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Tail bounds for the distribution of the deficit in the renewal risk model. (English) Zbl 1189.91080
Summary: We obtain upper and lower bounds for the tail of the deficit at ruin in the renewal risk model, which are (i) applicable generally; and (ii) based on reliability classifications. We also derive two-side bounds, in the general case where a function satisfies a defective renewal equation, and we apply them to the renewal model, using the function \(\Lambda_u\) introduced by G. Psarrakos and K. Politis [Stoch. Models 25, No. 1, 90–109 (2009; Zbl 1159.91412)]. Finally, we construct an upper bound for the integrated function \(\int_y^{\infty} \Lambda_u(z)\text dz\) and an asymptotic result when the adjustment coefficient exists.

MSC:
91B30 Risk theory, insurance (MSC2010)
60K10 Applications of renewal theory (reliability, demand theory, etc.)
62P05 Applications of statistics to actuarial sciences and financial mathematics
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