On the ruin probabilities in a general economic environment

Let {An | n=1,2,…} and {Bn | n=1,2,…} be sequences of random variables andYn=B1+A1B2+A1A2B3+⋯+A1⋯An−1Bn.Let M be a positive real number. Define the time of ruin by TM=inf{n | Yn>M} (TM=+∞, if Yn⩽M for n=1,2,…). We are interested in the ruin probabilities for large M. We assume that the sequences {An} and {Bn} are independent and that the variables A1,A2,… are strictly positive. The sequences are allowed to be general in other respects. Our main objective is to give reasons for the crude estimate P(TM<∞)≈M−w where w is a positive parameter. In the particular case where both {An} and {Bn} are sequences of independent and identically distributed random variables, we prove an asymptotic equivalence P(TM<∞)∼CM−w with a strictly positive constant C.