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Large deviations and ruin probabilities for solutions to stochastic recurrence equations with heavy-tailed innovations. (English) Zbl 1085.60017

The authors consider the stochastic recurrence equation \(Y_t=A_tY_{t-1}+B_t\) for an i.i.d. sequence of paris \((A_t,B_t)\) of nonnegative random variables, where \(B_t\) is regularly varying with index \(\kappa>0\) and \(EA_t^{\kappa}<1\). The authors show that the stationary solution \((Y_t)\) to this equation has regularly varying finite-dimensional distributions with index \(\kappa\). This implies that the particular sums \(S_n=Y_1+\ldots+Y_n\) of this process are regularly varying. In particular, the relation \(P(S_n>x)\sim c_1nP(Y_1>x)\) as \(x\to\infty\) holds for some constant \(c_1>0\).
For \(\kappa>1\), the large deviation probabilities \(P(S_n-ES_n>x)\), \(x\geq x_n\), are also studied, for some sequence \(x_n\to\infty\) whose growth depends on the heaviness of the tail of the distribution of \(Y_1\). The authors show that the relation \(P(S_n-ES_n>x)\sim c_2nP(Y_1>x)\) holds uniformly for \(x\geq x_n\) and some constant \(c_2>0\). Then the large deviation results are applied to derive bounds for the ruin probability \(\psi(u)=P(\sup_{n\geq1}((S_n-ES_n)-\mu n)>u)\) for any \(\mu>0\) and to show that \(\psi(u)\sim c_3 uP(Y_1>u)\mu^{-1}(\kappa-1)^{-1}\) for some constant \(c_3>0\). In contrast to the case of i.i.d. regularly varying \(Y_t\)’s, when the above results hold with \(c_1=c_2=c_3=1\), the constants \(c_1\), \(c_2\) and \(c_3\) are different from 1.

MSC:

60F10 Large deviations
91B30 Risk theory, insurance (MSC2010)
60G70 Extreme value theory; extremal stochastic processes
60G35 Signal detection and filtering (aspects of stochastic processes)
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