scholarly journals On the metrical theory of a non-regular continued fraction expansion

2015 ◽  
Vol 23 (2) ◽  
pp. 147-160
Author(s):  
Dan Lascu ◽  
George Cîrlig
Keyword(s):  
Dynamical System ◽  
Continued Fraction ◽  
Probability Function ◽  
Transition Probability ◽  
Continued Fraction Expansion ◽  
Metrical Theory ◽  
Unilateral Shift ◽  
Symbolic Dynamical System ◽  
Continued Fraction Expansions ◽  
Transition Probability Function

Abstract We introduced a new continued fraction expansions in our previous paper. For these expansions, we show the Brodén-Borel-Lévy type formula. Furthermore, we compute the transition probability function from this and the symbolic dynamical system of the natural number with the unilateral shift.

scholarly journals The Strassen invariance principle for certain non-stationary Markov–Feller chains

2020 ◽  
Vol 121 (1) ◽  
pp. 1-34 ◽  
Author(s):  
Dawid Czapla ◽  
Katarzyna Horbacz ◽  
Hanna Wojewódka-Ściążko
Keyword(s):  
Gene Expression ◽  
Mathematical Model ◽  
Population Dynamics ◽  
Invariance Principle ◽  
Probability Function ◽  
Stochastic Dynamics ◽  
Transition Probability ◽  
Type Condition ◽  
Iterated Logarithm ◽  
Transition Probability Function

We propose certain conditions implying the functional law of the iterated logarithm (the Strassen invariance principle) for some general class of non-stationary Markov–Feller chains. This class may be briefly specified by the following two properties: firstly, the transition operator of the chain under consideration enjoys a non-linear Lyapunov-type condition, and secondly, there exists an appropriate Markovian coupling whose transition probability function can be decomposed into two parts, one of which is contractive and dominant in some sense. Our criterion may serve as a useful tool in verifying the functional law of the iterated logarithm for certain random dynamical systems, developed e.g. in biology and population dynamics. In the final part of the paper we present an example application of our main theorem to a mathematical model describing stochastic dynamics of gene expression.

scholarly journals Renewal-type limit theorem for continued fractions with even partial quotients

2009 ◽  
Vol 29 (5) ◽  
pp. 1451-1478 ◽  
Author(s):  
FRANCESCO CELLAROSI
Keyword(s):  
Limit Theorem ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Type Theorem ◽  
Natural Extension ◽  
Gauss Map ◽  
Continued Fraction Expansion ◽  
Special Flow ◽  
Partial Quotients ◽  
Continued Fraction Expansions

AbstractWe prove the existence of the limiting distribution for the sequence of denominators generated by continued fraction expansions with even partial quotients, which were introduced by Schweiger [Continued fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg4 (1982), 59–70; On the approximation by continues fractions with odd and even partial quotients. Arbeitsberichte Math. Institut Universtät Salzburg1–2 (1984), 105–114] and studied also by Kraaikamp and Lopes [The theta group and the continued fraction expansion with even partial quotients. Geom. Dedicata59(3) (1996), 293–333]. Our main result is proven following the strategy used by Sinai and Ulcigrai [Renewal-type limit theorem for the Gauss map and continued fractions. Ergod. Th. & Dynam. Sys.28 (2008), 643–655] in their proof of a similar renewal-type theorem for Euclidean continued fraction expansions and the Gauss map. The main steps in our proof are the construction of a natural extension of a Gauss-like map and the proof of mixing of a related special flow.

The distribution of the largest digit in continued fraction expansions

2009 ◽  
Vol 146 (1) ◽  
pp. 207-212 ◽  
Author(s):  
JUN WU ◽  
JIAN XU
Keyword(s):  
Hausdorff Dimension ◽  
Continued Fraction ◽  
Continued Fraction Expansion ◽  
Image Position ◽  
Continued Fraction Expansions

AbstractLet [a1(x), a2(x), . . .] be the continued fraction expansion of x ∈ [0,1). Write Tn(x)=max{ak(x):1 ≤ k ≤ n}. Philipp [6] proved that Okano [5] showed that for any k ≥ 2, there exists x ∈ [0, 1) such that T(x)=1/log k. In this paper we show that, for any α ≥ 0, the set is of Hausdorff dimension 1.

Slow increasing functions and the largest partial quotients in continued fraction expansions

2016 ◽  
Vol 164 (1) ◽  
pp. 1-14 ◽  
Author(s):  
JINHUA CHANG ◽  
HAIBO CHEN
Keyword(s):  
Level Sets ◽  
Limit Theorems ◽  
Continued Fraction ◽  
Positive Function ◽  
Continued Fraction Expansion ◽  
Hausdorff Dimensions ◽  
Partial Quotients ◽  
Continued Fraction Expansions ◽  
Increasing Functions

AbstractLet 0 ⩽ α ⩽ ∞ and ψ be a positive function defined on (0, ∞). In this paper, we will study the level sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) which are related respectively to the sequence of the largest digits among the first n partial quotients {Ln(x)}n≥1, the increasing sequence of the largest partial quotients {Bn(x)}n⩾1 and the sequence of successive occurrences of the largest partial quotients {Tn(x)}n⩾1 in the continued fraction expansion of x ∈ [0,1) ∩ ℚc. Under suitable assumptions of the function ψ, we will prove that the sets L(α, {ψ(n)}), B(α, {ψ(n)}) and T(α, {ψ(n)}) are all of full Hausdorff dimensions for any 0 ⩽ α ⩽ ∞. These results complement some limit theorems given by J. Galambos [4] and D. Barbolosi and C. Faivre [1].

