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A146207
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Number of paths of the simple random walk on condition that the [n/2]th ordered value S_([n/2]) of the partial sums S_0=0, S_1,...,S_n, n odd (n=15 and S_(7) in this example), is equal to k, [ -n/2]-1<=k<=[n/2].
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4
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35, 70, 336, 602, 1456, 2310, 3760, 5210, 6435, 5210, 3270, 2310, 966, 602, 126, 70
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OFFSET
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0,1
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COMMENTS
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1) Suppose n is odd, the convolution of the probability distribution of the maximum of a simple random walk up to [n/2] and the minimum of a simple random walk up to [n/2]+1 is equal to the probability distribution of this ordered value. (see Mathematica program and references).
2) Relationship between median and the [n/2]th ordered value S_([n/2]) of partial sums for the odd case: A146207=A146205+(0,A146206); see lemma 2 in Pfeifer (2010).
3) 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 (Pfeifer et al. 2009).
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REFERENCES
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Pfeifer, C. (2010) Probability distribution of the median taken on partial sums of the simple random walk. Submitted to Stochastic Analysis and Applications.
Wendel, J.G. (1960) Order Statistics of Partial Sums. 31 Ann.Math.Statist. 31, pp. 1034-1044.
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LINKS
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EXAMPLE
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All possible different paths (sequences of partial sums) in case of n=3:
{0,-1,-2,-3}; S_(1)=-2
{0,-1,-2,-1}; S_(1)=-1
{0,-1,0,-1}; S_(1)=-1
{0,-1,0,1}; S_(1)=0
{0,1,0,-1}; S_(1)=0
{0,1,0,1}; S_(1)=0
{0,1,2,1}; S_(1)=1
{0,1,2,3}; S_(1)=1
sequence of integers in case of n=3: 1,2,3,2
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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]; minimum[n_, r_] := p[n, -r] + p[n, -r + 1];
(*distr. [k/2]th ordered value*)
k := 15; (*k odd integer*) n = Floor[k/2]; (*k=2n+1*) listmin = Table[If[r < -(n + 1) || r > 0, 0, minimum[n + 1, r]], {r, -(n + 1), n + 1}]; (*dist. minimum*) listmax = Table[If[r > n || r < 0, 0, maximum[n, r]], {r, -n, n}]; (*distr. maximum*) listsort = ListConvolve[listmax, listmin, {1, -1}]; (*convolution*)
listsort[[n + 1 ;; 3 n + 2]](*result ordered value*)
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CROSSREFS
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KEYWORD
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easy,fini,full,nonn
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AUTHOR
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Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Oct 28 2008, May 04 2010
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EXTENSIONS
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STATUS
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approved
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