From Osher Doctorow
Notice that for any set A:
1) P(A-->A) = 1
because P(A-->A) = P(A ' U A) = P(universe) = 1.
Therefore, for an arbitrary sequence of set/events in the probability
space, say A1, A2, A3, ..., An, ..., we have:
2) P(A1-->A1) = P(A2-->A2) = ... = P(An-->An) = ... = 1
Thus the diagonal of a square matrix with A1, A2, A3, ..., as row
labels and A1, A2, A3, ..., as column labels and entries P(Ai --> Aj)
for i, j = 1, 2, ..., n ,... consists only of 1's.
Next we convert to continuous random variables X, Y, Z, etc., as
follows. Instead of X, Y, Z, etc., let's use Z1, Z2, Z3, ..., to
match the set A1, A2, A3, ... numbering, and define:
3) A1(x) = {Z1 < = x}
4) A2(y) = {Z2 < = y}
....
and so on, defined for all real numbers x, y. Then:
5) P(A1(x) --> A2(y)) = P((Z1 < = x) --> (Z2 < = y)) = 1 - FZ1(x) +
F(x, y)
where we understand by F(x, y) the quantity:
6) F(x, y) = P(Z1 < = x, Z2 < = y)
Since we can theoretically be dealing with infinitely many sets of
types A1(x), A1(y), A1(u), etc., and similarly A2(x), A2(y), A2(z),
etc., ..., An(x), An(y), An(z), etc., we could keep track of the
correspond F values by writing:
7) Fij(x, y) = P((Zi < = x) --> (Zj < = y))
However, I'll write F(x, y) when (7) is understood, as before, and
when no confusion is likely.
Under what conditions does the diagonal of a matrix with row and
column labels Z1, Z2, ..., Zn, ..., and entries Fij(x, y) consist only
a row of 1's?
Well, the requirement is that the entry has to be of the form P(A-->A)
= 1 for some set A of the form {Zi < = x}, but notice carefully that
{Zi < = x} must then obey the condition:
8) {Zi < = x} = {Zj < = y} iff Zi = Zj and x = y
in order to be absolutely sure that the sets are identical (otherwise
{Zi < = x} might or might not equal {Zj < = y} and if Zi does not
equal Zj then this certainly won't be true for all (x, y)).
So in forming a matrix with row and column labels Z1, Z2, Z3, ...,
Zn, ..., and with (i, j)th entry P((Zi < = x) --> (Zj < = y)), we
should set y = x throughout the matrix if we want to be certain of
having a diagonal of 1's. The other elements will not necessarily be
1.
Under these conditions, the apparatus previous developed in this
section goes through without modification.
Osher Doctorow
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