pokey909
Feb17-11, 01:59 PM
I encountered the following problem in the context of infinite Markov chains which seems to be somehow related to a random-walk problem.
Suppose we have a banded matrix M of the following form:
M=
[
P P 0 0 0 ...
Q 0 P 0 0 ...
0 Q 0 P 0 ...
0 0 Q 0 P ...
0 0 0 Q 0 ...
. . . . .
. . . . . ]
So with the exception of column 1, all others are equal. The "P 0 Q" element is only shifted one down (or row-wise, "Q 0 P" is shifted to the right).
It is impossible to construct a stochastic matrix with finite size unless you set the bottom-right element to Q. (For reference, all columns must sum to 1 to be stochastic).
P and Q denote the transition probabilities.
For a finite stochastic matrix of this form, one can raise it to high powers and the rows will converge to a certain value. This means that at high powers M^n each column will be equal.
For a matrix of size i x i, I will denote this vector with
s=(s_1, s_2, ... , s_i).
The vector elements represent the state probabilities in the chain. The vector can also be obtained by calculating the eigenvectors of the matrix.
I am interested in the sum of s, i.e. s_1+s_2+...+s_i.
However, for an infinite matrix, it is impossible to calculate in the same way as i did above.
Does anyone know a way to calculate the limit value of s_1+s_2+...+s_i given M is infinitely large?
Suppose we have a banded matrix M of the following form:
M=
[
P P 0 0 0 ...
Q 0 P 0 0 ...
0 Q 0 P 0 ...
0 0 Q 0 P ...
0 0 0 Q 0 ...
. . . . .
. . . . . ]
So with the exception of column 1, all others are equal. The "P 0 Q" element is only shifted one down (or row-wise, "Q 0 P" is shifted to the right).
It is impossible to construct a stochastic matrix with finite size unless you set the bottom-right element to Q. (For reference, all columns must sum to 1 to be stochastic).
P and Q denote the transition probabilities.
For a finite stochastic matrix of this form, one can raise it to high powers and the rows will converge to a certain value. This means that at high powers M^n each column will be equal.
For a matrix of size i x i, I will denote this vector with
s=(s_1, s_2, ... , s_i).
The vector elements represent the state probabilities in the chain. The vector can also be obtained by calculating the eigenvectors of the matrix.
I am interested in the sum of s, i.e. s_1+s_2+...+s_i.
However, for an infinite matrix, it is impossible to calculate in the same way as i did above.
Does anyone know a way to calculate the limit value of s_1+s_2+...+s_i given M is infinitely large?