the usual stochastic order (see, for example, [@SS07 Sec.6.B]).

\[lem54\] Let $(R_1,R_2,\dots,R_n)$ be distributed according to the probability measure ${\widetilde}{\Pi}_n^{\varrho}(\cdot):=\Pi_n^{\varrho}(\cdot)/\Pi_n^{\varrho}({\mathbb{N}}^n)$.

-   For $\Gamma=\Gamma(M)=(1-e^{-M})^{-2}$ $$(R_1,R_2,\dots,R_n)\le_{\mathrm{st}} (Y_1,Y_2,\dots,Y_n),$$ where $(Y_i)_{1\le i\le n}$ are i.i.d., $Y_i\in{\mathbb{N}}$, and for all $x\in{\mathbb{N}}$ $$\label{y}
          P(Y_1\ge x)=1\wedge\left(
          \Gamma\sum_{r=x}^\infty\frac{{\varrho}\,(1-q)^r}{r}\right).$$

-   For $\gamma=1/\Gamma(M)=(1-e^{-M})^2$ $$(R_1,R_2,\dots,R_n) \ge_{\mathrm{st}} (Z_1,Z_2,\dots,Z_n),$$ where $(Z_i)_{1\le i\le n}$ are i.i.d., $Z_i\in{\mathbb{N}}\cup\{0\}$, and for all $x\in{\mathbb{N}}$ $$\label{z}
          P(Z_1\ge
        x)=\gamma\sum_{r=x}^\infty\frac{{\varrho}\,(1-q)^r}{r}.$$

Since proofs of both parts are very similar, we prove only part (a). By [@SS07 Th. 6.B.B], it is enough to check that there is $\Gamma$ such that for all $m\ge 1$ and $r_1,\dots,r_{m-1}\in{\mathbb{N}}$ $$[R_m\,|\,R_1=r_1,\dots
  R_{m-1}=r_{m-1}]\le_{\mathrm{st}} Y_1,$$ where for $m=1$ we agree to drop the conditioning. By the definition of ${\widetilde}{\Pi}_n^{\varrho}$, (\[co2\]), and (\[defF\]) we have for each $x\in{\mathbb{N}}$ $$\begin{aligned}
    &{\widetilde}{\Pi}_n^{\varrho}(R_m\ge x\,|\,R_1=r_1,\dots
    R_{m-1}=r_{m-1})\\&=\frac{\sum\limits_{r_m\ge
        x}\sum\limits_{r_{m+1},\dots,r_n\in{\mathbb{N}}} \prod\limits_{i=m}^n
      {\varrho}(1-q)^{r_i}F_M(r_{i-1},r_i,u_{i-1})}{\sum\limits_{r_m,\dots,r_n\in{\mathbb{N}}}
      \prod\limits_{i=m}^n
      {\varrho}(1-q)^{r_i}F_M(r_{i-1},r_i,u_{i-1})}\\&\le 1\wedge \left(
      (1-e^{-M})^{-2}\sum\limits_{r_m\ge
        x}{\varrho}(1-q)^{r_m}F_M(r_{i-1},r_i,u_{i-1})\right)\\&\le
    1\wedge \left( (1-e^{-M})^{-2}\sum\limits_{r_m\ge
        x}\frac{{\varrho}(1-q)^{r_m}}{2r_m(e^M-1)}\right)\le 1\wedge
      \left(\Gamma(M)\sum\limits_{r\ge x}\frac{{\varrho}(1-q)^{r}}{r}\right).\qedhere
  \end{aligned}$$

[*Observe that due to Proposition \[propcalc\] $EZ_1\to \infty$ when ${\varrho}\to 0$ or $M\to\infty$.*]{}

The next corollary follows from the last lemma by Cramér’s theorem.

\[corexp\] For $ \theta_0 \in(0,1]$, for each $H> EY_1$ (see (\[y\])), and each $h < EZ_1$ (see (\[z\])) there exist strictly positive $c(H)$ and $c'(h)$ such that for all $n$ large and $\theta\in[\theta_0,1]$ $${\widetilde}{\Pi}_n^{\varrho}\left(\sum_{1\le i\le \theta
    