Testing equality of standardized generalized variances of k multivariate normal populations with arbitrary dimensions

Abstract

For a p-variate normal distribution with covariance matrix ${\varvec{\Sigma }}$, the standardized generalized variance (SGV) is defined as the positive pth root of $|{\varvec{\Sigma }}|$ and used as a measure of variability. Testing equality of the SGVs, for comparing the variability of multivariate normal distributions with different dimensions, is still regarded as matter of interest. The most classical test for this problem is the likelihood ratio test (LRT). In this article, testing equality of the SGVs of k multivariate normal distributions with possibly unequal dimensions is studied. To test this hypothesis, two approximations for the null distribution of the LRT statistic are proposed based on the well known Welch–Satterthwaite and Bartlett adjustment distribution approximation methods. Furthermore, the high-dimensional behavior of these approximated distributions is also investigated. Through a wide simulation study: first, the performance of the proposed tests with the classical LRT is compared in terms of type I error, power, and alpha adjusted equivalents; second, the robustness of the procedures with respect to departures from normality assumption is evaluated. Finally, the proposed methods are illustrated with two real data examples.

Appendix

Here we present the proofs of Lemmas 2.2 and 3.2 as well as Theorems 3.1 and 3.3.

Proof of Lemma 2.2

Let ${\varvec{W}}_{1}^*, {\varvec{W}}_{2}^*,\ldots ,{\varvec{W}}_{k}^*$ be the corresponding values of ${\varvec{W}}_{1}, {\varvec{W}}_{2},\ldots ,{\varvec{W}}_{k}$ for transformed observations, respectively. Simply, it can be shown that ${\varvec{W}}_{i}^*= {\varvec{\Psi }}_i{\varvec{W}}_i {\varvec{\Psi }}_i^\prime$ and since that ${\varvec{\Psi }}_i^\prime {\varvec{\Psi }}_i={\varvec{I}}_{p_i}$, we have ${\root p_i \of {{\left| {{{\varvec{W}}_{i}^*}} \right| }}}={\root p_i \of {{\left| {\varvec{\Psi }}_i{\varvec{W}}_i {\varvec{\Psi }}_i^\prime \right| }}}={\root p_i \of {{\left| {{{\varvec{W}}_{i} }} \right| }}}$. So, by (3), $T_{LRT}\left( {{\varvec{x}}^{*}_1},{{\varvec{x}}^{*}_2},\ldots ,{{\varvec{x}}^{*}_k} \right) = T_{LRT}\left( {{\varvec{x}}_1},{{\varvec{x}}_2},\ldots ,{{\varvec{x}}_k} \right)$. The proof is complete. $\square$

Proof of Lemma 3.2

For any $|b|<\infty$, it can be justified (Jiang et al. 2012, Lemma 2.1) that

$$\begin{aligned} \frac{{\Gamma (x + b)}}{{\Gamma (x)}} = {x^b}{e^{\frac{{b(b - 1)}}{{2x}} + O({x^{ - 2}})}} \end{aligned}$$

(19)

as $x \longrightarrow \infty$. Hence,

$$\begin{aligned} \frac{{\Gamma \left( {\frac{{{n_j-t}}}{2} + h} \right) }}{{\Gamma \left( {\frac{{{n_j-t}}}{2} } \right) }} = {\left( {\frac{{{n_j-t}}}{2}} \right) ^h} e^{\frac{h(h-1)}{n_j-t}}e^{ O(n_j^{-2})} , \end{aligned}$$

as $n_j \longrightarrow \infty , \ j=1,2,\ldots , k$. Consequently,

$$\begin{aligned} \Gamma _j\left( h \right) = \prod \limits _{t = 1}^{{p_j}} {\frac{{\Gamma \left( {\frac{{{n_j} - t}}{2} + h} \right) }}{{\Gamma \left( {\frac{{{n_j} - t}}{2}} \right) }}} = \prod \limits _{t = 1}^{{p_j}} {{\left( {\frac{{{n_j-t}}}{2}} \right) ^h} e^{\frac{h(h-1)}{n_j-t}} } e^{ O(n_j^{-2})}, \end{aligned}$$

