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Large deviations for randomly weighted sums with dominantly varying tails and widely orthant dependent structure
Journal of Inequalities and Applications volume 2014, Article number: 140 (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; on the other hand, for arbitrarily fixed (c is an arbitrarily given real number), there are positive constant and such that
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
We say that a random variable X or its distribution function is heavy-tailed if for all . An important class of heavy-tailed distributions is , which consists of all distributions with dominated variation in the sense that the relation holds for some (hence for all) . Recall the upper/lower Matuszewska index of distribution F, defined as and , where and for any . From Lemma 3.5 of Tang and Tsitsiashvili , we know that if , then , and for arbitrary and , there exist positive constant and , , such that
holds for all , and
holds for all . Hence, for any , we have
and for any ,
Furthermore, if the distribution has a finite mean, then .
Now, we present some new dependence structures introduced in Wang et al..
Definition 2.1 We say that the r.v.s are widely upper orthant dependent (WUOD) if there exists a finite real sequence satisfying for each and for all , ,
we say that the r.v.s are widely lower orthant dependent (WLOD) if there exists a finite real sequence satisfying for each and for all , ,
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, (, ).
(2) Ifare non-negative and WUOD with dominating coefficients, , then for each,
In particular, ifare WUOD with dominating coefficients, , then for eachand any,
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.
(A1) There exist a real-valued random variable Y with its d.f. , and some positive integer , positive constants and T such that for all ,
holds uniformly for all , and for some .
(A2) There exist a real-valued random variable Z with its d.f. , and positive constants , and T such that for every ,
holds uniformly for all , and for n large enough,
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.
Theorem 2.1 Let the random variablesintroduced in Section 1 be WOD and, for some, , . If the assumptions (A1)-(A4) hold and there exists a positive numbersuch that
then (1.1) and (1.2) hold, respectively.
Theorem 2.2 In addition to the conditions of Theorem 2.1, if one of the following two conditions is satisfied:
when, we have for arbitrarily fixedand some(2.13)
when, we have for all(2.14)
then (1.3) and (1.4) hold, respectively.
Remark 2.2 According to (3.13), (3.15), and (3.16), we can take
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.
Lemma 3.1 Iffor some, then the relation
holds for arbitrarily fixed.
Lemma 3.2 If, , andfor some, and (2.8), (2.9) hold, then there exists positive constantsuch that
holds for any .
Lemma 3.3 Letbe a sequence of real-valued and WUOD r.v.s with’s d.f. andfor every, and letbe a sequence of real numbers satisfying, . If the assumptions (A1)-(A3) hold and there exists a positive numbersuch that
then there exists a constant such that the relation
holds for arbitrarily fixed constant, where.
Proof For arbitrarily fixed , let , and . Using the standard truncation technique, we have
Now, we deal with the second term on the right-hand side of (3.3). Let , then we can obtain according to (2.8) and Lemma 3.1. Write . For a positive number , which we shall specify later, by Chebyshev’s inequality and Proposition 2.1, we have
where is an arbitrarily fixed constant. Take , where . By (2.4) and (2.7),
For , we have
Denote and . By (2.9), we see that there exists such that for and all , . And for any , , for all . Hence,
From the definition of , we know that for every n and all , and , for all . So,
Combining (3.6), (3.7) with (3.5), we obtain
where . For and , we have
where according to (2.4) and (2.7). Plugging (3.8) and (3.9) into (3.4) yields
where in the first step we apply (2.8), in the second step we use Lemma 3.2 and (2.2), and is an appropriately chosen positive number. By the value of h, we have
Using (2.3) and (2.8), we obtain
where . Taking , from (3.11), we have
Applying (2.7), (2.8), and (2.10), we know that there exists such that
Combining (3.12), (3.13) with (3.3), we can obtain (3.2). □
Lemma 3.4 Letbe a sequence of real-valued and WOD r.v.s with’s d.f. andfor every, and letbe a sequence of real numbers satisfying, . If the assumptions (A1)-(A4) hold and there exists a positive numbersuch that
then there exists a constant such that the relation
holds for arbitrarily fixed constant, where.
