A146207 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].

35, 70, 336, 602, 1456, 2310, 3760, 5210, 6435, 5210, 3270, 2310, 966, 602, 126, 70

Offset: 0

Author

Christian Pfeifer (christian.pfeifer(AT)uibk.ac.at), Oct 28 2008, May 04 2010

Keywords

• easy
• fini
• full
• nonn

Comments

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).

Examples

			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
		

References

• 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.

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.

Crossrefs

A137272, A146205, A146206

Programs

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]; 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*)

Extensions

Keyword:full added by R. J. Mathar, Sep 17 2009