as $n_j \longrightarrow \infty , \ j=1,2,\ldots , k$. So, by replacing $\Gamma _j\left( h \right)$ by $\prod \limits _{t = 1}^{{p_j}} {{\left( {\frac{{{n_j-t}}}{2}} \right) ^h} e^{\frac{h(h-1)}{n_j-t}} } e^{ O(n_j^{-2})}$ in (24), we obtain

$$\begin{aligned} E\left[ {{{\left| {{{\varvec{W}}_j}} \right| }^h}} \right] = {\left( {2\root {p_j} \of {{|{{\varvec{\Sigma }}_j}|}}} \right) ^{h{p_j}}} \prod \limits _{t = 1}^{{p_j}} {{\left( {\frac{{{n_j-t}}}{2}} \right) ^h} e^{\frac{h(h-1)}{n_j-t}} } e^{ O(n_j^{-2})}, \end{aligned}$$

as $n_j \longrightarrow \infty , \ j=1,2,\ldots , k$. This implies (17). The proof is complete. $\square$

Proof of Theorem 3.1

Using the multinomial theorem (Gossett 2009, Theorem 5.15), for any given nonnegative integer r,

$$\begin{aligned} Z_i^r&\!=\! {\left( {\sum \limits _{j = 1,j \ne i }^k {\frac{{{p_j}}}{{{p_i}}}\frac{{\root {p_j} \of {{\left| {{{\varvec{W}}_j}} \right| }}}}{{\root {p_i} \of {{\left| {{{\varvec{W}}_i}} \right| }}}}} }\right) ^r} \!\!=\! p_i^{-r } {\left| {{{\varvec{W}}_{i}}} \right| ^{ - \frac{r}{p_i}}}\sum \limits _{\mathop {{r_1},\ldots ,{r_{i - 1}},{r_{i + 1}},\ldots ,{r_k}\ge 0}\limits _{\sum \limits _{j \ne i} {{r_j}} = r} } {\left[ r!{ \prod \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {\frac{{{p_j^{r_j}{\left| {{{{\varvec{W}}_j}}{}} \right| }^{\frac{{{r_j}}}{p_j}}}}}{{{r_j}!}}} } \right] }. \end{aligned}$$

(20)

Also, for $i \ne j \in \left\{ {1,2, \ldots ,k} \right\}$, using again the multinomial theorem,

$$\begin{aligned} Z_i^mZ_j^n&= {\left( {\sum \limits _{t = 1,t \ne i }^k {\frac{{{p_t}}}{{{p_i}}}\frac{{\root {p_t} \of {{\left| {{{\varvec{W}}_t}} \right| }}}}{{\root {p_i} \of {{\left| {{{\varvec{W}}_i}} \right| }}}}} }\right) ^m} {\left( {\sum \limits _{u = 1,u \ne j }^k {\frac{{{p_u}}}{{{p_j}}}\frac{{\root {p_u} \of {{\left| {{{\varvec{W}}_u}} \right| }}}}{{\root {p_j} \of {{\left| {{{\varvec{W}}_j}} \right| }}}}} }\right) ^n} \nonumber \\&= \dfrac{ p_i^{-m } {\left| {{{\varvec{W}}_{i}}} \right| ^{ - \frac{m}{p_i}}}}{p_j^{n } {\left| {{{\varvec{W}}_{j}}} \right| ^{ \frac{n}{p_j}}}} \sum \limits _{\begin{array}{c} \scriptstyle {r_t} \ge 0,t \ne i\\ \scriptstyle \sum \limits _{t \ne i} {{r_t}} = m \end{array}} {\sum \limits _{\begin{array}{c} \scriptstyle {s_u} \ge 0,u\ne j\\ \scriptstyle \sum \limits _{u \ne j} {{s_u}} = n \end{array}} { \left[ \frac{{m!}}{{\prod \limits _{\begin{array}{c} \scriptstyle t = 1\\ \scriptstyle t \ne i \end{array}}^k {{r_t}!} }} { \prod \limits _{\begin{array}{c} \scriptstyle t = 1\\ \scriptstyle t \ne i \end{array}}^k { {{{p_t^{r_t}{\left| {{{{\varvec{W}}_t}}{}} \right| }^{\frac{{{r_t}}}{p_t}}}}}} } \right] \left[ \frac{{n!}}{{\prod \limits _{\begin{array}{c} \scriptstyle u = 1\\ \scriptstyle u \ne j \end{array}}^k {{s_u}!} }} { \prod \limits _{\begin{array}{c} \scriptstyle u = 1\\ \scriptstyle u \ne j \end{array}}^k { {{{p_u^{s_u}{\left| {{{{\varvec{W}}_u}}{}} \right| }^{\frac{{{s_u}}}{p_u}}}}}} } \right] } } \nonumber \\&= \sum \limits _{\begin{array}{c} \scriptstyle {r_t} \ge 0,t \ne i\\ \scriptstyle \sum \limits _{t \ne i} {{r_t}} = m \end{array}} {\sum \limits _{\begin{array}{c} \scriptstyle {s_u} \ge 0,u\ne j\\ \scriptstyle \sum \limits _{u \ne j} {{s_u}} = n \end{array}} { \left[ \frac{{n!m!}}{{\prod \limits _{\begin{array}{c} \scriptstyle u = 1\\ \scriptstyle u \ne j \end{array}}^k {{s_u}!} }{\prod \limits _{\begin{array}{c} \scriptstyle t = 1\\ \scriptstyle t \ne i \end{array}}^k {{r_t}!} }} \right] } \left[ { \frac{ {{{p_i^{s_i-m}{\left| {{{{\varvec{W}}_i}}{}} \right| }^{\frac{{{s_i-m}}}{p_i}}}}}}{ {{{p_j^{n-r_j}{\left| {{{{\varvec{W}}_j}}{}} \right| }^{\frac{{{n-r_j}}}{p_j}}}}}} { \prod \limits _{\begin{array}{c} \scriptstyle u = 1\\ \scriptstyle u \ne {i,j} \end{array}}^k { {{{ \frac{ {\left| {{{{\varvec{W}}_u}}{}} \right| }^{\frac{{{s_u+r_u}}}{p_u}}}{p_u^{-(s_u+r_u)}}}}}} } } \right] }. \end{aligned}$$

