Self-similar processes as weak limits of a risk reserve process (Q2772052)

The traditional collective risk theory is expressed as \(R(t)=u+ct-\sum_{k=1}^{N(t)} Y_k,\) where \(u\) means the initial capital, \(c\) is the interest and \(Y_k\)'s are independent risks with finite variance which are controlled by a Poisson point process \(N(t)\). The author tries to weaken the asumptions. He treats the above formula: \(N(t)\) is not Poisson, \(Y_k\)'s are identical distributed but are strongly dependent. He obtains an estimation of the ruin probability NEWLINE\[NEWLINE\Psi(u,T)=P(R(t)<0 \text{ for some } t \leq T) \sim_{n\rightarrow \infty} P\{\inf(u+\theta\lambda\mu s -\lambda^H X_H(s))<0\},NEWLINE\]NEWLINE where \(\theta=c/(\lambda\mu) -1,\) under the assumption that \(\sum(Y_k-\mu)\) converges to a self-similar process \(X_H\).