scholarly journals Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk

2004 ◽  
Vol 70 (4) ◽  
Author(s):  
Gunter M. Schütz ◽  
Steffen Trimper
Keyword(s):  
Random Walk ◽  
Long Range ◽  
Memory Effects

scholarly journals Differences in fractal patterns and characteristic periodicities between word salads and normal sentences: Interference of meaning and sound

PLoS ONE ◽  
2021 ◽  
Vol 16 (2) ◽  
pp. e0247133
Author(s):  
Jun Shimizu ◽  
Hiromi Kuwata ◽  
Kazuo Kuwata
Keyword(s):  
Fractal Dimension ◽  
Random Walk ◽  
Long Range ◽  
Mental State ◽  
Fractal Dimensions ◽  
Social Robots ◽  
Fractal Patterns

Fractal dimensions and characteristic periodicities were evaluated in normal sentences, computer-generated word salads, and word salads from schizophrenia patients, in both Japanese and English, using the random walk patterns of vowels.　In normal sentences, the walking curves were smooth with gentle undulations, whereas computer-generated word salads were rugged with mechanical repetitions, and word salads from patients with schizophrenia were unreasonably winding with meaningless repetitive patterns or even artistic cohesion. These tendencies were similar in both languages. Fractal dimensions between normal sentences and word salads of schizophrenia were significantly different in Japanese [1.19 ± 0.09 (n = 90) and 1.15 ± 0.08 (n = 45), respectively] and English [1.20 ± 0.08 (n = 91), and 1.16 ± 0.08 (n = 42)] (p < 0.05 for both). Differences in long-range (>10) periodicities between normal sentences and word salads from schizophrenia patients were predominantly observed at 25.6 (p < 0.01) in Japanese and 10.7 (p < 0.01) in English. The differences in fractal dimension and characteristic periodicities of relatively long-range (>10) presented here are sensitive to discriminate between schizophrenia and healthy mental state, and could be implemented in social robots to assess the mental state of people in care.

scholarly journals How linear reinforcement affects Donsker’s theorem for empirical processes

2020 ◽  
Vol 178 (3-4) ◽  
pp. 1173-1192 ◽  
Author(s):  
Jean Bertoin
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Long Range ◽  
Limit Theorems ◽  
Brownian Bridge ◽  
Empirical Processes ◽  
Constant Factor ◽  
Empirical Measure ◽  
Reinforced Random Walks ◽  
Bernoulli Processes

Abstract A reinforcement algorithm introduced by Simon (Biometrika 42(3/4):425–440, 1955) produces a sequence of uniform random variables with long range memory as follows. At each step, with a fixed probability $$p\in (0,1)$$ p ∈ ( 0 , 1 ) , $${\hat{U}}_{n+1}$$ U ^ n + 1 is sampled uniformly from $${\hat{U}}_1, \ldots , {\hat{U}}_n$$ U ^ 1 , … , U ^ n , and with complementary probability $$1-p$$ 1 - p , $${\hat{U}}_{n+1}$$ U ^ n + 1 is a new independent uniform variable. The Glivenko–Cantelli theorem remains valid for the reinforced empirical measure, but not the Donsker theorem. Specifically, we show that the sequence of empirical processes converges in law to a Brownian bridge only up to a constant factor when $$p<1/2$$ p < 1 / 2 , and that a further rescaling is needed when $$p>1/2$$ p > 1 / 2 and the limit is then a bridge with exchangeable increments and discontinuous paths. This is related to earlier limit theorems for correlated Bernoulli processes, the so-called elephant random walk, and more generally step reinforced random walks.

Analytical and empirical fluctuation functions of the EEG microstate random walk - Short-range vs. long-range correlations

NeuroImage ◽  
2016 ◽  
Vol 141 ◽  
pp. 442-451 ◽  
Author(s):  
F. von Wegner ◽  
E. Tagliazucchi ◽  
V. Brodbeck ◽  
H. Laufs
Keyword(s):  
Random Walk ◽  
Long Range ◽  
Short Range ◽  
Long Range Correlations

Schramm–Loewner evolution of the accessible perimeter of isoheight lines of correlated landscapes

2018 ◽  
Vol 29 (01) ◽  
pp. 1850008 ◽  
Author(s):  
N. Posé ◽  
K. J. Schrenk ◽  
N. A. M. Araújo ◽  
H. J. Herrmann
Keyword(s):  
Fractal Dimension ◽  
Random Walk ◽  
Long Range ◽  
Hurst Exponent ◽  
Diffusion Constant ◽  
Strongly Correlated ◽  
Loewner Evolution ◽  
Positive Formula ◽  
Practical Implications ◽  
Negative Formula

Real landscapes exhibit long-range height–height correlations, which are quantified by the Hurst exponent [Formula: see text]. We give evidence that for negative [Formula: see text], in spite of the long-range nature of correlations, the statistics of the accessible perimeter of isoheight lines is compatible with Schramm–Loewner evolution curves and therefore can be mapped to random walks, their fractal dimension determining the diffusion constant. Analytic results are recovered for [Formula: see text] and [Formula: see text] and a conjecture is proposed for the values in between. By contrast, for positive [Formula: see text], we find that the random walk is not Markovian but strongly correlated in time. Theoretical and practical implications are discussed.

scholarly journals Memory effects in a random walk description of protein structure ensembles

2019 ◽  
Vol 150 (6) ◽  
pp. 064911
Author(s):  
Gerald R. Kneller ◽  
Konrad Hinsen
Keyword(s):  
Random Walk ◽  
Protein Structure ◽  
Memory Effects

LONG RANGE CORRELATION IN HUMAN WRITINGS

Fractals ◽  
1993 ◽  
Vol 01 (01) ◽  
pp. 47-57 ◽  
Author(s):  
ALAIN SCHENKEL ◽  
JUN ZHANG ◽  
YI-CHENG ZHANG
Keyword(s):  
Random Walk ◽  
Power Spectrum ◽  
Long Range ◽  
Computer Programs ◽  
Random Walk Model ◽  
The Other ◽  
Scaling Exponent ◽  
The Bible ◽  
Range Correlation ◽  
Roman Letter

We study long range correlation in various human writings by mapping them into a simple 1d random walk model. This approach allows us to obtain better quality scaling data than the traditional power spectrum methods. Optimally written computer programs seem to have the highest scaling exponent, i.e., close to that of ideal 1/f noise. Literature, on the other hand, leads to smaller exponents. The Bible, however, has the strongest correlation among the Roman letter writings examined.

scholarly journals Random Walk in Dynamic Random Environment with Long-Range Space Correlations

2016 ◽  
Vol 16 (4) ◽  
pp. 621-640
Author(s):  
C. Boldrighini ◽  
R. A. Minlos ◽  
A. Pellegrinotti
Keyword(s):  
Random Walk ◽  
Long Range ◽  
Random Environment ◽  
Range Space
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