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Relevant bibliographies by topics / Random walk

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Academic literature on the topic 'Random walk'

Author: Grafiati

Published: 4 June 2021

Last updated: 21 September 2024

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Journal articles on the topic "Random walk"

1

Blondel, Oriane, Marcelo R. Hilário, Renato S. dos Santos, Vladas Sidoravicius, and Augusto Teixeira. "Random walk on random walks: Low densities." Annals of Applied Probability 30, no. 4 (August 2020): 1614–41. http://dx.doi.org/10.1214/19-aap1537.

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Van Doorn, Erik A., and Pauline Schrijner. "Random walk polynomials and random walk measures." Journal of Computational and Applied Mathematics 49, no. 1-3 (December 1993): 289–96. http://dx.doi.org/10.1016/0377-0427(93)90162-5.

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Tran, Loc Hoang, Linh Hoang Tran, Hoang Trang, and Le Trung Hieu. "Combinatorial and Random Walk Hypergraph Laplacian EigenmapsCombinatorial and Random Walk Hypergraph Laplacian Eigenmaps." International Journal of Machine Learning and Computing 5, no. 6 (December 2015): 462–66. http://dx.doi.org/10.18178/ijmlc.2015.5.6.553.

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Boissard, Emmanuel, Serge Cohen, Thibault Espinasse, and James Norris. "Diffusivity of a random walk on random walks." Random Structures & Algorithms 47, no. 2 (April 16, 2014): 267–83. http://dx.doi.org/10.1002/rsa.20541.

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Mundy, John N. "Random Walk." Defect and Diffusion Forum 353 (May 2014): 1–7. http://dx.doi.org/10.4028/www.scientific.net/ddf.353.1.

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Random walk is the central concept in the mathematical formalization of the diffusion coefficient and so when asked a year ago to present a talk at a diffusion conference it appeared to be a totally appropriate topic. I spent most of my career studying diffusion and even after twenty years in retirement I believed I could write an interesting story about the importance of random walk to diffusion. Unfortunately when I sat down to write I discovered two problems: in the majority of materials that I investigated atoms did follow a random walk; and the history of random walk has been well documented and shows little connection to diffusion. The phrase was coined in 1905 at a time of rapid changes in physics. Scientists are not accustomed to writing history and as Henry Ford said around the same period of time “History is bunk”. He also remarked, "You can have any color (car) as long as it's black". This essay presents my story (not his- or her- tory) of why the use of the phrase random walk in discussions of diffusion in solids is also bunk.

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Butler, Mary Odell. "Random Walk." NAPA Bulletin 26, no. 1 (January 8, 2008): 20–31. http://dx.doi.org/10.1525/napa.2006.26.1.20.

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van der Hofstad, Remco, Tim Hulshof, and Jan Nagel. "Random walk on barely supercritical branching random walk." Probability Theory and Related Fields 177, no. 1-2 (September 25, 2019): 1–53. http://dx.doi.org/10.1007/s00440-019-00942-0.

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Croydon, David A. "Random Walk on the Range of Random Walk." Journal of Statistical Physics 136, no. 2 (July 2009): 349–72. http://dx.doi.org/10.1007/s10955-009-9785-2.

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Kutner, R. "Biased random walk on a biased random walk." Physica A: Statistical Mechanics and its Applications 171, no. 1 (February 1991): 43–46. http://dx.doi.org/10.1016/0378-4371(91)90356-h.

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KOJIMA, Seiji, Tomoko MASAIKE, Tohru MINAMINO, and Makoto MIYATA. "Following the Random Walk." Seibutsu Butsuri 54, no. 4 (2014): 226–29. http://dx.doi.org/10.2142/biophys.54.226.

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Dissertations / Theses on the topic "Random walk"

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Renlund, Henrik. "Reinforced Random Walk." Thesis, Uppsala University, Department of Mathematics, 2005. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-121389.

