## Approximation of the tail probability of randomly weighted sums and applications.(English)Zbl 1271.62030

Summary: Consider the problem of approximating the tail probability of randomly weighted sums $$\sum^n_{i=1}\Theta_iX_i$$ and their maxima, where $$\{X_i,i\geq 1\}$$ is a sequence of identically distributed but not necessarily independent random variables from the extended regular variation class, and $$\{\Theta_i,\;i\geq 1\}$$ is a sequence of nonnegative random variables, independent of $$\{X_i,\;i\geq 1\}$$ and satisfying certain moment conditions. Under the assumption that $$\{X_i,\;i\geq 1\}$$ has no bivariate upper tail dependence along with some other mild conditions, this paper establishes the following asymptotic relations: $\text{Pr} \left(\max_{1\leq k\leq n}\sum^k_{i=1}\Theta_iX_i>x\right)\sim \text{Pr} \left (\sum^n_{i=1}\Theta_iX_i>x\right)\sim\sum^n_{i=1}\text{Pr} (\Theta_iX_i>x),$ and $\text{Pr}\left(\max_{1\leq k\leq \infty} \sum^k_{i=1}\Theta_i X_i>x\right)\sim \text{Pr}\left( \sum^\infty_{i=1}\Theta_iX_i^+>x\right)\sim\sum^\infty_{i=1}\text{Pr}(\Theta_iX_i>x),$ as $$x\to\infty$$. In doing so, no assumption is made on the dependence structure of the sequence $$\{\Theta_i,i\geq 1\}$$.

### MSC:

 6.2e+21 Asymptotic distribution theory in statistics
Full Text:

### References:

  Alam, K.; Saxena, K.M.L., Positive dependence in multivariate distributions, Communications in statistics — theory and methods, 10, 1183-1196, (1981) · Zbl 0471.62045  H. Albrecher, Dependent risks and ruin probabilities in insurance, IIASA Interim Report, IR-98-072, 1998  Asmussen, S., Ruin probabilities, (2000), World Scientific Singapore  Bingham, N.H.; Goldie, C.M.; Teugels, J.L., Regular variation, (1987), Cambridge University Press Cambridge · Zbl 0617.26001  Block, H.W.; Savits, T.H.; Shaked, M., Some concepts of negative dependence, Annals of probability, 10, 765-772, (1982) · Zbl 0501.62037  Chen, Y.; Ng, K.W.; Xie, X., On the maximum of rnadomly weighted sums with regularly varying tails, Statistics and probability letters, 76, 971-975, (2006) · Zbl 1090.62046  Chen, Y.; Su, C., Finite time ruin probability with heavy-tailed insurance and financial risks, Statistics and probability letters, 76, 1812-1820, (2006) · Zbl 1171.91348  Cline, D.B.H.; Samorodnitsky, G., Subexponentiality of the product of independent random variables, Stochastic processes and their applications, 49, 75-98, (1994) · Zbl 0799.60015  Cossette, H.; Marceau, É., The discrete-time risk model with correlated classes of business, Insurance: mathematics and economics, 26, 133-149, (2000) · Zbl 1103.91358  Davis, R.A.; Resnick, S.I., Limit theory for bilinear processes with heavy- tailed noise, Annals of applied probability, 6, 1191-1210, (1996) · Zbl 0879.60053  Ebrahimi, N.; Ghosh, M., Multivariate negative dependence, Communications in statistics — theory and methods, 10, 307-337, (1981) · Zbl 0506.62034  Embrechts, P.; Kluppelberg, C.; Mikosch, T., Modelling extremal events for insurance and finance, (1997), Springer Berlin · Zbl 0873.62116  Geluk, J.L.; De Vries, C.G., Weighted sums of subexponential random variables and asymptotic dependence between returns on reinsurance equities, Insurance: mathematics and economics, 38, 39-56, (2006) · Zbl 1112.62011  Goovaerts, M.J.; Kaas, R.; Laeven, R.J.A.; Tang, Q.; Vernic, R., The tail probability of discounted sums of Pareto-like losses in insurance, Scandinavian actuarial journal, 446-461, (2005) · Zbl 1144.91026  Grandell, J., Aspects of risk theory, (1991), Springer New York · Zbl 0717.62100  Joag-Dev, K.; Proschan, F., Negative association of random variables, with applications, Annals of statistics, 11, 286-295, (1983) · Zbl 0508.62041  Joe, H., Multivariate models and dependence concepts, (1997), Chapman & Hall London · Zbl 0990.62517  Klugman, S.; Panjer, H.H.; Willmot, G.E., Loss models: from data to decisions, (2004), John Wiley & Sons New York, NY · Zbl 1141.62343  Nelsen, R.B., ()  Nyrhinen, H., On the ruin probabilities in a general economic environment, Stochastic processes and their applications, 83, 319-330, (1999) · Zbl 0997.60041  Nyrhinen, H., Finite and infinite time ruin probabilities in a stochastic economic environment, Stochastic processes and their applications, 92, 265-285, (2001) · Zbl 1047.60040  Panjer, H.H., Operational risk, (2006), John Wiley & Sons New York · Zbl 1258.62101  Resnick, S.I.; Willehens, E., Moving averages with random coefficients and random coefficient autoregressive models, Communications in statistics — stochastic models, 7, 511-525, (1991) · Zbl 0747.60062  Tang, Q., Asymptotic ruin probabilities of the renewal model with constant interest force and regular variation, Scandinivaian actuarial journal, 1, 1-5, (2005) · Zbl 1144.91030  Tang, Q.; Tsitsiashvili, G., Precise estimates for the ruin probability in finite horizon in a discrete-time model with heavy-tailed insurance and financial risks, Stochastic processes and their applications, 108, 299-325, (2003) · Zbl 1075.91563  Tang, Q.; Tsitsiashvili, G., Finite and infinite time ruin probabilities in the presence of stochastic return on investments, Advances in applied probability, 36, 1278-1299, (2004) · Zbl 1095.91040  Vervaat, W., On a stochastic difference equation and a representation of non-negative infinitely divisible random variables, Advances in applied probability, 11, 750-783, (1979) · Zbl 0417.60073  Wang, D.; Tang, Q., Tail probabilities of randomly weighted sums of random variables with variables with dominated variation, Stochastic models, 22, 253-272, (2006) · Zbl 1095.60008  Wang, D.; Su, C.; Zeng, Y., Uniform estimate for maximum of randomly weighted sums with applications to insurance risk theory, Science in China-series A, 48, 1379-1394, (2005) · Zbl 1112.62123  C. Weng, Y. Zhang, K.S. Tan, Ruin probability of a discrete time risk model under constant interest rate and dependent risks with heavy tails, manuscript, 2007
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