Ruin probability of an insurance company which also performs as a bank (Q2784986)

Let an insurance company has \(N\) clients and performs as a bank with the constant interest rate \(r\) if a client does not bring claims during the insurance period. The discount capital of the \(i\)-th client during the insurance period has the form \(\eta_{i}=I(\nu(T)=0)c+\sum_{n=1}^{\infty}I(\nu(T)=n)\sum_{k=1}^{n} \xi_{k}\nu^{\tau_1+\ldots+\tau_{k}}\), where \(I(B)\) is the indicator of an event \(B\); \(\nu(T)\) is the number of claims during the insurance period \([0,T]\); \(\nu=(1+r)^{-1}\) is a discount coefficient; \(c\) is an insurance premium; claims values \(\xi_{k}, k=1,2,\ldots\) are independent identically distributed random variables; \(\tau_1,\tau_2,\ldots\) are independent moments of claims. The author obtains the estimate NEWLINE\[NEWLINEP\left(\sum_{i=1}^{N}\eta_{i}>Nc\right)\leq 1-\Phi\left(\sqrt{{N\over D\eta}}(c-E\eta)\right)+{AE(\eta+E\eta)^3\over\sqrt{N(D\eta)^3}},NEWLINE\]NEWLINE where \(\Phi(x)\) is a normal \((0,1)\) distribution function; \(A\) is a positive constant; \(\eta\) is a random variable. Explicit forms of \(E\eta, D\eta, E\eta^3\) are presented.