(21)

Now, taking the expectation of (20) and (21), and then using the independence of ${{\varvec{W}}_{j}}$s, immediately yields

$$\begin{aligned} {{\,\mathrm{E}\,}}\left[ {Z_i^r} \right] = p_i^{-r } E\left[ {\left| {{{\varvec{W}}_{i}}} \right| ^{ - \frac{r}{p_i}}}\right] \sum \limits _{\mathop {{r_1},\ldots ,{r_{i - 1}},{r_{i + 1}},\ldots ,{r_k}\ge 0}\limits _{\sum \limits _{j \ne i} {{r_j}} = r} } {\left[ r!{ \prod \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {\frac{{{p_j^{r_j}E[ {\left| {{{{\varvec{W}}_j}}{}} \right| }^{\frac{{{r_j}}}{p_j}} ] }}}{{{r_j}!}}} } \right] } \end{aligned}$$

(22)

and

$$\begin{aligned} E\left[ Z_i^mZ_j^n\right]&= \sum \limits _{\begin{array}{c} \scriptstyle {r_t} \ge 0,t \ne i\\ \scriptstyle \sum \limits _{t \ne i} {{r_t}} = m \end{array}} \sum \limits _{\begin{array}{c} \scriptstyle {s_u} \ge 0,u\ne j\\ \scriptstyle \sum \limits _{u \ne j} {{s_u}} = n \end{array}} { \left[ \frac{{n!m!}}{{\prod \limits _{\begin{array}{c} \scriptstyle u = 1\\ \scriptstyle u \ne j \end{array}}^k {{s_u}!} }{\prod \limits _{\begin{array}{c} \scriptstyle t = 1\\ \scriptstyle t \ne i \end{array}}^k {{r_t}!} }} \right] } \nonumber \\&\quad \left[ { \frac{ {{{p_i^{s_i-m} E\left[ {\left| {{{{\varvec{W}}_i}}{}} \right| }^{\frac{{{s_i-m}}}{p_i}}\right] }}}}{ {{{p_j^{n-r_j} E^{-1}\left[ {\left| {{{{\varvec{W}}_j}}{}} \right| }^{\frac{{{r_j-n}}}{p_j}}\right] }}}} { \prod \limits _{\begin{array}{c} \scriptstyle u = 1\\ \scriptstyle u \ne {i,j} \end{array}}^k { {{{ \frac{ E\left[ {\left| {{{{\varvec{W}}_u}}{}} \right| }^{\frac{{{s_u+r_u}}}{p_u}}\right] }{p_u^{-(s_u+r_u)}}}}}} } } \right] . \end{aligned}$$

