Large deviations for randomly weighted sums with dominantly varying tails and widely orthant dependent structure
© Bai et al.; licensee Springer. 2014
Received: 11 May 2013
Accepted: 25 March 2014
Published: 4 April 2014
We prove large deviation inequalities for the randomly weighted partial and random sums , ; , , where is a counting process, is a sequence of positive random variables with two-sided bounds, and is a sequence of non-identically distributed real-valued random variables, while the three random sources above are mutually independent. Special attention is paid to the distribution of dominated variation and the widely orthant dependence structure.
MSC:62E20, 62H20, 62P05.
Let be a sequence of real-valued random variables (r.v.s) with ’s distribution function (d.f.) and for every , and be another sequence of positive random variables, satisfying , , where . denotes a counting process (that is, a non-negative, non-decreasing, and integer-valued stochastic process) with a finite mean function for and as . Besides, the three random sources above are mutually independent. Denote , and , , . By convention, the summation over an empty set of indices produces a value of 0. In the present paper, we are interested in the probabilities of large deviations of and in the situation that are heavy-tailed and widely orthant dependent.
Since the theory of large deviations with heavy tails is widely used in insurance and finance, in recent decades, there have been a series of articles devoted to related problems. For more details, please refer to Embrechts et al., Klüppelberg and Mikosch , Mikosch and Nagaev  and references therein. Recently Tang  extended the asymptotic behavior of large deviation probabilities of partial sums of heavy-tailed random variables to the case of negatively dependent ones. Under the assumption that random variables are non-identically distributed and extended negatively dependent, Liu  obtained a result similar to the one in the above paper, which was promoted to random sums in various situations later by Chen et al.. Specially, Shen and Lin  investigated large deviations of randomly weighted partial sums with negatively dependent and consistently varying-tailed random variables, but, unfortunately, there are some flaws in their proofs.
hold uniformly for all as , respectively.
The paper is organized as follows. Section 2 presents our main results after recalling some preliminaries. Sections 3 and 4 prove Theorems 2.1 and 2.2, respectively.
2 Main results
Furthermore, if the distribution has a finite mean, then .
Now, we present some new dependence structures introduced in Wang et al..
if they are both WUOD and WLOD, then we say that the r.v.s are widely orthant dependent (WOD). WUOD, WLOD, and WOD r.v.s are called, by a joint name, wide dependence (WD) r.v.s, and , , , are called dominating coefficients.
Wang et al. also gave some examples of WD r.v.s with various dominating coefficients which show that WD r.v.s contain some common negatively dependent r.v.s, some positively dependent r.v.s and some others.
From the definitions of WD, the following proposition can be obtained directly (see, e.g., Wang et al.).
Proposition 2.1 (1) Letbe WUOD (WLOD) with dominating coefficients, (, ). Ifare non-decreasing, thenare still WUOD (WLOD) with dominating coefficients, (, ); ifare non-increasing, thenare WLOD (WUOD) with dominating coefficients, (, ).
For convenience, we introduce some notation. For two positive functions and , we write if . For two positive bivariate functions and , we say that the asymptotic relation or holds uniformly over all x in a nonempty set Δ, if or . For a real number x, we write and .
Before we state our main results, we will introduce some basic assumptions, to be used in this paper.
holds uniformly for all , and for some .
holds uniformly for all ; and , .
Remark 2.1 According to (2.7), (2.8), and (2.10), we can see that the r.v. Z’s and Y’s right tails are weak equivalent, i.e., . The assumption (A4), which is equivalent to , shows the r.v. Z’s left tails are lighter than the r.v. Y’s right tails. It is clear that all assumptions (A1)-(A4) are easily satisfied.
The main results of this paper are given below.
then (1.1) and (1.2) hold, respectively.
- (I)when, we have for arbitrarily fixedand some(2.13)
- (II)when, we have for all(2.14)
then (1.3) and (1.4) hold, respectively.
in Theorem 2.1, respectively, where a, b, u, ν, , and are some fixed positive constants. For the given distribution functions and , we can obtain the sharp lower and upper bound , . Hence, though the above expressive forms are not nice-looking, causes no trouble for real applications. For and , by the proof of Theorem 2.2, we can make a similar remark.
3 Proof of Theorem 2.1
We start with a series of lemmas based on which Theorem 2.1 will be proved. The proofs of the following Lemmas 3.1 and 3.2 are straightforward and are therefore omitted.
holds for arbitrarily fixed.
holds for any .
holds for arbitrarily fixed constant, where.
Combining (3.12), (3.13) with (3.3), we can obtain (3.2). □
holds for arbitrarily fixed constant, where.
where at the third step we used Proposition 2.1.
Combing (3.17)-(3.20) with (3.25), we can obtain (3.16). This ends the proof of Lemma 3.4. □
According to Lemma 3.4 and using a similar method of proof as in (3.26), we can obtain the remainder of Theorem 2.1. □
4 Proof of Theorem 2.2
For proving Theorems 2.2, we first give two lemmas.
holds for large n and all.
Proof Using the techniques similar to Lemma 3.3 with some obvious modifications, we can prove the lemma. □
for every fixed, ;
for every fixed, ;
for every fixed, .
where is some positive constant.
Combing (4.6), (4.7), (4.9), (4.12), (4.13), and (4.15) with (4.16), we finish the proof under condition (I).
Combing (4.19)-(4.22), we finish the proof under condition (II). □
The authors would like to thank the two referees for careful reading of our manuscript and for helpful and valuable comments and suggestions, which helped us improve the earlier version of the paper. The research of the authors was supported by the Natural Science Foundation of the Inner Mongolia Autonomous Region (No. 2013MS0101), the National Natural Sciences Foundation of China (No. 11201317), and the Beijing Municipal Education Commission Foundation (No. KM201210028005).
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