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Search: id:A137272
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| A137272 |
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Number of paths of the simple random walk on condition that the median applied to the partial sums S_0=0, S_1,...,S_n, n even (n=8 in this example), is equal to k, -n/2<=k<=n/2. |
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+0
1
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OFFSET
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0,1
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COMMENT
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1) Suppose n is even, the convolution of the probability distribution of the minimum and the maximum of a simple random walk up to n/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 asymetrically informed traders is traded (see Pfeifer et al. 2008).
3) Transformation T007 gave a match with first differences of A089877 (superseeker).
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REFERENCES
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Feller, W., (1968) An Introduction to Probability Theory and its Applications I. New York: Wiley
Pfeifer, C., Schredelseker, K. and Seeber G. U. H., (2008) On the negative value of information in informationally inefficient markets. Calculations for large number of traders. European Journal of Operational Research, in press.
Wendel, J. G., (1960) Order Statistics of Partial Sums, 31 Ann.Math.Statist. 31, pp. 1034-1044.
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EXAMPLE
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The possible different paths (sequences of partial sums) in the case n=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 n=2: 1,2,1
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MATHEMATICA
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(*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
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Cf. A089877.
Adjacent sequences: A137269 A137270 A137271 this_sequence A137273 A137274 A137275
Sequence in context: A025129 A093559 A130440 this_sequence A121801 A032740 A025629
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KEYWORD
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easy,fini,nonn,uned
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AUTHOR
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Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Mar 13 2008
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