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

0,1

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).

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

Table of n, a(n) for n=0..15.

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.

Example

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

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*)

Crossrefs

A137272, A146205, A146206

Sequence in context: A229357 A376713 A254364 * A135802 A043390 A031481

Adjacent sequences: A146204 A146205 A146206 * A146208 A146209 A146210

Keyword

easy,fini,full,nonn

Author

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

Extensions

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

Status

approved