(23)

But, for any $j = 1,2,\ldots , k$, it is well known that $\frac{ \left| {{{\varvec{W}}_{j}}} \right| }{\left| {{{\varvec{\Sigma }}_j}} \right| }$ is distributed as a product of the chi-square distributions (see, Muirhead 2009), that is,

$$\begin{aligned} \dfrac{ \left| {{{\varvec{W}}_{j}}} \right| }{\left| {{{\varvec{\Sigma }}_j}} \right| }\sim \chi _{{n_j} - 1}^2\chi _{{n_j} - 2}^2\ldots \chi ^2_{{n_j} - p_{j}}, \ j = 1,2,{\ldots }, k, \end{aligned}$$

where $\chi _{{n_j} - r}^2$, $r = 1,2,\ldots , p_{j}$ are independently chi-square random variables with ${{n_j} - r}$ degrees of freedom. So, for any real number h and ${n_j} > \max (p_j ,p_j-2h)$, $j=1,2,\ldots ,k$, we have:

$$\begin{aligned} {{\,\mathrm{E}\,}}\left[ {{{\left| {{{\varvec{W}}_{j}}} \right| }^h}} \right] = { \left( 2\root p_j \of {|{\varvec{\Sigma }}_j|}\right) }^{h{p_{j}}} \prod \limits _{t= 1}^{p_{j}} {\frac{{ {\Gamma \left[ {\frac{{{n_j} -t}}{2} + h} \right] } }}{{ {\Gamma \left[ {\frac{{{n_j} - t}}{2} } \right] } }}}= { \left( 2\root p_j \of {|{\varvec{\Sigma }}_j|}\right) }^{h{p_{j}}} \Gamma _j(h). \end{aligned}$$

(24)

Replacing the mathematical expectations in the equations (22) and (23) by their respective values calculated from (24) under the ${H_0}$ in (1), yield the desired equations (15) and (16). The proof is complete. $\square$

Proof of Theorem 3.3

From (20) with $r=1$, we have

$$\begin{aligned} E{[Z_i]}&=\! p_i^{ - 1}{\Gamma _i}\left( { - \frac{1}{{{p_i}}}} \right) \sum \limits _{\begin{array}{c} \scriptstyle {r_j} \ge 0,j \ne i\\ \scriptstyle \sum \limits _{j \ne i} {{r_j} = 1} \end{array}} {\prod \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {p_j^{{r_j}}{\Gamma _j}\left( {\frac{{{r_j}}}{{{p_j}}}} \right) } } \!=\! p_i^{ - 1}{\Gamma _i}\left( { - \frac{1}{{{p_i}}}} \right) \sum \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {{p_j}{\Gamma _j}\left( {\frac{1}{{{p_j}}}} \right) } \nonumber \\&= \sum \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {\frac{{{p_j}}}{{{p_i}}}{\Gamma _j}\left( {\frac{1}{{{p_j}}}} \right) } {\Gamma _i}\left( { - \frac{1}{{{p_i}}}} \right) . \end{aligned}$$

(25)

Since that $p_j \longrightarrow \infty$ and $n_{j}>p_{j} +1$, we have $n_j \longrightarrow \infty , \ j=1,2,\ldots , k$ and by (19),

$$\begin{aligned} \ln \frac{{\Gamma \left( {\frac{{{n_j}}}{2} + h - \frac{t}{2}} \right) }}{{\Gamma \left( {\frac{{{n_j}}}{2} - \frac{t}{2}} \right) }}&= \ln \frac{{\Gamma \left( {\frac{{{n_j}}}{2} + h - \frac{t}{2}} \right) }}{{\Gamma \left( {\frac{{{n_j}}}{2}} \right) }} - \ln \frac{{\Gamma \left( {\frac{{{n_j}}}{2} - \frac{t}{2}} \right) }}{{\Gamma \left( {\frac{{{n_j}}}{2}} \right) }} \nonumber \\&= \left( {h - \frac{t}{2}} \right) \ln \left( {\frac{{{n_j}}}{2}} \right) + \frac{t}{2}\ln \left( {\frac{{{n_j}}}{2}} \right) + o(1) \nonumber \\&= \ln {\left( {\frac{{{n_j}}}{2}} \right) ^h} + o(1). \end{aligned}$$

