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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 $T_{V}(x)$?

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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

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'.

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