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Yip, Siu Kwan. "On the Pfaffian property of annihilating random walk and coalescing random walk." Thesis, University of Warwick, 2014. http://wrap.warwick.ac.uk/63894/.

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In this thesis we are to investigate two discrete interacting particle systems, namely annihilating random walk and coalescing random walk. By mapping the annihilating random walk to Glauber model and employing empty interval method respectively, we prove there is a similar structure behind them albeit their apparent differences, that is, they are both Pfaffian point process under a special initial condition. Then we extend the result to investigate whether the Pfaffian property preserves in the case of multi-time correlation function, which is called extended Pfaffian property. And we also investigate the case which the initial condition is changed from independent particles to another peculiar one-sided initial condition and proved it also preserved the Pfaffian property.

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nl, jvdberg@cwi. "Randomly Coalescing Random Walk in Dimension $ge$ 3." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1051.ps.

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Lo, Chak Hei. "On some random walk problems." Thesis, Durham University, 2017. http://etheses.dur.ac.uk/12498/.

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We consider several random walk related problems in this thesis. In the first part, we study a Markov chain on R₊ x S, where R₊ is the non-negative real numbers and S is a finite set, in which when the R₊-coordinate is large, the S-coordinate of the process is approximately Markov with stationary distribution πi on S. Denoting by μi(x) the mean drift of the R₊-coordinate of the process at (x,i) Ε R₊ x S, we give an exhaustive recurrence classification in the case where Σiπiμi(x) → 0, which is the critical regime for the recurrence-transience phase transition. If μi(x) → 0 for all i, it is natural to study the Lamperti case where μi(x) = O(1/x); in that case the recurrence classification is known, but we prove new results on existence and non-existence of moments of return times. If μi(x) → di for di ≠0 for at least some i, then it is natural to study the generalized Lamperti case where μi(x) = di + O(1/x). By exploiting a transformation which maps the generalized Lamperti case to the Lamperti case, we obtain a recurrence classification and an existence of moments result for the former. The generalized Lamperti case is seen to be more subtle, as the recurrence classification depends on correlation terms between the two coordinates of the process. In the second part of the thesis, for a random walk Sn on R^d we study the asymptotic behaviour of the associated centre of mass process Gn = n⁻¹Σ^n i=1 Si. For lattice distributions we give conditions for a local limit theorem to hold. We prove that if the increments of the walk have zero mean and finite second moment, Gn is recurrent if d=1 and transient if d≥2. In the transient case we show that Gn has diffusive rate of escape. These results extend work of Grill, who considered simple symmetric random walk. We also give a class of random walks with symmetric heavy-tailed increments for which Gn is transient in d=1.

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Zhao, Kai. "QUANTUM RANDOM WALK ON FRACTALS." Diss., Temple University Libraries, 2018. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/490714.

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Mathematics
Ph.D.
Quantum walks are the quantum mechanical analogue of classical random walks. Discrete-time quantum walks have been introduced and studied mostly on the line Z or higher dimensional space Z d but rarely defined on graphs with fractal dimensions because the coin operator depends on the position and the Fourier transform on the fractals is not defined. Inspired by its nature of classical walks, different quantum walks will be defined by choosing different shift and coin operators. When the coin operator is uniform, the results of classical walks will be obtained upon measurement at each step. Moreover, with measurement at each step, our results reveal more information about the classical random walks. In this dissertation, two graphs with fractal dimensions will be considered. The first one is Sierpinski gasket, a degree-4 regular graph with Hausdorff di- mension of df = ln 3/ ln 2. The second is the Cantor graph derived like Cantor set, with Hausdorff dimension of df = ln 2/ ln 3. The definitions and amplitude functions of the quantum walks will be introduced. The main part of this dissertation is to derive a recursive formula to compute the amplitude Green function. The exiting probability will be computed and compared with the classical results. When the generation of graphs goes to infinity, the recursion of the walks will be investigated and the convergence rates will be obtained and compared with the classical counterparts.
Temple University--Theses

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Zhang, Zhihan. "Random walk on simplicial complexes." Electronic Thesis or Diss., université Paris-Saclay, 2020. http://www.theses.fr/2020UPASM010.

