Large deviations of the aggregate amount of claims are an important topic, which was initiated by Klüppelberg and Mikosch [4], reactivated by Tang et al. [6], and revisited by many researchers afterward. Due to its potential applications to insurance and finance, this topic is receiving increasing attention from academia.

In a recent work by Zhang et al. [7] the authors studied large deviations of the aggregate amount of claims in a size-dependent renewal risk model. In their notation, let \(\{X_{k},k\in \mathbb{N}\}\) and \(\{\theta _{k},k\in \mathbb{N}\}\) be claim sizes and interarrival times, respectively. Assume that the pairs \((X_{k},\theta _{k})\), \(k\in \mathbb{N}\), form a sequence of independent and identically distributed (i.i.d.) copies of a generic random pair \((X,\theta )\) with marginal distribution functions \(F=1-\overline{F}\) on \([0,\infty )\) and *G* on \([0,\infty )\) and with arbitrary dependence between *X* and *θ*. Define an integer-valued stochastic process

$$ N_{t}^{\ast }=\inf \{ k\in \mathbb{N}:\theta _{1}+ \cdots +\theta _{k} \geq t \} ,\quad t\geq 0. $$

Note that \(N_{t}^{\ast }\) is slightly different from the commonly used renewal counting process

$$ N_{t}=\sup \{ k\in \mathbb{N}:\theta _{1}+\cdots +\theta _{k} \leq t \}. $$

Then the aggregate amount of claims is defined by

$$ S_{t}^{\ast }=\sum_{k=1}^{N_{t}^{\ast }}X_{k},\quad t\geq 0, $$

where the sum is understood as 0 when \(N_{t}^{\ast }=0\).

The authors consider the case of subexponential claims. By definition a distribution function *F* on \([0,\infty )\) is subexponential, denoted by \(F\in \mathcal{S}\), if

$$ \lim_{x\rightarrow \infty }\frac{\overline{F^{\ast n}}(x)}{ \overline{F}(x)}=n $$

(1.1)

for all \(n\geq 2\), where \(F^{\ast n}\) denotes the *n*-fold convolution of *F*. As the authors pointed out, (1.1) implies

$$ \lim_{x\rightarrow \infty }\frac{P ( X_{1}+\cdots +X_{n}>x ) }{P ( \max \{X_{1},\ldots ,X_{n}\}>x ) }=1, $$

(1.2)

where \(X_{1}, X_{2}, \ldots\) are i.i.d. random variables with common distribution function *F*.

Then the authors stated the following precise large-deviation result.

### Theorem ZWY

*Assume that*
\(F\in \mathcal{S}\), \(E[X]=\mu \in (0,\infty )\), *and*
\(E[\theta ]=1/\lambda \in (0,\infty )\). *Then for arbitrarily given*
\(\gamma >0\), *it holds uniformly for all*
\(x\geq \gamma t\)
*that*

$$ P \bigl( S_{t}^{\ast }-\mu \lambda t>x \bigr) \sim \lambda t \overline{F}(x),\quad t\rightarrow \infty. $$

(1.3)

*Here the uniformity is understood as*

$$ \lim_{t\rightarrow \infty }\sup_{x\geq \gamma t} \biggl\vert \frac{P ( S_{t}^{\ast }-\mu \lambda t>x ) }{\lambda t\overline{F}(x)}-1 \biggr\vert =0. $$

This result is claimed to hold for the whole subexponential class \(\mathcal{S}\), and, in particular, it squarely removes a condition on the dependence structure of \((X,\theta )\) originally proposed by Chen and Yuen [2] and recently used by many researchers. Thus this result, if correct, would be an important contribution to the theory of large deviations. Unfortunately, the counterexample given in the following section disproves it.