Comments - 2-D random walks: simulation, video with R source code, curious facts - Data Science Central 2021-09-24T13:16:01Z https://www.datasciencecentral.com/profiles/comment/feed?attachedTo=6448529%3ABlogPost%3A213755&xn_auth=no Also, E[M(n)] = SQRT(n/2) is… tag:www.datasciencecentral.com,2014-10-21:6448529:Comment:216276 2014-10-21T18:54:53.594Z Vincent Granville https://www.datasciencecentral.com/profile/VincentGranville <p>Also, <span>E[M(n)] = SQRT(n/2) is <span style="text-decoration: underline;">not</span> the law of the iterative algorithm: my result is about a max observed over n steps, not a lim sup. And it is <span style="text-decoration: underline;">not</span> the expectation of the absolute value observed at step n either: that one uses a factor 2/pi, rather than 1/2.</span></p> <p>Also, <span>E[M(n)] = SQRT(n/2) is <span style="text-decoration: underline;">not</span> the law of the iterative algorithm: my result is about a max observed over n steps, not a lim sup. And it is <span style="text-decoration: underline;">not</span> the expectation of the absolute value observed at step n either: that one uses a factor 2/pi, rather than 1/2.</span></p> Regarding M(n), there is an i… tag:www.datasciencecentral.com,2014-10-13:6448529:Comment:214119 2014-10-13T16:33:17.664Z Mirko Krivanek https://www.datasciencecentral.com/profile/MirkoKrivanek <p>Regarding M(n), there is an interesting formula for P(M(n) = r), for any non-negative integer r. Denoting as S(n) the value observed at step n in a one-dimensional symmetrical random walk starting starting with S(0) = 0, moving by increments or +1 or -1 at each new step, we have</p> <p style="text-align: center;">P(M(n) = r) = P(S(n) = r) + P(S(n) = r+1) = max{P(S(n) = r), P(S(n) = r+1)}.</p> <p>Since S(n) is a sum of n binary random variables, its distribution is well known and approximated…</p> <p>Regarding M(n), there is an interesting formula for P(M(n) = r), for any non-negative integer r. Denoting as S(n) the value observed at step n in a one-dimensional symmetrical random walk starting starting with S(0) = 0, moving by increments or +1 or -1 at each new step, we have</p> <p style="text-align: center;">P(M(n) = r) = P(S(n) = r) + P(S(n) = r+1) = max{P(S(n) = r), P(S(n) = r+1)}.</p> <p>Since S(n) is a sum of n binary random variables, its distribution is well known and approximated by a normal distribution with standard deviation SQRT(n).</p> <p>Details are found <a href="http://www2.math.uu.se/~sea/kurser/stokprocmn1/slumpvandring_eng.pdf" target="_blank">in this PDF document</a>, page 19. Note that M(n) is equal to max{S(0), S(1), ... , S(n)}.</p>