. \newline
\textbf{Step 1: }\textit{Enlargement of the probabilistic framework.} The new elements, to be considered together with $\tilde{\theta}$, $\tilde{i}$, $\tilde{\nu}$, are denoted by 
\begin{itemize}
\item $\tilde{b}=(\tilde{b}_n)_{n\geq1}$
\item $U=(U_n)_{n\geq1}$
\item $Z=(Z_n)_{n\geq1}$
\end{itemize} 
and they are defined in combination with a family $\{\ptm:\; t>0,\; m=1,2,\dots\}$ of p.m.'s on $(\Omega,\mathcal{F})$ in such a way they meet the following conditions:
\begin{itemize}
\item $\tilde{\theta}$, $\tilde{i}$, $\tilde{\nu}$, $\tilde{b}$, $U$, $Z$ are mutually independent with respect to each $\ptm$.
\item $\tilde{\theta}$, $\tilde{i}$, $\tilde{\nu}$ preserve, under each $\ptm$, the same distributions as those defined at the beginning of Section \ref{sec:preliminaries}.
\item $b$ is a Bernoulli sequence with parameter $1/K_{1,m}$, with respect to each $\ptm$, where $1/K_{1,m}:=2\theta_m/(\alpha x_m^\alpha)+2F^*_0(x_m)-1$.
\item both the $U$'s and the $Z$'s are, under each $\ptm$, i.i.d. random numbers with common p.d.f.'s $G_{1,m}$ and $G_{2,m}$, respectively, defined by\newline
\[
\begin{split}
&G_{1,m}(x):=\mathbb{I}_{(-\infty,-x_m)}(x)\Big[\dfrac{K_{1,m}\theta_m}{\alpha}\cdot \frac{1}{(-x)^\alpha} \Big]+\mathbb{I}_{[-x_m,x_m)}(x)K_{1,m}\Big[F_0^*(x)\\
&\qquad\qquad\qquad+\dfrac{\theta_m}{\alpha x_m^\alpha}+F^*_0(x_m)-1\Big]+\mathbb{I}_{[x_m,+\infty)}(x)\Big[1-\dfrac{K_{1,m}\theta_m}{\alpha}\cdot \frac{1}{x^\alpha}\Big]
\end{split}
\]
and
\[
\begin{split}
G_{2,m}(x)&:=\mathbb{I}_{(-\infty,-x_m)}(x)K_{2,m}\Big[F_0^*(x)- \dfrac{\theta_m}{\alpha(-x)^\alpha}\Big]+\frac{1}{2}\mathbb{I}_{[-x_m,x_m)}(x)\\
&+\mathbb{I}_{[x_m,+\infty)}(x)K_{2,m}\Big[F_0^*(x)+\dfrac{\theta_m}{\alpha x^\alpha}+1-2F_0^*(x_m)-\frac{2\theta_m}{\alpha x_m^\alpha}\Big].
\end{split}
\]
\end{itemize} 
To verify that they are p.d.f.'s, it is enough to recall, apropos of $G_{2,m}$, condition \eqref{cond2}. Moreover, the random numbers $\tilde{b}_jU_j+(1-\tilde{b}_j)Z_j$ ($j=1,2,\dots$) turn out to be i.i.d. with common p.d.f. $F^*_0$, with respect to each $\ptm$. Hence, the c.f. of the random sum $\sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(\tilde{b}_jU_j+(1-\tilde{b}_j)Z_j)$, under each $\ptm$, is the solution of problem \eqref{eqCauchy} with initial datum $\varphi^*_0$. Thus, we can now aim at verifying the validity of \eqref{megabound} under the additional information that the LHS therein coincides with
\begin{equation}\label{bound}
\mathcal{P}_{t,m(t)}\Big\{-x\leq \sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(\tilde{b}_jU_j+(1-\tilde{b}_j)Z_j)\leq x\Big\}.
\end{equation}
\newline
\textbf{Step 2: }\textit{Analysis of \eqref{bound}.} We begin by verifying the following two inequalities, in which the symbol comma between two conditions stands for their intersection: 
\begin{equation*}
\begin{split}
&\ptm\Big\{-x\leq \sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(\tilde{b}_jU_j+(1-\tilde{b}_j)Z_j)\leq x\Big\}\\
&\leq \ptm\Big\{-x\leq \sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|U_j+ \sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(1-\tilde{b}_j)Z_j-\sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(1-\tilde{b}_j) U_j\leq x,\\ 
&\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Big|\sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(1-\tilde{b}_j) U_j\Big|\leq A_m\Big\}
\end{split}
\end{equation*}
\begin{equation}\label{2termini}
\begin{split}
&+ \ptm\Big\{\Big|\sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(1-\tilde{b}_j) U_j\Big|> A_m\Big\}\\
&\leq \ptm\Big\{\Big|\sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(1-\tilde{b}_j) U_j\Big|> A_m\Big\}\\
&+ \ptm\Big\{-x-A_m\leq \sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|U_j+ \sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(1-\tilde{b}_j)Z_j\leq x+A_m\Big\}\\
& =: \mathcal{L}_1+\mathcal{L}_2.
\end{split}
\end{equation}
Throughout the paper, we preserve the symbols $\theta_m:=c_m x_m^\alpha[(1-F^*_0(x_m))]$, $a_m:=\dfrac{2}{\alpha}K_{1,m}\theta_m\int_0^{+\infty} (\sin x)/x^\alpha dx$, $k_m:=a_m^{1/\alpha}(1-F_0^*(x_m))^{\frac{1-\delta}{\alpha}}$ for $\delta$ in $(0,1)$, $A_m:=\frac{k_m}{(1+2\pi)^2}$, $m\geq1$. \newline
\newline

\textit{Bound for $\mathcal{L}_1$.} We prove that 
\begin{equation}\label{tesidoob}
\mathcal{L}_1\leq \dfrac{1}{\alpha+1}\frac{k_m(a_m+\eta)}{A_m^{\alpha+1}K_{2,m}}+\ptm\{\max_{j=1,\dots,\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}(1-\tilde{b}_j)|>\varepsilon\}
\end{equation}
holds for every $\varepsilon$ not greater than a suitable positive $\varepsilon_m$ and $\eta$ in $(0,a_m)$. Given the $\sigma$-algebra $\mathcal{S}$ generated by $(\tilde{\nu},\tilde{\theta},\tilde{i},\tilde{b})$, we write $\mathcal{L}_1=\mathbb{E}_{t,m}[ \mathcal{A}_{1,\varepsilon}+ \mathcal{A}_{2,\varepsilon}]$ where 
\[
\begin{split}
 \mathcal{A}_{1,\varepsilon}&= \mathcal{A}_{1,\varepsilon}(\omega)\\
&:=\ptm\Big\{\Big|\sum_{j=1}^{\tilde{\nu}}|\tilde{\beta}_{j,\tilde{\nu}}|(1-\tilde{b}_j) U_j\Big|> A_m \Big|\mathcal{S}\Big\}(\omega)\mathbb{I}_{\{\max