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La notion de laplacien d’un graphe peut être généralisée aux complexes simpliciaux et aux hypergraphes. Cette notion contient des informations sur la topologie de ces structures. Dans la première partie de cette thèse,nous définissons une nouvelle chaîne de Markov sur les complexes simpliciaux. Pour un degré donné k de simplexes, l’espace d’états n’est pas les k-simplexes comme dans les articles précédents sur ce sujet mais plutôt l’ensemble des k-chaines ou k-co-chaines. Ce nouveau cadre est la généralisation naturelle sur les chaînes de Markov canoniques sur des graphes. Nous montrons que le générateur de notre chaîne de Markov est lié au Laplacien supérieur défini dans le contexte de la topologie algébrique pour structure discrète. Nous établissons plusieurs propriétés clés de ce nouveau procédé. Nous montrons que lorsque les complexes simpliciaux examinés sont une séquence de triangulation du tore plat, les chaînes de Markov tendent vers une forme différentielle valorisée processus continu.Dans la deuxième partie de cette thèse, nous explorons quelques applications de la marche aléatoire, i.e. la détection de trous basée sur la marche aléatoire et les noyaux complexes simpliciaux. Pour la détection de trous basée sur la marche aléatoire, nous introduisons un algorithme pour faire des simulations effectuées pour la marche aléatoire à valeur cyclique (k = 1) sur un complexe simplicial avec trous. Pour les noyaux de complexes simpliciaux, nous étendons la définition des noyaux de graphes basés sur la marche aléatoire afin de mesurer la similitude entre deux complexes simpliciaux
The notion of Laplacian of a graph can be generalized to simplicial complexes and hypergraphs. This notion contains information on the topology of these structures. In the first part of this thesis, we define a new Markov chain on simplicial complexes. For a given degree k of simplices, the state space is not thek-simplices as in previous papers about this subject but rather the set of k-chains or k-cochains.This new framework is the natural generalization on the canonical Markov chains on graphs.We show that the generator of our Markov chainis related to the upper Laplacian defined in the context of algebraic topology for discrete structure. We establish several key properties of this new process. We show that when the simplicial complexes under scrutiny are a sequence of ever refining triangulation of the flat torus, the Markov chains tend to a differential form valued continuous process.In the second part of this thesis, we explore some applications of the random walk, i.e., random walk based hole detection and simplicial complexes kernels. For the random walk based hole detection, we introduce an algorithm tomake simulations carried for the cycle-valuedrandom walk (k = 1) on a simplicial complex with holes. For the simplicial complexes kernels,we extend the definition of random walk based graph kernels in order to measure the similarity between two simplicial complexes

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Shiraishi, Daisuke. "Random walk on non-intersecting two-sided random walk trace is subdiffusive in low dimensions." 京都大学 (Kyoto University), 2012. http://hdl.handle.net/2433/157839.

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Phetpradap, Parkpoom. "Intersections of random walks." Thesis, University of Bath, 2011. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.548100.

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We study the large deviation behaviour of simple random walks in dimension three or more in this thesis. The first part of the thesis concerns the number of lattice sites visited by the random walk. We call this the range of the random walk. We derive a large deviation principle for the probability that the range of simple random walk deviates from its mean. Our result describes the behaviour for deviation below the typical value. This is a result analogous to that obtained by van den Berg, Bolthausen, and den Hollander for the volume of the Wiener sausage. In the second part of the thesis, we are interested in the number of lattice sites visited by two independent simple random walks starting at the origin. We call this the intersection of ranges. We derive a large deviation principle for the probability that the intersection of ranges by time n exceeds a multiple of n. This is also an analogous result of the intersection volume of two independent Wiener sausages.

