login
A146206
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 odd (n=15 in this example), is equal to integer values k, -[n/2]<=k<=[n/2].
4
35, 91, 357, 721, 1575, 2535, 3985, 5210, 3985, 2535, 1575, 721, 357, 91, 35
OFFSET
0,1
COMMENTS
1) A146207=A146205+(0,A146206), see lemma 2 in Pfeifer (2010).
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 (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.
LINKS
EXAMPLE
All possible different paths (sequences of partial sums) in case of n=3:
{0,-1,-2,-3}; median=-1.5
{0,-1,-2,-1}; median=-1
{0,-1,0,-1}; median=-0.5
{0,-1,0,1}; median=0
{0,1,0,-1}; median=0
{0,1,0,1}; median=0.5
{0,1,2,1}; median=1
{0,1,2,3}; median=1.5
sequence of integers in case of n=3: 1,2,1
CROSSREFS
KEYWORD
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