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Comment on the work of Zhang et al. (2017, Journal of Inequalities and Applications)
Journal of Inequalities and Applications volumeÂ 2019, ArticleÂ number:Â 186 (2019)
Abstract
In a recent work by Zhang et al. (J. Inequal. Appl. 2017:82, 2017) the authors studied large deviations of the aggregate amount of claims in a sizedependent renewal risk model. They claimed that they obtained a precise largedeviation formula for the case with subexponential claims and with arbitrary dependence between each claim size and its waiting time. In this short note, we point out that their main result is unfortunately not correct in general.
1 Review of the work by Zhang et al. [7]
Large deviations of the aggregate amount of claims are an important topic, which was initiated by KlÃ¼ppelberg 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 sizedependent 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 betweenÂ X and Î¸. Define an integervalued stochastic process
Note that \(N_{t}^{\ast }\) is slightly different from the commonly used renewal counting process
Then the aggregate amount of claims is defined by
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
for all \(n\geq 2\), where \(F^{\ast n}\) denotes the nfold convolution of F. As the authors pointed out, (1.1) implies
where \(X_{1}, X_{2}, \ldots\) are i.i.d. random variables with common distribution function F.
Then the authors stated the following precise largedeviation 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
Here the uniformity is understood as
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.
2 A counterexample
Assume that:

(i)
the generic random pair \((X,\theta )\) contains comonotonic and identical components, that is, \(X=\theta \);

(ii)
the common distribution of X and Î¸ is the Weibull distribution
$$ F(x)=1e^{\sqrt{x}},\quad x\geq 0; $$ 
(iii)
\(x=t\rightarrow \infty \).
Then
Our proof given in the next section shows that we have plenty of room to generalize this counterexample, but we will not do so to save space. Thus Theorem ZWY is fatally wrong. The erroneous step appears on the first two lines of their proof, where the authors claimed that, by the assumption \(F\in \mathcal{S}\) and relation (1.2), to prove (1.3), one needs only to prove
They seem to have overlooked an essential difference between (1.2) and (2.2): The indexÂ n in (1.2) is arbitrarily fixed, whereas the index \(N_{t}^{\ast }\) in (2.2) varies in t and almost surely diverges to âˆž as \(t\rightarrow \infty \).
Nevertheless, in the paper the authors developed a martingale approach to the study of precise large deviations, which is novel to us.
3 Proof of (2.1)
By conditions (i) and (ii), \(\mu =1/\lambda =12\). By condition (ii), F has a long and rapidly varying tail; see Embrechts et al. [3] for these and related terminologies. Furthermore, for a distribution F with a long and rapidly varying tail, it is easy to verify the following:
See Su and Tang [5] for closely related discussions; for example, the first relation in (3.1) can be found in their Theorem 3.1(i). Thus, it remains to prove the first step inÂ (2.1). Keeping in mind condition (iii), we derive
where \(\int _{0}^{x}\) is understood as \(\int _{(0,x)}\), and \(\lambda (y)=E[N_{y}]\) for \(y\geq 0\) is the renewal function. The last step can be verified as follows: for \(y>0\),
Recall Blackwellâ€™s renewal theorem:
see, e.g., page 155 of Asmussen [1]. Thus, for arbitrarily fixed small \(\varepsilon >0\), there is some large \(x_{0}\in \mathbb{N}\) such that, for all \(x\geq x_{0}\),
We continue the derivation:
where \(\lfloor x\rfloor \) is the commonly used floor function. Since F has a long tail,
For the other two terms in (3.3), we derive
where in step (3.4) we applied the upper bound in (3.2), and in step (3.5) we used the long tail property of F again and the two asymptotic relations in (3.1). Plugging these estimates into (3.3) yields
where the last step is due to \(\overline{F} ( 2x ) \leq \overline{F} ( x ) =o ( \int _{x}^{2x}\overline{F}(y)\,dy ) \). A similar asymptotic lower bound can also be established. Finally, by the arbitrariness of Îµ we prove the first step in (2.1).
References
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Chen, Y., Yuen, K.C.: Precise large deviations of aggregate claims in a sizedependent renewal risk model. Insurance Math. Econom. 51(2), 457â€“461 (2012)
Embrechts, P., KlÃ¼ppelberg, C., Mikosch, T.: Modelling Extremal Events: For Insurance and Finance. Springer, Berlin (1997)
KlÃ¼ppelberg, C., Mikosch, T.: Large deviations of heavytailed random sums with applications in insurance and finance. J. Appl. Probab. 34(2), 293â€“308 (1997)
Su, C., Tang, Q.: Characterizations on heavytailed distributions by means of hazard rate. Acta Math. Appl. Sin. 19(1), 135â€“142 (2003)
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Zhang, S., Wang, D., Yu, S.: Precise large deviations of aggregate claims in a sizedependent renewal risk model with stopping time claimnumber process. J. Inequal. Appl. 2017, 82 (2017)
Acknowledgements
The author wishes to thank the two anonymous referees for their very careful reading of the previous version of the paper.
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The research was supported by a Center of Actuarial Excellence (CAE) Research Grant (2018â€“2021) from the Society of Actuaries (SOA) and a Summer Research Grant from the College of Business and Public Administration, Drake University.
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Chen, Y. Comment on the work of Zhang et al. (2017, Journal of Inequalities and Applications). J Inequal Appl 2019, 186 (2019). https://doi.org/10.1186/s1366001921423
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DOI: https://doi.org/10.1186/s1366001921423