So, we have

$$\begin{aligned} {\Gamma _j}\left( {\frac{1}{{{p_j}}}} \right)= & {} \prod \limits _{t = 1}^{{p_j}} {{{\left( {\frac{{{n_j}}}{2}} \right) }^{\frac{1}{{{p_j}}}} }{e^{o(1)}}} = {\left( {\frac{{{n_j}}}{2}} \right) }{e^{o(1)}} \ \ \text{ and } \\ {\Gamma _i}\left( -{\frac{1}{{{p_i}}}} \right)= & {} \prod \limits _{t = 1}^{{p_i}} {{{\left( {\frac{{{n_i}}}{2}} \right) }^{-\frac{1}{{{p_i}}}} }{e^{o(1)}}} = {\left( {\frac{{{n_i}}}{2}} \right) ^{-1}}{e^{o(1)}}. \end{aligned}$$

Consequently, for $i \ne j \in \left\{ {1,2, \ldots ,k} \right\}$,

$$\begin{aligned} {\Gamma _j}\left( {\frac{1}{{{p_j}}}} \right) {\Gamma _i}\left( { - \frac{1}{{{p_i}}}} \right) = \left( \frac{{{n_j}}}{{{n_i}}} \right) {e^{o(1)}}, \end{aligned}$$

as $p_j \longrightarrow \infty$, $j=1,2,\ldots ,k$. Now, by the assumptions of the theorem,

$$\begin{aligned} {\Gamma _j}\left( {\frac{1}{{{p_j}}}} \right) {\Gamma _i}\left( { - \frac{1}{{{p_i}}}} \right) \longrightarrow 1, \end{aligned}$$

as $p_j \longrightarrow \infty$, $j=1,2,\ldots ,k$. By taking the limit on both sides of (25) as $p_j \longrightarrow \infty$, $j=1,2,\ldots ,k$ and using the assumptions of the theorem, we get

$$\begin{aligned} \mathop {\lim }\limits _{{p_1},{p_2}, \ldots ,{p_k} \rightarrow \infty } E{[Z_i]} = \mathop {\lim }\limits _{{p_1},{p_2}, \ldots ,{p_k} \rightarrow \infty } \sum \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {\frac{{{p_j}}}{{{p_i}}}{\Gamma _j}\left( {\frac{1}{{{p_j}}}} \right) } {\Gamma _i}\left( { - \frac{1}{{{p_i}}}} \right) = k - 1.\nonumber \\ \end{aligned}$$

(26)

Setting $r=2$ in (20), we obtain

$$\begin{aligned} E[Z_i^2]&= p_i^{ - 2}{\Gamma _i}\left( { - \frac{2}{{{p_i}}}} \right) \sum \limits _{\begin{array}{c} \scriptstyle {r_j} \ge 0,j \ne i\\ \scriptstyle \sum \limits _{j \ne i} {{r_j} = 2} \end{array}} {\left[ {2!\prod \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {\frac{{p_j^{{r_j}}}}{{{r_j}!}}{\Gamma _j}\left( {\frac{{{r_j}}}{{{p_j}}}} \right) } } \right] } \nonumber \\&= p_i^{ - 2}{\Gamma _i}\left( { - \frac{2}{{{p_i}}}} \right) \left[ {\sum \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {p_j^2{\Gamma _j}\left( {\frac{2}{{{p_j}}}} \right) } + 2\sum \limits _{\begin{array}{c} \scriptstyle r = 1\\ \scriptstyle r \ne i \end{array}}^{k - 1} {\sum \limits _{\begin{array}{c} \scriptstyle s = r + 1\\ \scriptstyle s \ne i \end{array}}^k {{p_r}{\Gamma _r}\left( {\frac{1}{{{p_r}}}} \right) {p_s}{\Gamma _s}\left( {\frac{1}{{{p_s}}}} \right) } } } \right] \nonumber \\&= \sum \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {{{\left( {\frac{{{p_j}}}{{{p_i}}}} \right) }^2}{\Gamma _j}\left( {\frac{2}{{{p_j}}}} \right) } {\Gamma _i}\left( { - \frac{2}{{{p_i}}}} \right) \\&\quad + 2\sum \limits _{\begin{array}{c} \scriptstyle r = 1\\ \scriptstyle r \ne i \end{array}}^{k - 1} {\sum \limits _{\begin{array}{c} \scriptstyle s = r + 1\\ \scriptstyle s \ne i \end{array}}^k {\left( {\frac{{{p_r}}}{{{p_i}}}} \right) \left( {\frac{{{p_s}}}{{{p_i}}}} \right) {\Gamma _r}\left( {\frac{1}{{{p_r}}}} \right) {\Gamma _s}\left( {\frac{1}{{{p_s}}}} \right) {\Gamma _i}\left( { - \frac{2}{{{p_i}}}} \right) } }. \end{aligned}$$

