scholarly journals Sufficient Conditions for Convergence of the Sum–Product Algorithm

2007 ◽  
Vol 53 (12) ◽  
pp. 4422-4437 ◽  
Author(s):  
Joris M. Mooij ◽  
Hilbert J. Kappen
Keyword(s):  
Sufficient Conditions ◽  
Sufficient Conditions For Convergence

scholarly journals Continued fractions built from convex sets and convex functions

2015 ◽  
Vol 17 (05) ◽  
pp. 1550003 ◽  
Author(s):  
Ilya Molchanov
Keyword(s):  
Convex Functions ◽  
Continued Fractions ◽  
Convex Sets ◽  
Sufficient Conditions ◽  
Ordered Semigroup ◽  
Minkowski Addition ◽  
Partially Ordered ◽  
The Family ◽  
Sufficient Conditions For Convergence

In a partially ordered semigroup with the duality (or polarity) transform, it is possible to define a generalization of continued fractions. General sufficient conditions for convergence of continued fractions are provided. Two particular applications concern the cases of convex sets with the Minkowski addition and the polarity transform and the family of non-negative convex functions with the Legendre–Fenchel and Artstein-Avidan–Milman transforms.

scholarly journals Return- and hitting-time distributions of small sets in infinite measure preserving systems

2019 ◽  
Vol 40 (8) ◽  
pp. 2239-2273
Author(s):  
SIMON RECHBERGER ◽  
ROLAND ZWEIMÜLLER
Keyword(s):  
Sufficient Conditions ◽  
Finite Measure ◽  
Hitting Time ◽  
Independent Random Variables ◽  
Limit Laws ◽  
Interval Maps ◽  
Infinite Measure ◽  
Sufficient Conditions For Convergence ◽  
Set Up ◽  
The One

We study convergence of return- and hitting-time distributions of small sets $E_{k}$ with $\unicode[STIX]{x1D707}(E_{k})\rightarrow 0$ in recurrent ergodic dynamical systems preserving an infinite measure $\unicode[STIX]{x1D707}$. Some properties which are easy in finite measure situations break down in this null-recurrent set-up. However, in the presence of a uniform set $Y$ with wandering rate regularly varying of index $1-\unicode[STIX]{x1D6FC}$ with $\unicode[STIX]{x1D6FC}\in (0,1]$, there is a scaling function suitable for all subsets of $Y$. In this case, we show that return distributions for the $E_{k}$ converge if and only if the corresponding hitting-time distributions do, and we derive an explicit relation between the two limit laws. Some consequences of this result are discussed. In particular, this leads to improved sufficient conditions for convergence to ${\mathcal{E}}^{1/\unicode[STIX]{x1D6FC}}{\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$, where ${\mathcal{E}}$ and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ are independent random variables, with ${\mathcal{E}}$ exponentially distributed and ${\mathcal{G}}_{\unicode[STIX]{x1D6FC}}$ following the one-sided stable law of order $\unicode[STIX]{x1D6FC}$ (and ${\mathcal{G}}_{1}:=1$). The same principle also reveals the limit laws (different from the above) which occur at hyperbolic periodic points of prototypical null-recurrent interval maps. We also derive similar results for the barely recurrent $\unicode[STIX]{x1D6FC}=0$ case.

A simple proof of a random stable limit theorem

1970 ◽  
Vol 7 (2) ◽  
pp. 502-504 ◽  
Author(s):  
Stephen R. Kimbleton
Keyword(s):  
Limit Theorem ◽  
Elementary Proof ◽  
General Theorem ◽  
Natural Extension ◽  
Sufficient Conditions ◽  
Domain Of Attraction ◽  
Stable Limit ◽  
Stable Law ◽  
Sufficient Conditions For Convergence ◽  
Necessary And Sufficient

Random stable limit theorems have been obtained by several authors, e.g., [3], [4]. The purpose of this note is to give a rather elementary proof of the basic version of this theorem. Our proof may be viewed as the natural extension to stable laws of the method used by Rényi [2] in obtaining a random central limit theorem. Indeed, the only “outside” theorems used are Kolmogorov's inequality (which Rényi also uses) and a general theorem on necessary and sufficient conditions for convergence of a triangular array. It will also be observed that in the present theorem, the consideration of random variables in the domain of attraction of a stable law of index α = 1, introduces no additional difficulties.

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