Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Joint types can often be given in terms of the type of x and a stochastic matrix \begin{equation} V:X\rightarrow Y \end{equation}such that $ P_{x,y}(a,b)=P_{x}(a)V(b|a)$ for every $a\in X$ , $b\in Y$. The question is that how can we define $V(b|a)$ as conditional type given x and what is the V-shell of x denoted by $T_{V}(x)$?

share|improve this question
    
The type of a sequence is usually defined to be empirical frequency counts of a sequence, but you seem to be referring to probability distributions as types - can you clarify? Also, what is a $V$-shell? –  sai Jul 7 '12 at 2:50
add comment

1 Answer

Lets take an example. I will talk in terms of empirical probabilities (Probability or relative frequency observed in a given sequence). Lets have X={a,b} and Y = {c,d}. Lets have the length of the sequence to be 7. Now for a given 7-length sequence of x^7 = (aababba} lets consider y^7 = (dccddcd). More graphically :

| x^7 = | a | a | b | a | b | b | a |
| y^7 = | d | c | c | d | d | c | d |

Conditional type will be a matrix with X as rows and Y|X as columns:

| Y/X | 'c' | 'd' |
| 'a' | 1/4 | 3/4 |
| 'b' | 2/3 | 1/3 |

This table or matrix is V(b|a) is conditional type of a sequence y^7 given a particular x^7. It is defined for each pair {X, Y} and behaves just like a type 'P' will behave for a sequence x^n in isolation. Now V-shell T_V(x) of Y^n given x^n is counterpart of type class of x^n. SO V-shell is a set of all such y^7 which will have same conditional type V(b|a) given a x^7. One such y^7 for above example would be 'cdddccd'.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.