Similar to the way that we used in the proof of (26), it can be justified that

$$\begin{aligned} {{\Gamma _j}\left( {\frac{2}{{{p_j}}}} \right) {\Gamma _i}\left( { - \frac{2}{{{p_i}}}} \right) \rightarrow 1}\ \text{ and } \ {{\Gamma _r}\left( {\frac{1}{{{p_r}}}} \right) {\Gamma _s}\left( {\frac{1}{{{p_s}}}} \right) {\Gamma _i}\left( { - \frac{2}{{{p_i}}}} \right) \rightarrow 1}, \end{aligned}$$

as $p_j \longrightarrow \infty$, $j=1,2,\ldots ,k$. Therefore,

$$\begin{aligned}&\mathop {\lim }\limits _{{p_1},{p_2}, \ldots ,{p_k} \rightarrow \infty } E[Z_i^2] \nonumber \\&\quad = \mathop {\lim }\limits _{{p_1},{p_2}, \ldots ,{p_k} \rightarrow \infty } \sum \limits _{\begin{array}{c} \scriptstyle j = 1\\ \scriptstyle j \ne i \end{array}}^k {{{\left( {\frac{{{p_j}}}{{{p_i}}}} \right) }^2}{\Gamma _j}\left( {\frac{2}{{{p_j}}}} \right) } {\Gamma _i}\left( { - \frac{2}{{{p_i}}}} \right) \nonumber \\&\qquad + \mathop {\lim }\limits _{{p_1},{p_2}, \ldots ,{p_k} \rightarrow \infty } 2\sum \limits _{\begin{array}{c} \scriptstyle r = 1\\ \scriptstyle r \ne i \end{array}}^{k - 1} {\sum \limits _{\begin{array}{c} \scriptstyle s = r + 1\\ \scriptstyle s \ne i \end{array}}^k {\left( {\frac{{{p_r}}}{{{p_i}}}} \right) \left( {\frac{{{p_s}}}{{{p_i}}}} \right) {\Gamma _r}\left( {\frac{1}{{{p_r}}}} \right) {\Gamma _s}\left( {\frac{1}{{{p_s}}}} \right) {\Gamma _i}\left( { - \frac{2}{{{p_i}}}} \right) } } \nonumber \\&\quad = (k - 1) + 2\frac{{(k - 2)(k - 1)}}{2} = {(k - 1)^2}. \end{aligned}$$

(27)

Hence, by (26) and (27),

$$\begin{aligned} \mathop {\lim }\limits _{{p_1},{p_2}, \ldots ,{p_k} \rightarrow \infty } Var({Z_i}) = \mathop {\lim }\limits _{{p_1},{p_2}, \ldots ,{p_k} \rightarrow \infty } \left( {E[Z_i^2] - {E^2}[{Z_i}]} \right) = 0. \end{aligned}$$

(28)

So, $E{[Z_i]}\longrightarrow k-1$ and $Var({Z_i}) \longrightarrow 0$, as $p_j \longrightarrow \infty$, $j=1,2,\ldots ,k$. In other words, $Z_i$ converge in probability to $E[Z_i]$, $i=1,2,\ldots ,k$ (Billingsley 2008). The proof is complete. $\square$