-4,1

Consider the 1D random walk starting at position 0, with equal probability to move

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 range between -s/2 and +s/2.

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).

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).

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. 2008).

3) Transformation T007 gave a match with first differences of A089877 (superseeker).

Feller, W., (1968) An Introduction to Probability Theory and its Applications I. New York: Wiley

C. Pfeifer, K. Schredelseker, G. U. H. Seeber, On the negative value of information in informationally inefficient markets. Calculations for large number of traders, Eur. J. Operat. Res., 195 (1) (2009) 117-126.

J. G. Wendel, J. G., Order Statistics of Partial Sums, Ann. Math. Statist. 31 (4) (1960) pp. 1034-1044.

The possible different paths (sequences of partial sums) in the case s=2:

{0,-1,-2}; median=-1

{0,-1,0}; median=0

{0,1,0}; median=0

{0,1,2}; median=1

Sequence of integers in the case s=2: 1,2,1

In the current case s=8, we have 6 trajectories with median -4, 10 trajectories with median -3 etc.

(*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*)

Cf. A089877, A146205, A146206, A146207.

Sequence in context: A093559 A163478 A130440 this_sequence A121801 A032740 A167330

Adjacent sequences: A137269 A137270 A137271 this_sequence A137273 A137274 A137275

easy,fini,full,nonn

Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Mar 13 2008

Variables names normalized, offset set to -4 - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 17 2009