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Bui, Hoai Thang Computer Science &amp Engineering Faculty of Engineering UNSW. "Guided random-walk based model checking." Awarded by:University of New South Wales. Computer Science & Engineering, 2009. http://handle.unsw.edu.au/1959.4/44829.

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The ever increasing use of computer systems in society brings emergent challenges to companies and system designers. The reliability of software and hardware can be financially critical, and lives can depend on it. The growth in size and complexity of software, and increasing concurrency, compounds the problem. The potential for errors is greater than ever before, and the stakes are higher than ever before. Formal methods, particularly model checking, is an approach that attempts to prove mathematically that a model of the behaviour of a product is correct with respect to certain properties. Certain errors can therefore be proven never to occur in the model. This approach has tremendous potential in system development to provide guarantees of correctness. Unfortunately, in practice, model checking cannot handle the enormous sizes of the models of real-world systems. The reason is that the approach requires an exhaustive search of the model to be conducted. While there are exceptions, in general model checkers are said not to scale well. In this thesis, we deal with this scaling issue by using a guiding technique that avoids searching areas of the model, which are unlikely to contain errors. This technique is based on a process of model abstraction in which a new, much smaller model is generated that retains certain important model information but discards the rest. This new model is called a heuristic. While model checking using a heuristic as a guide can be extremely effective, in the worst case (when the guide is of no help), it performs the same as exhaustive search, and hence it also does not scale well in all cases. A second technique is employed to deal with the scaling issue. This technique is based on the concept of random walks. A random walk is simply a `walk' through the model of the system, carried out by selecting states in the model randomly. Such a walk may encounter an error, or it may not. It is a non-exhaustive technique in the sense that only a manageable number of walks are carried out before the search is terminated. This technique cannot replace the conventional model checking as it can never guarantee the correctness of a model. It can however, be a very useful debugging tool because it scales well. From this point of view, it relieves the system designer from the difficult task of dealing with the problem of size in model checking. Using random walks, the effort goes instead into looking for errors. The effectiveness of model checking can be greatly enhanced if the above two techniques are combined: a random walk is used to search for errors, but the walk is guided by a heuristic. This in a nutshell is the focus of this work. We should emphasise that the random walk approach uses the same formal model as model checking. Furthermore, the same heuristic technique is used to guide the random walk as a guided model checker. Together, guidance and random walks are shown in this work to result in vastly improved performance over conventional model checking. Verification has been sacrificed of course, but the new technique is able to find errors far more quickly, and deal with much larger models.

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Thomson, D. J. "Random walk models of turbulent dispersion." Thesis, Brunel University, 1988. http://bura.brunel.ac.uk/handle/2438/5549.

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An understanding of the dispersion of contaminants in turbulent flows is important in many fields ranging from air pollution to chemical engineering, and random walk models provide one approach to understanding and calculating aspects of dispersion. Two types of random walk model are investigated in this thesis. The first type, so-called "one-particle models", are capable of predicting only mean concentrations while the second type, "two-particle models", are able to give some information on the fluctuations in concentration as well. Many different one-particle random walk models have been proposed previously and several criteria have emerged to distinguish good models from bad. In this thesis, the relationships between the various criteria are examined and it is shown that most of the criteria are equivalent. It is also shown how a model can be designed to (i) satisfy the criteria exactly and (ii) be consistent with inertial subrange theory. Some examples of models which obey the criteria are described. The theory developed for one-particle models is then extended to the two-particle case and used to design a two-particle model suitable for modelling dispersion in high Reynolds number isotropic turbulence. The properties of this model are investigated in detail and compared with previous models. In contrast to most previous models, the new model is three-dimensional and leads to a prediction for the particle separation probability density function which is in agreement with inertial subrange theory. The values of concentration variance from the new model are compared with experimental data and show encouraging agreement.