Proof It is sufficient to prove that
holds for arbitrarily fixed . We write and . Observing that , , are mutually disjoint, we have
where at the third step we used Proposition 2.1.
For , by (2.3), (2.8), and (3.14), we have for arbitrarily fixed
Now we deal with . For fixed , Let and , then
For , by (2.7), (2.9)-(2.11), we have
For , using Chebyshev’s inequality and Proposition 2.1 again, we have
where and . Using similar techniques as in (3.8), we can obtain
where holds uniformly for , as . By (2.9) and (2.11),
Take sufficiently large n such that . Combining (3.21)-(3.23) and observing the monotonicity of for all , we have
where is some positive constant. For fixed , we take such that . By (2.4) and (2.7), we have
Combing (3.17)-(3.20) with (3.25), we can obtain (3.16). This ends the proof of Lemma 3.4. □
Proof of Theorem 2.1 By Lemma 3.3, for arbitrarily fixed , we have uniformly for
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.
Lemma 4.1 Letbe a sequence of real-valued and WUOD r.v.s with’s d.f. andfor every, and Letbe a sequence of non-negative r.v.s satisfying, , and independent of. If (2.8) andhold andfor some positive number β, then for every fixed, there issuch that
holds for large n and all.
Proof Using the techniques similar to Lemma 3.3 with some obvious modifications, we can prove the lemma. □
Lemma 4.2 If a non-negative random processsatisfies, , then (i)-(iv) are mutually equivalent:
for every fixed, ;
for every fixed, ;
for every fixed, .
By Lemma 4.2 and (2.13), we know that
Proof of Theorem 2.2 Now, we prove Theorem 2.2 under condition (I). Using Theorem 2.1 and (2.8), we obtain for any fixed the result that there exists a positive integral number N such that when , for sufficiently large x,
It is clear that for every and all large x,
Hence, by (4.3) and (4.4), there exists some positive number D, for every , and all sufficiently large x,
Take such that . Throughout this proof, we suppose that . Consider the following decomposition:
Firstly, we deal with . For sufficiently large t, by (4.5), we have
For convenience, write . According to , (2.7), (2.10), and (iv) of Lemma 4.2, we have
where is some positive constant.
Secondly, we deal with . On the one hand, by Theorem 2.1, for arbitrary and sufficiently large t,
By (2.8), we have . Using (2.10) and (iii) of Lemma 4.2, we know that there is a positive constant such that
On the other hand, using similar techniques as in (4.8), for any and sufficiently large t, we have
By (2.7), we have
Then, by (4.10) and (4.11), there exists a positive number such that
Finally, we deal with . Taking and splitting into two parts, we obtain
where is understood as 0 in case . For , taking in (4.1) and letting , we have
Hence, according to (ii) of Lemma 4.2, (2.13), and (2.4), we have
For , we have
Combing (4.6), (4.7), (4.9), (4.12), (4.13), and (4.15) with (4.16), we finish the proof under condition (I).
Finally, we prove Theorem 2.2 under condition (II). Without loss of generality, we assume . We still take such that and use the decomposition (4.6). For , we take and divide the interval into two parts, which are and . When , we have
When , we have
Applying (2.14), (4.5), (4.17), and (4.18), we obtain
For , since , then according to Theorem 2.1, (2.10), and Lemma 4.2, we have
For , observing and , by Theorem 2.1, (2.14), and (iii) of Lemma 4.2, we know that, on the one hand, there exists a positive number such that
on the other hand, there exists a positive number such that
Combing (4.19)-(4.22), we finish the proof under condition (II). □
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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).
The authors declare that they have no competing interests.
All authors read and approved the final manuscript.
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Bai, X., Song, L. & Hu, T. Large deviations for randomly weighted sums with dominantly varying tails and widely orthant dependent structure. J Inequal Appl 2014, 140 (2014). https://doi.org/10.1186/1029-242X-2014-140
- large deviations
- randomly weighted sums
- widely orthant dependence
- dominated variation