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It is easy to check that the ( ) is a special case of Theorem 2 with $$r_{n,i} = 1; n=1,2,\ldots ; i = 1,2,\ldots n$$ .

Theorem 3
\begin{aligned} - {{\bar{{\lambda _n}}} ^{ - 1}}\left( {{e^{\bar{{\lambda _n}}} } - 1} \right) \sum \limits _{i = 1}^n {\min \left\{ {\alpha _i ,\frac{{\beta _i - \alpha _i }}{{{w_0} + 1}}} \right\} } \le P\left( {W_{n} \le {w_0}} \right) - \sum \limits _{k = 0}^{{w_0}} {\frac{{{\bar{{\lambda _n}} ^k}{e^{ - {\bar{{\lambda _n}}} }}}}{{k!}}} \le 0, \end{aligned}

With $$\alpha _i = 1 - {p^{r_{n,i}}_{n,i}} - {r_{n,i}}{{q_{n,i}}}{p^{r_{n,i}}_{n,i}}$$ , $$\beta _i = {r_{n,i}}\left( {{p^{ - {r_{n,i}}}_{n,i}} - 1 - {r_{n,i}}{{q_{n,i}}}{p^{r_{n,i}}_{n,i}}} \right) .$$

\begin{aligned} {h_{w_0}}\left( w \right) - \sum \limits _{k =0}^{w_0} {{e^{ - {\bar{{\lambda _n}}} }}\frac{{{\bar{{\lambda _n}} ^k}}}{{k!}}} = {\bar{{\lambda _n}}} f\left( {w + 1} \right) - wf\left( w \right) . \end{aligned}
\begin{aligned}\displaystyle P\left( {W_n \le {w_0}} \right) - \sum \limits _{k = 0}^{{w_0}} {\frac{{{\bar{{\lambda _n}} ^k}{e^{ - {\bar{{\lambda _n}}} }}}}{{k!}}}\nonumber \\\quad = \displaystyle E\left[ {{\bar{{\lambda _n}}} f\left( {W_n + 1} \right) - W_n f\left( W_n \right) } \right] \nonumber \\\quad = \sum \limits _{i = 1}^n {E\left[ {{{r_{n,i}}{q_{n,i}}}f\left( {W_n + 1} \right) - {X_{n,i}}f\left( W_n \right) } \right] }. \end{aligned}
(12)
\begin{aligned}\displaystyle E[{r_{n,i}}{{q_{n,i}}}f(W_{n} + 1) - {X_{n,i}}f(W_{n})]\\\quad =\displaystyle E[E[({r_{n,i}}{{q_{n,i}}}f({W_i} + {X_{n,i}} + 1) - {X_{n,i}}f({W_i} + {X_{n,i}}))|{X_{n,i}}]] \\\quad = \displaystyle E\left[ {{r_{n,i}}{{q_{n,i}}}{p^{r_{n,i}}_{n,i}}f({W_i} + 1)} \right] + \displaystyle E[{r_{n,i}^2}{q^2_{n,i}}{p^{r_{n,i}}_{n,i}}f({W_i} + 2) - {r_{n,i}}{{q_{n,i}}}{p^{r_{n,i}}_{n,i}}f({W_i} + {X_{n,i}})]\\\qquad + \displaystyle \sum \limits _{k \ge 2} {E[{r_{n,i}}C_{{r_{n,i}} + k - 1}^k{q^{k + 1}_{n,i}}{p^{r_{n,i}}_{n,i}}f({W_i} + k + 1) - kC_{{r_{n,i}} + k - 1}^k{q^k_{n,i}}{p^{r_{n,i}}_{n,i}}f({W_i} + k)]} \\\quad = \displaystyle \sum \limits _{k \ge 2} {E[{r_{n,i}}C_{{r_{n,i}} + k - 2}^{k - 1}{q^k_{n,i}}{p^{r_{n,i}}_{n,i}}f({W_i} + k) - kC_{{r_{n,i}} + k - 1}^k{q^k_{n,i}}{p^{r_{n,i}}_{n,i}}f({W_i} + k)]} \\\quad =\displaystyle \sum \limits _{k \ge 2} {E[\frac{{{r_{n,i}}k}}{{{r_{n,i}} + k - 1}}C_{{r_{n,i}} + k - 1}^k{q^k_{n,i}}{p^{r_{n,i}}_{n,i}}f({W_i} + k) - kC_{{r_{n,i}} + k - 1}^k{q^k_{n,i}}{p^{r_{n,i}}_{n,i}}f({W_i} + k)]} \\\quad =\displaystyle \sum \limits _k {\frac{{k\left( {1 - k} \right) }}{{{r_{n,i}} + k - 1}}} C_{{r_{n,i}} + k - 1}^k{q^k_{n,i}}{p^{r_{n,i}}_{n,i}}f({W_i} + k) \\\quad \ge \displaystyle - \sum \limits _k {\frac{{k\left( {k - 1} \right) }}{{{r_{n,i}} + k - 1}}} C_{{r_{n,i}} + k - 1}^k{q^k_{n,i}}{p^{r_{n,i}}_{n,i}}\mathop {\sup }\limits _{w \ge k} f\left( w \right) . \end{aligned}