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Books on the topic "Random walk"

1

Nicolas, Christian. Random walk. London: Architectural Association Students Union, 1998.

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ill, Freeman Stephanie, ed. Rory's random walk down Wall Street. Charlotte, N.C: Playgroup Press, 1999.

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Lo, Andrew W. A non-random walk down Wall Street. Princeton, N.J: Princeton University Press, 2002.

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1955-, MacKinlay Archie Craig, ed. A non-random walk down Wall Street. Princeton, N.J: Princeton University Press, 1999.

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Cencini, Massimo, Andrea Puglisi, Davide Vergni, and Angelo Vulpiani. A Random Walk in Physics. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-72531-0.

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Schwarz, Wolf. Random Walk and Diffusion Models. Cham: Springer International Publishing, 2022. http://dx.doi.org/10.1007/978-3-031-12100-5.

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E, Mendoza, ed. A random walk in science. Bristol: Institute of Physics Pub., 1999.

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Cribari-Neto, Francisco. Canadian economic growth: Random walk or just a walk? Champaign: University of Illinois at Urbana-Champaign, 1992.

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Kaye, Brian H. A random walk through fractal dimensions. 2nd ed. Weinheim: VCH, 1994.

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Kaye, Brian H. A random walk through fractal dimensions. Weinheim, Germany: VCH Verlagsgesellschaft, 1989.

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Book chapters on the topic "Random walk"

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Meester, Ronald. "Random Walk." In A Natural Introduction to Probability Theory, 67–74. Basel: Birkhäuser Basel, 2003. http://dx.doi.org/10.1007/978-3-0348-7786-2_3.

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Fristedt, Bert, and Lawrence Gray. "Random Walk." In A Modern Approach to Probability Theory, 163–84. Boston, MA: Birkhäuser Boston, 1997. http://dx.doi.org/10.1007/978-1-4899-2837-5_11.

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Shreve, Steven E. "Random Walk." In Stochastic Calculus for Finance I, 119–42. New York, NY: Springer New York, 2005. http://dx.doi.org/10.1007/978-0-387-22527-2_5.

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Weik, Martin H. "random walk." In Computer Science and Communications Dictionary, 1411. Boston, MA: Springer US, 2000. http://dx.doi.org/10.1007/1-4020-0613-6_15460.

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Bhattacharya, Rabi. "Random Walk." In International Encyclopedia of Statistical Science, 1178–80. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-04898-2_475.

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Cencini, Massimo, Andrea Puglisi, Davide Vergni, and Angelo Vulpiani. "Random Walk." In A Random Walk in Physics, 7–8. Cham: Springer International Publishing, 2021. http://dx.doi.org/10.1007/978-3-030-72531-0_2.

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Korosteleva, Olga. "Random Walk." In Stochastic Processes with R, 43–60. Boca Raton: Chapman and Hall/CRC, 2022. http://dx.doi.org/10.1201/9781003244288-2.

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Chen, Xia. "Basics on large deviations." In Random Walk Intersections, 1–24. Providence, Rhode Island: American Mathematical Society, 2010. http://dx.doi.org/10.1090/surv/157/01.

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Chen, Xia. "Brownian intersection local times." In Random Walk Intersections, 25–58. Providence, Rhode Island: American Mathematical Society, 2010. http://dx.doi.org/10.1090/surv/157/02.

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Chen, Xia. "Mutual intersection: large deviations." In Random Walk Intersections, 59–89. Providence, Rhode Island: American Mathematical Society, 2010. http://dx.doi.org/10.1090/surv/157/03.

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Conference papers on the topic "Random walk"

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Rashid, Mahmood A., Swakkhar Shatabda, M. A. Hakim Newton, Md Tamjidul Hoque, Duc Nghia Pham, and Abdul Sattar. "Random-walk." In the ACM Conference. New York, New York, USA: ACM Press, 2012. http://dx.doi.org/10.1145/2382936.2383043.

