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A137272
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Number of 1D random walks with 8 steps where the median of the positions is n.
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3
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OFFSET
| -4,1
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COMMENTS
| 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. 2009).
3) Transformation T007 gave a match with first differences of A089877 (superseeker).
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REFERENCES
| Feller, W., (1968) An Introduction to Probability Theory and its Applications I. New York: Wiley
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LINKS
| 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.
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EXAMPLE
| 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.
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MATHEMATICA
| (*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*)
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CROSSREFS
| Cf. A089877, A146205, A146206, A146207.
Sequence in context: A163478 A130440 A178676 * A121801 A192774 A032740
Adjacent sequences: A137269 A137270 A137271 * A137273 A137274 A137275
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KEYWORD
| easy,fini,full,nonn
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AUTHOR
| Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Mar 13 2008, May 03 2010
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EXTENSIONS
| Variables names normalized, offset set to -4. - R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Sep 17 2009
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