A method for the calculation of characteristics for the solution to stochastic differential equations

2017 ◽  
Vol 23 (3) ◽  
Author(s):  
Alexander Egorov ◽  
Victor Malyutin
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Numerical Method ◽  
Numerical Experiments ◽  
Probability Function ◽  
Transition Probability ◽  
Planck Equation ◽  
Fokker Planck Equation ◽  
Transition Probability Function ◽  
Fokker Planck

AbstractIn this work, a new numerical method to calculate the characteristics of the solution to stochastic differential equations is presented. This method is based on the Fokker–Planck equation for the transition probability function and the representation of the transition probability function by means of eigenfunctions of the Fokker–Planck operator. The results of the numerical experiments are presented.

Computational and estimation procedures in multidimensional right-shift processes and some applications

1975 ◽  
Vol 7 (2) ◽  
pp. 349-382 ◽  
Author(s):  
Richard J. Kryscio ◽  
Norman C. Severo
Keyword(s):  
Queueing Theory ◽  
Probability Function ◽  
Complete System ◽  
Transition Probability ◽  
Finite State ◽  
Computational Procedures ◽  
Forward Equations ◽  
Kolmogorov Forward Equations ◽  
Transition Probability Function ◽  
Finite State Space

A right-shift process is a Markov process with multidimensional finite state space on which the infinitesimal transition movement is a shifting of one unit from one coordinate to some other to its right. A multidimensional right-shift process consists of v ≧ 1 concurrent and dependent right-shift processes. In this paper applications of multidimensional right-shift processes to some well-known examples from epidemic theory, queueing theory and the Beetle probblem due to Lucien LeCam are discussed. A transformation which orders the Kolmogorov forward equations into a triangular array is provided and some computational procedures for solving the resulting system of equations are presented. One of these procedures is concerned with the problem of evaluating a given transition probability function rather than obtaining the solution to the complete system of forward equations. This particular procedure is applied to the problem of estimating the parameters of a multidimensional right-shift process which is observed at only a finite number of fixed timepoints.

scholarly journals Continued fraction expansions for q-tangent and q-cotangent functions

2010 ◽  
Vol Vol. 12 no. 2 ◽  
Author(s):  
Helmut Prodinger
Keyword(s):  
Continued Fraction ◽  
Similar Type ◽  
Recursive Relation ◽  
Continued Fraction Expansion ◽  
International Audience ◽  
Continued Fraction Expansions

International audience For 3 different versions of q-tangent resp. q-cotangent functions, we compute the continued fraction expansion explicitly, by guessing the relative quantities and proving the recursive relation afterwards. It is likely that these are the only instances with a ''nice'' expansion. Additional formulae of a similar type are also provided.

scholarly journals Rational Approximations for Irrational Functions using Collocating Continued Fraction Expansions

2021 ◽  
Author(s):  
philip olivier
Keyword(s):  
Complex Variable ◽  
Continued Fraction ◽  
Rational Approximations ◽  
Mathematical Tool ◽  
Continued Fraction Expansion ◽  
Continued Fraction Expansions ◽  
Irrational Function

<div> <div> <div> <p>This paper is motivated by the need in certain engineering contexts to construct approximations for irrational functions in the complex variable z. The main mathematical tool that will be used is a special continued fraction expansion that cause the rational approximant to collocate the irrational function at specific important values of z. This paper introduces two theorems that facilitate the construction of the rational approxima- tion. </p> </div> </div> </div>

scholarly journals Equilibrium in misspecified Markov decision processes

2021 ◽  
Vol 16 (2) ◽  
pp. 717-757
Author(s):  
Ignacio Esponda ◽  
Demian Pouzo
Keyword(s):  
Probability Function ◽  
Transition Probability ◽  
Transition Function ◽  
Equilibrium Concept ◽  
State Behavior ◽  
State Variable ◽  
Transition Functions ◽  
Markov Decision ◽  
Previous State ◽  
Transition Probability Function

We provide an equilibrium framework for modeling the behavior of an agent who holds a simplified view of a dynamic optimization problem. The agent faces a Markov decision process, where a transition probability function determines the evolution of a state variable as a function of the previous state and the agent's action. The agent is uncertain about the true transition function and has a prior over a set of possible transition functions; this set reflects the agent's (possibly simplified) view of her environment and may not contain the true function. We define an equilibrium concept and provide conditions under which it characterizes steady‐state behavior when the agent updates her beliefs using Bayes' rule.

Optimal approximation by continued fractions

1991 ◽  
Vol 50 (3) ◽  
pp. 481-504 ◽  
Author(s):  
Wieb Bosma ◽  
Cor Kraaikamp
Keyword(s):  
Continued Fractions ◽  
Best Approximation ◽  
Continued Fraction ◽  
Irrational Number ◽  
Approximation Properties ◽  
Continued Fraction Expansion ◽  
The Best Approximation ◽  
Natural Sense ◽  
The One ◽  
Continued Fraction Expansions

AbstractAmong all possible semiregular continued fraction expansions of an irrational number the one with the best approximation properties, in a well-defined and natural sense, is determined. Some properties of this so called optimal continued fraction expansion are described.

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