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Multivariate Cases

In this subsection, we shall extend the results studied in Section 6.2.1 for three or more random variables. These are given by the following properties:

Property 6.8. We have 

$$^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X,Y\vert Z)\geq \, ^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Z)\,\,$$   or$$\,^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(Y\vert Z),\,\, (\alpha =1,2,3\,\,$$and$$\,\, 4).$$

Property 6.9. We have 

$${\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X_1,X_2,...,X_\delta)......,{^4{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X_i\vert X_1,X_2,...,X_{i-1})}.$$ Property 6.10. We have

(i) $^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y,Z)\left\{\begin{array}{......Y),& r<0\ \mbox{with}\ s<1\ \mbox{or}\ 1<s<2-{1\over r}\end{array}\right.$

(ii) $^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y,Z)\left\{\begin{ar......ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(X\vert Y),& r<0\end{array}\right.$ for $\alpha$=2 and 3;

The equality sign holds in (i) and (ii) iff $X$ and $Y$ are independent given Z, i.e., iff $p(x_i\vert z_k)=p(x_i\vert y_j,z_k)$ or$p(x_i,y_j\vert z_k)=p(x_i\vert z_k)\cdot p(y_j\vert z_k)$, $\forall$ i,j,k.

Property 6.11. We have

$$^1{\ensuremath{\boldsymbol{\mathscr{H}}}}^s_r(Y,Z\vert X)\left\{\......Z\vert X,Y),& {{s-1}\over{r-1}}>1\geqr{{s-1}\over{r-1}}\end{array}\right.$$

Property 6.12. For all $r \geq s \geq2-\frac{1}{r} \geq 1$, we have

(i) ${\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X,Y) \leq {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X) + {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(Y);$

(ii) ${\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X,Y,Z) \leq {\ensuremath{\boldsymb......dsymbol{\mathscr{H}}}}_r^s(Y) + {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(Z);$

(iii) $${\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X_1,X_2,\cdots,x_\de......^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X_i\vert X_1,X_2,\cdots,X_{i-1});$$

(iv) $^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(Y\vert Z) \leq \, ^1{\ensuremath......}}}}_r^s(Y\vert X) +\, ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X\vert Z).$

Property 6.13. If  $$d_r^s(X,Y) = \, ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X\vert Y)+ \, ^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(Y\vert X),$$ then for all $r \geq s \geq2-\frac{1}{r} \geq 1$, we have

(i) $d_r^s(X,Z) \leq d_r^s(X,Y) + d_r^s(Y,Z);$

(ii) $\vert{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X) - {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(Y)\vert \leq {\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X,Y);$

(iii) $\vert\,^1{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s(X_1\vert Y_1) - \, ^1{\en......bol{\mathscr{H}}}}_r^s(X_1\vert Y_2)\vert \leq d_r^s(X_1,X_2) + d_r^s(Y_1,Y_2).$

Note 6.3. It is interesting to verify the properties 6.11, 6.12 and 6.13 for the measures $^\alpha{\ensuremath{\boldsymbol{\mathscr{H}}}}_r^s$($\alpha=2,3$ and $4$) and to find the conditions of their validites.