\begin{aligned}-\displaystyle \sum \limits _k {\frac{{k\left( {k - 1} \right) }}{{{r_{n,i}} + k - 1}}} C_{{r_{n,i}} + k - 1}^k{q^k_{n,i}}{p^{r_{n,i}}_{n,i}}\mathop {\sup }\limits _{w \ge k} f\left( w \right) \nonumber \\\quad \ge - {{\bar{{\lambda _n}}} ^{ - 1}}\left( {{e^{\bar{{\lambda _n}}} } - 1} \right) {p^{r_{n,i}}_{n,i}}\min \left\{ {\sum \limits _k {\frac{{k - 1}}{{{r_{n,i}} + k - 1}}} C_{{r_{n,i}} + k - 1}^k{q^k_{n,i}},} \right. \nonumber \\\qquad \left. { \frac{1}{{{w_0} + 1}}\sum \limits _k {\frac{{k\left( {k - 1} \right) }}{{{r_{n,i}} + k - 1}}} C_{{r_{n,i}} + k - 1}^k{q^k_{n,i}}} \right\} . \end{aligned}
(13)
\begin{aligned} {p^{r_{n,i}}_{n,i}}\sum \limits _{k} {\frac{{k - 1}}{{{r_{n,i}} + k - 1}}} C_{{r_{n,i}} + k - 1}^k{q^k_{n,i}} \le 1 - {p^{r_{n,i}}_{n,i}} - {r_{n,i}}{{q_{n,i}}}{p^{r_{n,i}}_{n,i}} \end{aligned}
(14)
\begin{aligned}\displaystyle {p^{r_{n,i}}_{n,i}}\sum \limits _{k} {\frac{{k\left( {k - 1} \right) }}{{{r_{n,i}} + k - 1}}} C_{{r_{n,i}} + k - 1}^k{q^k_{n,i}}\nonumber \\\quad \le {r_{n,i}}\left( {{p^{ - {r_{n,i}}}_{n,i}} - 1 - {r_{n,i}}{{q_{n,i}}}{p^{r_{n,i}}_{n,i}}} \right) - \left( {1 - {p^{r_{n,i}}_{n,i}} - {r_{n,i}}{{q_{n,i}}}{p^{r_{n,i}}_{n,i}}} \right) . \end{aligned}
(15)
\begin{aligned} - {{\bar{{\lambda _n}}} ^{ - 1}}\left( {{e^{\bar{{\lambda _n}}} } - 1} \right) \sum \limits _{i = 1}^n {\min \left\{ {\alpha _i ,\frac{{\beta _i - \alpha _i }}{{{w_0} + 1}}} \right\} } \le P\left( {W_{n} \le {w_0}} \right) - \sum \limits _{k = 0}^{{w_0}} {\frac{{{\bar{{\lambda _n}} ^k}{e^{ - {\bar{{\lambda _n}}} }}}}{{k!}}} \le 0. \end{aligned}
\begin{aligned} \begin{array}{l} \alpha _i = 1 - {{p_{n,i}}} - {{q_{n,i}}}{{p_{n,i}}} = \displaystyle \left( {1 - {{p_{n,i}}}} \right) \left( {1 - {{p_{n,i}}}} \right) = {q^2_{n,i}}, \\ \beta _i = {p^{-1}_{n,i}} - 1 - {{q_{n,i}}}{{p_{n,i}}} = \displaystyle \frac{{1 - {{p_{n,i}}}}}{{{{p_{n,i}}}}} - \left( {1 - {{p_{n,i}}}} \right) {{p_{n,i}}} = \displaystyle \frac{{\left( {1 - {{p_{n,i}}}} \right) \left( {1 - p_{n,i}^2} \right) }}{{{{p_{n,i}}}}} = {q^2_{n,i}}\frac{{1 + {{p_{n,i}}}}}{{{{p_{n,i}}}}}, \\ \beta _i - \alpha _i = \displaystyle {q^2_{n,i}}\left( {\frac{{1 + {{p_{n,i}}}}}{{{{p_{n,i}}}}} - 1} \right) = \frac{{{q^2_{n,i}}}}{{{{p_{n,i}}}}}. \\ \end{array} \end{aligned}

It is clear that the ( ) is a special case of Theorem 3 with $$r_{n,i} = 1; n=1,2,\ldots ; i = 1,2,\ldots n$$ .

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