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Stutz, Philip, Bibek Paudel, Mihaela Verman, and Abraham Bernstein. "Random Walk TripleRush." In WWW '15: 24th International World Wide Web Conference. Republic and Canton of Geneva, Switzerland: International World Wide Web Conferences Steering Committee, 2015. http://dx.doi.org/10.1145/2736277.2741687.

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Murai, Shogo, and Yuichi Yoshida. "Estimating Walk-Based Similarities Using Random Walk." In The World Wide Web Conference. New York, New York, USA: ACM Press, 2019. http://dx.doi.org/10.1145/3308558.3313421.

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Zabot, Alexandre, Diogo Nardelli Siebert, and Danilo Silva. "Random Walk in Petrophysics." In 24th ABCM International Congress of Mechanical Engineering. ABCM, 2017. http://dx.doi.org/10.26678/abcm.cobem2017.cob17-0820.

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Hosaka, Tadaaki, and Toru Ohira. "Repulsive delayed random walk." In Second International Symposium on Fluctuations and Noise, edited by Derek Abbott, Sergey M. Bezrukov, Andras Der, and Angel Sanchez. SPIE, 2004. http://dx.doi.org/10.1117/12.548492.

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Yilmaz, Erdal. "Squeezing angle random walk." In 2016 IEEE International Symposium on Inertial Sensors and Systems. IEEE, 2016. http://dx.doi.org/10.1109/isiss.2016.7435570.

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Sinha, Kalyan B. "Quantum random walk revisited." In Quantum Probability. Warsaw: Institute of Mathematics Polish Academy of Sciences, 2006. http://dx.doi.org/10.4064/bc73-0-30.

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Shulman, Ami, and Jorge Soto-Andrade. "A random walk in stochastic dance." In LINK 2021. Tuwhera Open Access, 2021. http://dx.doi.org/10.24135/link2021.v2i1.71.

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Stochastic music, developed last century by Xenakis, has older avatars, like Mozart, who showed how to compose minuets by tossing dice, in a similar way that contemporary choreographer Cunningham took apart the structural elements of what was considered to be a cohesive choreographic work (including movement, sound, light, set and costume) and reconstructed them in random ways. We intend to explore an enactive and experiential analogue of stochastic music, in the realm of dance, where the poetry of a choreographic spatial/floor pattern is elicited by a mathematical stochastic process, to wit a random walk – a stochastic dance of sorts. Among many possible random walks, we consider two simple examples, embodied in the following scenarios, proposed to the students/dancers: - a frog, jumping randomly on a row of stones, choosing right and left as if tossing a coin, - a person walking randomly on a square grid, starting a given node, and choosing each time randomly, equally likely N, S, E or W, and walking non-stop along the corresponding edge, up to the next node, and so on.When the dancers encounter these situations, quite natural questions arise for the choreographer, like: Where will the walker/dancer be after a while? Several ideas for a choreography emerge, which are more complex than just having one or more dancers perform the random walk, and which surprisingly turn our random process into a deterministic one!For instance, for the first random walk, 16 dancers start at the same node of a discrete line on the stage, and execute, each one, a different path of the 16 possible 4 – jump paths the frog can follow. They would need to agree first on how to carry this out. Interestingly, they may proceed without a Magister Ludi handing out scripts to every dancer. After arriving to their end node/position, they could try to retrace their steps, to come back all to the starting node.Analogously for the grid random walk, where we may have now 16 dancers enacting the 16 possible 2-edge paths of the walker. The dancers could also enter the stage (the grid or some other geometric pattern to walk around), one by one, sequentially, describing different random paths, or deterministic intertwined paths, in the spirit of Beckett’s Quadrat. Also, the dancers could choose their direction ad libitum, after some spinning, each time, on a grid-free stage, but keeping the same step length, as in statistician Pearson’s model for a mosquito random flight.We are interested in various possible spin-offs of these choreographies, which intertwine dance and mathematical cognition: For instance, when the dancers choose each one a different path, they will notice that their final distribution on the nodes is uneven (interesting shapes emerge). In this way, just by moving, choreographer and dancers can find a quantitative answer to the impossible question: where will the walker/dancer be after a while? Indeed, the percentage of dancers ending up at each node gives the probability of the random walker landing there.

