715 142:1<201 2011 142 10955 https://doi.org/10.1007/s10955-010-0087-5 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s10955-010-0087-5 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Ramos A. Department of Statistics, Federal University of Pernambuco, 50740-540, Recife, PE, Brazil aut NATIONALLICENCE 700 1- Toom A. Department of Statistics, Federal University of Pernambuco, 50740-540, Recife, PE, Brazil aut NATIONALLICENCE 773 0- Journal of Statistical Physics Springer US; http://www.springer-ny.com 142/1(2011-01-01), 201-219 0022-4715 142:1<201 2011 142 10955 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer