MULTIFRACTAL ANALYSIS OF ONE-DIMENSIONAL BIASED WALKS

Fractals ◽  
2018 ◽  
Vol 26 (03) ◽  
pp. 1850030 ◽  
Author(s):  
YUFEI CHEN ◽  
MEIFENG DAI ◽  
XIAOQIAN WANG ◽  
YU SUN ◽  
WEIYI SU
Keyword(s):  
Random Walk ◽  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Infinite Sequence ◽  
Real Numbers ◽  
One Dimensional ◽  
Dimension Formula ◽  
Sum Of Digits ◽  
Packing Dimensions ◽  
Dyadic System

For an infinite sequence [Formula: see text] of [Formula: see text] and [Formula: see text] with probability [Formula: see text] and [Formula: see text], we mainly study the multifractal analysis of one-dimensional biased walks. Let [Formula: see text] and [Formula: see text]. The Hausdorff and packing dimensions of the sets [Formula: see text] are [Formula: see text], which is the development of the theorem of Besicovitch [On the sum of digits of real numbers represented in the dyadic system, Math. Ann. 110 (1934) 321–330] on random walk, saying that: For any [Formula: see text], the set [Formula: see text] has Hausdorff dimension [Formula: see text].

scholarly journals Good’s theorem for Hurwitz continued fractions

2020 ◽  
Vol 16 (07) ◽  
pp. 1433-1447
Author(s):  
Gerardo Gonzalez Robert
Keyword(s):  
Hausdorff Dimension ◽  
Complex Plane ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Analogous Result ◽  
Complex Numbers ◽  
Real Numbers ◽  
Dimension Formula

Good’s Theorem for regular continued fraction states that the set of real numbers [Formula: see text] such that [Formula: see text] has Hausdorff dimension [Formula: see text]. We show an analogous result for the complex plane and Hurwitz Continued Fractions: the set of complex numbers whose Hurwitz Continued fraction [Formula: see text] satisfies [Formula: see text] has Hausdorff dimension [Formula: see text], half of the ambient space’s dimension.

scholarly journals On the Discrepancy of Two Families of Permuted Van der Corput Sequences

2018 ◽  
Vol 13 (1) ◽  
pp. 47-64 ◽  
Author(s):  
Florian Pausinger ◽  
Alev Topuzoğlu
Keyword(s):  
Finite Fields ◽  
Infinite Sequence ◽  
Real Numbers ◽  
One Dimensional ◽  
Polynomials Over Finite Fields ◽  
Van Der Corput Sequence ◽  
Distribution Behavior

Abstract A permuted van der Corput sequence $S_b^\sigma$ in base b is a one-dimensional, infinite sequence of real numbers in the interval [0, 1), generation of which involves a permutation σ of the set {0, 1,..., b − 1}. These sequences are known to have low discrepancy DN, i.e. $t\left({S_b^\sigma } \right): = {\rm{lim}}\,{\rm{sup}}_{N \to \infty } D_N \left({S_b^\sigma } \right)/{\rm{log}}\,N$ is finite. Restricting to prime bases p we present two families of generating permutations. We describe their elements as polynomials over finite fields 𝔽p in an explicit way. We use this characterization to obtain bounds for $t\left({S_p^\sigma } \right)$ for permutations σ in these families. We determine the best permutations in our first family and show that all permutations of the second family improve the distribution behavior of classical van der Corput sequences in the sense that $t\left({S_p^\sigma } \right) < t\left({S_p^{id} } \right)$ .

BIRKHOFF SPECTRA FOR ONE-DIMENSIONAL MAPS WITH SOME HYPERBOLICITY

2010 ◽  
Vol 10 (01) ◽  
pp. 53-75 ◽  
Author(s):  
YONG MOO CHUNG
Keyword(s):  
Dynamical System ◽  
Hausdorff Dimension ◽  
Multifractal Analysis ◽  
Probability Measures ◽  
One Dimensional ◽  
System A ◽  
Topologically Mixing ◽  
One Dimensional Maps ◽  
Dimension One

We study the multifractal analysis for smooth dynamical systems in dimension one. It is given a characterization of the Hausdorff dimension of the level set obtained from the Birkhoff averages of a continuous function by the local dimensions of hyperbolic measures for a topologically mixing C2 map modeled by an abstract dynamical system. A characterization which corresponds to above is also given for the ergodic basins of invariant probability measures. And it is shown that the complement of the set of quasi-regular points carries full Hausdorff dimension.

