Self-information and surprisal variance for a finite law

Let $A$ be a nonempty finite alphabet identified with $[m]$, and let $p$ be a probability vector as in Finite distributions, entropy, divergence, and stochastic matrices. Its support is $A_{+}:=\{a\in A:p(a)>0\}$. On $A_{+}$ define self-information ${\mathrm{ı}}_p(a):=-\ln p(a)$; set ${\mathrm{ı}}_p(a)=0$ when $p(a)=0$ only to make a finite-valued function on all of $A$. Using the finite entropy $H(p)$ from Finite distributions, entropy, divergence, and stochastic matrices and the natural-logarithm/nats convention from Information units and logarithm-base convention, define the surprisal variance $V(p):={\sum }_{a\in A_{+}}p(a)({\mathrm{ı}}_p(a)-H(p){)}^2\mathrm{.}$ For $n\ge 1$ and $x^n=(x_1,\dots ,x_n)\in A^n$, define the product law $p^n(x^n):={\prod }_{i=1}^{n}p(x_i)$. On its support $A_{+}^n$, define block self-information ${\mathrm{ı}}_{p^n}(x^n):=-\ln p^n(x^n)$. Logarithms use the natural base, matching the entropy convention in Finite distributions, entropy, divergence, and stochastic matrices.