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Lu, Lin, Xiaohua Xu, Ping He, Yue Ma, Qi Chen, and Ling Chen. "Supervised Lazy Random Walk Classifier." In 2013 10th Web Information System and Application Conference (WISA). IEEE, 2013. http://dx.doi.org/10.1109/wisa.2013.60.

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Mastio, Guillaume, and Pérola Milman. "1D Discrete-Time Random Walk." In Quantum Information and Measurement. Washington, D.C.: OSA, 2017. http://dx.doi.org/10.1364/qim.2017.qt6a.31.

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Reports on the topic "Random walk"

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Dshalalow, Jewgeni H. Random Walk Analysis in Antagonistic Stochastic Games. Fort Belvoir, VA: Defense Technical Information Center, July 2010. http://dx.doi.org/10.21236/ada533481.

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Morris, Richard. Solving random walk problems using resistive analogues. Portland State University Library, January 2000. http://dx.doi.org/10.15760/etd.529.

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Young, Richard M. Modeling Random Walk Processes In Human Concept Learning. Fort Belvoir, VA: Defense Technical Information Center, May 2006. http://dx.doi.org/10.21236/ada462700.

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Bacchetta, Philippe, and Eric van Wincoop. Random Walk Expectations and the Forward Discount Puzzle. Cambridge, MA: National Bureau of Economic Research, June 2007. http://dx.doi.org/10.3386/w13205.

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Pan, L., H. H. Liu, M. Cushey, and G. S. Bodvarsson. DCPT: A dual-continua random walk particle tracker fortransport. Office of Scientific and Technical Information (OSTI), April 2000. http://dx.doi.org/10.2172/926691.

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Smith, John, Aaron Hill, Leah Reeder, Brian Franke, Richard Lehoucq, Ojas Parekh, William Severa, and James Aimone. Neuromorphic scaling advantages for energy-efficient random walk computations. Office of Scientific and Technical Information (OSTI), September 2020. http://dx.doi.org/10.2172/1671377.

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Lunsford, Kurt G., and Kenneth D. West. Random Walk Forecasts of Stationary Processes Have Low Bias. Federal Reserve Bank of Cleveland, August 2023. http://dx.doi.org/10.26509/frbc-wp-202318.

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Abstract:

We study the use of a zero mean first difference model to forecast the level of a scalar time series that is stationary in levels. Let bias be the average value of a series of forecast errors. Then the bias of forecasts from a misspecified ARMA model for the first difference of the series will tend to be smaller in magnitude than the bias of forecasts from a correctly specified model for the level of the series. Formally, let P be the number of forecasts. Then the bias from the first difference model has expectation zero and a variance that is O(1/P²), while the variance of the bias from the levels model is generally O(1/P). With a driftless random walk as our first difference model, we confirm this theoretical result with simulations and empirical work: random walk bias is generally one-tenth to one-half that of an appropriately specified model fit to levels.

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West, Kenneth. On the Interpretation of Near Random-Walk Behavior in GNP. Cambridge, MA: National Bureau of Economic Research, August 1987. http://dx.doi.org/10.3386/w2364.

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Shiller, Robert, and Pierre Perron. Testing the Random Walk Hypothesis: Power versus Frequency of Observation. Cambridge, MA: National Bureau of Economic Research, April 1985. http://dx.doi.org/10.3386/t0045.

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Roberts, S. Convergence of a random walk method for the Burgers equation. Office of Scientific and Technical Information (OSTI), October 1985. http://dx.doi.org/10.2172/6336620.

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