Let [i.sub.A]: A [right arrow] T and [i.sub.C] : C [right arrow] T be the inclusion maps sending elements of A and B into the direct

summands A and C of T, respectively, i.e., [i.sub.A](a) = a [member of] A [subset] T and [i.sub.C](c) = c [member of] C [subset] T for every a [member of] A and c [member of] C.

The last

summand should be (n - 1)-th because for n-th

summand we obtain a positive growth: for k = n we have [s.sub.n] = s - ae - n+1/2 = - n - [delta] + n + 1/2= -[delta] +1/2 > 0.

Excluding from (37) and (38) the

summand, which contains [s.sub.12], we define

In the first

summand I1 we integrate by parts via identity (19), to get

In particular, e(V) =0 if and only if V contains a trivial direct

summand.

The sign of the first

summand in squared brackets is determined by (p'q - pq') and depends on the relative efficiency of the two prevention opportunities.

[b.sub.m-1][parallel] each

summand of expression in parenthesis for A m+1(A), possibly except the first two, differs from the corresponding

summand for [[DELTA].sub.m+1](B) by the multiplier [c.sup.m+]1.

Let us then look at the first

summand on the last line of (73).

Some derivations employ characteristic functions in a variety of ways, since the characteristic function of a sum of independent random variables is the product of each

summand's characteristic function and the inverse transform is not intractable ([11, pages 188-189], [12-14], [15, pages 362-363], [16, 17]).

The next step in XApEn estimation is to average logarithms of the probability estimates (4) over all template values and to form a

summand [[??].sup.(m)]:

Let us consider the error [delta] [[Y.bar].sup.dis.sub.ph] taking into account that the second

summand in (19) is [delta] [[Y.bar].sup.dis.sub.ph] (21):