# Random difference equations with subexponential innovations

Qi He Tang, Zhongyi Yuan

Research output: Contribution to journalArticle

3 Citations (Scopus)

### Abstract

We consider the random difference equations S =d (X + S)Y and T =dX + 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) 2411-2426 16 Science China Mathematics 59 12 https://doi.org/10.1007/s11425-016-0146-0 Published - Dec 1 2016

### Fingerprint

Tail Probability
Difference equation
Subexponential Distribution
Dependence Structure
Conditional probability
Asymptotic Formula
Weak Solution
Equality
Random variable
Non-negative
Denote
Innovation

### All Science Journal Classification (ASJC) codes

• Mathematics(all)

### Cite this

title = "Random difference equations with subexponential innovations",
abstract = "We consider the random difference equations S =d (X + S)Y and T =dX + 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.",
author = "Tang, {Qi He} and Zhongyi Yuan",
year = "2016",
month = "12",
day = "1",
doi = "10.1007/s11425-016-0146-0",
language = "English (US)",
volume = "59",
pages = "2411--2426",
journal = "Science China Mathematics",
issn = "1674-7283",
publisher = "Science in China Press",
number = "12",

}

In: Science China Mathematics, Vol. 59, No. 12, 01.12.2016, p. 2411-2426.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Random difference equations with subexponential innovations

AU - Tang, Qi He

AU - Yuan, Zhongyi

PY - 2016/12/1

Y1 - 2016/12/1

N2 - We consider the random difference equations S =d (X + S)Y and T =dX + 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.

AB - We consider the random difference equations S =d (X + S)Y and T =dX + 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.

UR - http://www.scopus.com/inward/record.url?scp=85000350985&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85000350985&partnerID=8YFLogxK

U2 - 10.1007/s11425-016-0146-0

DO - 10.1007/s11425-016-0146-0

M3 - Article

VL - 59

SP - 2411

EP - 2426

JO - Science China Mathematics

JF - Science China Mathematics

SN - 1674-7283

IS - 12

ER -