g X1X3 (Fig. H1), with the argument I2 changed into I. Consider next the case of symbol strings with lengths up to N = 3, which correspond to all the intervals that have been highlighted in Fig. H2. The strings with lengths N = 3, namely X1X1X3, X1X2X3, X2X1X3, and X2X2X3, are associated with the subintervals I113, I123, I213, and I223. We do not need to consider all of them to prove the general relation that defines the start and stop values of each of these subintervals. Take, for instance, I223. Looking at Fig. H2, we readily obtain: u(I223) = p(x1) + p(x1/x2)p(x2) + p(x1/x2, x2) p(x2,x2 + p(x2/x2, X2) p(x2,x2) = p(x1) + p(x1,x2) + p(x2,x2) [p(x1/x2, x2) + p(x2/x2, x2)] = u(I22) v(I223) = p(x1) + p(x1/x2)p(x2) + p(x1/x2,x2) p(x2,x2 + p(x2/x2,x2), p(x2,x2 = p(x1) + p(x1,x2) + p(x2, X2) [[p(x1/x2,x2) + p(x2/x2, x2) + p(x3/x2, x2)] (H11) In this derivation, we used the property p(x2, x2) = w(I22) = v(I22) - u(I22), according to which the width of the subinterval corresponding to a preceding string sequence XiXj is equal to the joint probability p(xi, xj) = p(xj|xi)p(xi). We also used the fact that, by definition, u(I22) = p(x1) + p(x1,x2). Appendix H The previous analysis makes it possible to rewrite Eq. (H11) concerning any three- symbol string of the type XmXnX3 (m, n = 1 ..2) in the general form: (H12) In turn, Eq. (H12) suggests a general recurrence formula defining the interval I (UN, UN) corresponding to any string aja2 ...ak an of length N made from a source X = {x1,x2, ....,Xj,...,Xn} of size n, with the string being terminated by symbol Xn. The start and stop values of the corresponding interval are computed by successive iterations from k = 2 to k = N according to the following recurrence relations: Uk = Uk-1+Wk-1Q(X,a1,a2,...,ak-1) Uk-1 + Wk-1R(X,a1,a2,...,ak-1) (H13) Wk = Vk Uk, (H14) and with the initial conditions EXPRESS (H15) note that R(X, al.az,...,Ak-1) = 1, by application of the property EXEX p(x/y) = 1 for any event y (see Chapter 5), therefore, the sum involved in the definition of R does no