### 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.

Original language | English (US) |
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Pages (from-to) | 2411-2426 |

Number of pages | 16 |

Journal | Science China Mathematics |

Volume | 59 |

Issue number | 12 |

DOIs | |

State | Published - Dec 1 2016 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

*Science China Mathematics*,

*59*(12), 2411-2426. https://doi.org/10.1007/s11425-016-0146-0