## 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)

### Keywords:

regular variation
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### References:

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