Extensions of a renewal theorem

Source abstract

Let X 1 , X 2 , X 3 ,… be a sequence of independent, identically distributed, absolutely continuous random variables whose first moment is μ 1 . Let S k = X 1 + X 2 + … + X k , and let f k ( x ) be the frequency function of S k , defined as the k -fold convolution of f 1 ( x ). When f 1 ( x ) has been defined for all x , f k ( x ) is uniquely defined for all k , x . Write