scholarly journals Increasing subsequences of random walks

2016 ◽  
Vol 163 (1) ◽  
pp. 173-185 ◽  
Author(s):  
OMER ANGEL ◽  
RICHÁRD BALKA ◽  
YUVAL PERES
Keyword(s):  
Asymptotic Behavior ◽  
Random Walk ◽  
Lower Bound ◽  
Upper Bound ◽  
Simple Random Walk ◽  
Real Numbers ◽  
Record Times ◽  
One Dimensional ◽  
Expected Length ◽  
Longest Increasing Subsequence

AbstractGiven a sequence of n real numbers {Si}i⩽n, we consider the longest weakly increasing subsequence, namely i1 < i2 < . . . < iL with Sik ⩽ Sik+1 and L maximal. When the elements Si are i.i.d. uniform random variables, Vershik and Kerov, and Logan and Shepp proved that ${\mathbb E} L=(2+o(1)) \sqrt{n}$.We consider the case when {Si}i⩽n is a random walk on ℝ with increments of mean zero and finite (positive) variance. In this case, it is well known (e.g., using record times) that the length of the longest increasing subsequence satisfies ${\mathbb E} L\geq c\sqrt{n}$. Our main result is an upper bound ${\mathbb E} L\leq n^{1/2 + o(1)}$, establishing the leading asymptotic behavior. If {Si}i⩽n is a simple random walk on ℤ, we improve the lower bound by showing that ${\mathbb E} L \geq c\sqrt{n} \log{n}$.We also show that if {Si} is a simple random walk in ℤ2, then there is a subsequence of {Si}i⩽n of expected length at least cn1/3 that is increasing in each coordinate. The above one-dimensional result yields an upper bound of n1/2+o(1). The problem of determining the correct exponent remains open.

ON THE DECOMPOSITION OF CONTINUOUS FUNCTIONS AND DIMENSIONS

Fractals ◽  
2019 ◽  
Vol 28 (01) ◽  
pp. 2050007
Author(s):  
JIA LIU ◽  
DEZHI LIU
Keyword(s):  
Continuous Function ◽  
Hausdorff Dimension ◽  
Present Author ◽  
Continuous Functions ◽  
Box Dimension ◽  
Real Numbers ◽  
Dimension Formula

In this paper, we consider decomposition of continuous functions in [Formula: see text] in terms of Hausdorff dimension and lower box dimension. Precisely, we show that, given real numbers [Formula: see text], any real-valued continuous function in [Formula: see text] can be decomposed into a sum of two real-valued continuous functions each having a graph of Hausdorff dimension [Formula: see text] and lower box dimension [Formula: see text]. This generalizes a theorem of Wingren, also Wu and the present author. We also consider the arbitrary decomposition of continuous functions in terms of Hausdorff dimension and lower box dimension.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

On coupling of random walks and renewal processes

1996 ◽  
Vol 33 (1) ◽  
pp. 122-126
Author(s):  
Torgny Lindvall ◽  
L. C. G. Rogers
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Simple Random Walk ◽  
Renewal Processes ◽  
Renewal Theorem ◽  
One Dimensional ◽  
Continuous State Space ◽  
Continuous State ◽  
Efficient Coupling ◽  
Blackwell's Renewal Theorem

The use of Mineka coupling is extended to a case with a continuous state space: an efficient coupling of random walks S and S' in can be made such that S' — S is virtually a one-dimensional simple random walk. This insight settles a zero-two law of ergodicity. One more proof of Blackwell's renewal theorem is also presented.

scholarly journals Optimal Strategies for Prudent Investors

1998 ◽  
Vol 01 (04) ◽  
pp. 473-486 ◽  
Author(s):  
Roberto Baviera ◽  
Michele Pasquini ◽  
Maurizio Serva ◽  
Angelo Vulpiani
Keyword(s):  
Growth Rate ◽  
Random Walk ◽  
Exponential Growth ◽  
Optimal Strategies ◽  
Maximum Amount ◽  
Exponential Growth Rate ◽  
One Dimensional ◽  
Economic Level ◽  
Random Walk With Drift ◽  
Analytical Expressions

We consider a stochastic model of investment on an asset in a stock market for a prudent investor. she decides to buy permanent goods with a fraction α of the maximum amount of money owned in her life in order that her economic level never decreases. The optimal strategy is obtained by maximizing the exponential growth rate for a fixed α. We derive analytical expressions for the typical exponential growth rate of the capital and its fluctuations by solving an one-dimensional random walk with drift.

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