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%I #15 Jan 03 2021 00:59:55
%S 6,10,32,45,70,45,32,10,6
%N Number of 1D random walks with 8 steps where the median of the positions is n.
%C Consider the 1D random walk starting at position 0, with equal probability of moving one unit to the left or one unit to the right. This allows 2^s different trajectories if we consider a maximum of s steps (s=8 here). For each of the trajectories, compute the median position, which is in the interval [-s/2, +s/2].
%C The sequence shows the count of trajectories with median equal to n (so the sum over all elements of the sequence is again 2^s = 256).
%C 1) Suppose s is even, the convolution of the probability distribution of the minimum and the maximum of a simple random walk up to s/2 is equal to the probability distribution of the median (see Mathematica program and references).
%C 2) The median taken on partial sums of the simple random walk represents the market price in a simulation model wherein a single security among non-cooperating and asymmetrically informed traders is traded (see Pfeifer et al. 2009).
%C 3) Transformation T007 gave a match with first differences of A089877 (superseeker).
%D W. Feller, An Introduction to Probability Theory and its Applications I. New York: Wiley, 1968.
%H C. Pfeifer, <a href="https://doi.org/10.1080/07362994.2013.741359">Probability Distribution of the Median Taken on Partial Sums of a Simple Random Walk</a>, Stochastic Analysis and Applications, Volume 31, Issue 1, 2013, pp. 31-46; DOI:10.1080/07362994.2013.741359. - _N. J. A. Sloane_, Jan 04 2013
%H C. Pfeifer, K. Schredelseker, and G. U. H. Seeber, <a href="http://dx.doi.org/10.1016/j.ejor.2008.01.015">On the negative value of information in informationally inefficient markets. Calculations for large number of traders</a>, Eur. J. Operat. Res., 195 (1) (2009) 117-126.
%H J. G. Wendel, <a href="http://dx.doi.org/10.1214/aoms/1177705676">Order Statistics of Partial Sums</a>, Ann. Math. Statist. 31 (4) (1960) pp. 1034-1044.
%e The possible different paths (sequences of partial sums) in the case s=2:
%e {0,-1,-2}; median=-1
%e {0,-1,0}; median=0
%e {0,1,0}; median=0
%e {0,1,2}; median=1
%e Sequence of integers in the case s=2: 1,2,1.
%e In the current case s=8, we have 6 trajectories with median -4, 10 trajectories with median -3, etc.
%t (* calculation of distribution of median single random walk *) p[n_, r_] := If[Floor[(n + r)/2] - (n + r)/2 == 0, Binomial[n, (n + r)/2], 0] maximum[n_, r_] := p[n, r] + p[n, r + 1]; (* prob. maximum *) minimum[n_, r_] := p[n, -r] + p[n, -r + 1]; (* prob. minimum *) median[n_] := ( (* distr. median *) listmin = Table[If[r < -(n/2) || r > 0, 0, minimum[n/2, r]], {r, -n, n}] (* distr. minimum *); listmax = Table[If[r > n/2 || r < 0, 0, maximum[n/2, r]], {r, -n, n}] (* distr. maximum *); listmedian = ListConvolve[listmax, listmin, {1, -1}] (* convolution *); listmedian[[3 n/2 + 1 ;; 5 n/2 + 1]]); (* result median *) Table[median[2 n], {n, 1, 7}](* result up to n=14 *)
%Y Cf. A089877, A146205, A146206, A146207.
%K easy,fini,full,nonn
%O -4,1
%A Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Mar 13 2008, May 03 2010
%E Variables names normalized, offset set to -4 by _R. J. Mathar_, Sep 17 2009
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