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Random difference equations with subexponential innovations

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Abstract

We consider the random difference equations S = d (X + S)Y and T = d X + TY, where = d denotes equality in distribution, X and Y are two nonnegative random variables, and S and T on the right-hand side are independent of (X, Y). Under the assumptions that X follows a subexponential distribution with a nonzero lower Karamata index, that Y takes values in [0, 1] and is not degenerate at 0 or 1, and that (X, Y) fulfills a certain dependence structure via the conditional tail probability of X given Y, we derive some asymptotic formulas for the tail probabilities of the weak solutions S and T to these equations. In doing so we also obtain some by-products which are interesting in their own right.

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Authors and Affiliations

  1. School of Finance, Renmin University of China, Beijing, 100872, China

    QiHe Tang

  2. Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA, 52242, USA

    QiHe Tang

  3. Department of Risk Management, The Pennsylvania State University, University Park, PA, 16802, USA

    ZhongYi Yuan

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  1. QiHe Tang
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  2. ZhongYi Yuan
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Correspondence to QiHe Tang.

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In memory of Professor Xiru Chen (1934–2005)

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Tang, Q., Yuan, Z. Random difference equations with subexponential innovations. Sci. China Math. 59, 2411–2426 (2016). https://doi.org/10.1007/s11425-016-0146-0

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  • DOI: https://doi.org/10.1007/s11425-016-0146-0

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