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Branching random walk
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<h2 id="example">Example</h2> <p>An example of branching random walk can be constructed where the branching process generates exactly two descendants for each element, a <i>binary </i>branching random walk. Given the <a href="/facts/Initial_condition/sA4ohFP4">initial condition</a> that <i>X</i>ϵ = 0, we suppose that <i>X</i>1 and <i>X</i>2 are the two children of <i>X</i>ϵ. Further, we suppose that they are <a href="/facts/Independence_(probability_theory)/NUzQtnUL">independent</a> <a href="/facts/Normal_distribution/UapjjPyQ">N(0, 1)</a> random variables. Consequently, in generation 2, the random variables <i>X</i>1,1 and <i>X</i>1,2 are each the sum of <i>X</i>1 and a N(0, 1) random variable. In the next generation, the random variables <i>X</i>1,2,1 and <i>X</i>1,2,2 are each the sum of <i>X</i>1,2 and a N(0, 1) random variable. The same construction produces the values at successive times. </p><p>Each lineage in the infinite "genealogical tree" produced by this process, such as the sequence <i>X</i>ϵ, <i>X</i>1, <i>X</i>1,2, <i>X</i>1,2,2, ..., forms a conventional random walk. </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Discrete-time_dynamical_system/hcaspdq4">Discrete-time dynamical system</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Kaplan, Norman (1982). "A Note on the Branching Random Walk". Journal of Applied Probability. 19 (2): 421–424. doi:10.2307/3213494. ISSN 0021-9002. <a href="https://www.jstor.org/stable/3213494" target="_blank">https://www.jstor.org/stable/3213494</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Shi, Zhan (2015). Branching Random Walks. École d’Été de Probabilités de Saint-Flour XLII – 2012. Vol. 2151. Paris: Springer. doi:10.1007/978-3-319-25372-5. ISBN 978-3-319-25371-8. ISSN 0075-8434. <a href="978-3-319-25371-8" target="_blank">978-3-319-25371-8</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Bovier, Anton, ed. (2016), "Branching Brownian Motion", Gaussian Processes on Trees: From Spin Glasses to Branching Brownian Motion, Cambridge Studies in Advanced Mathematics, Cambridge: Cambridge University Press, pp. 60–75, ISBN 978-1-107-16049-1, retrieved 2024-11-25 <a href="978-1-107-16049-1" target="_blank">978-1-107-16049-1</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> </ol>