We consider the random deposition of objects of variable width and height over a line. The successive additions of these structures create a random interface. We focus on the regime of heavy tailed distributions of the structure width. When the structure center is chosen at random, the problem is exactly solvable and we prove that the interfaces generically tend towards self-affine random curves. The asymptotic behavior reached after a large number of iterations is universal in the sense that it depends on only three parameters: the shape of the added structure at its maximum, the power-law exponent of the width distribution and the exponent that relates height and width. The parameter space displays several transitions that separate different asymptotic behaviors. In particular for a set of parameters, the interface tends towards a fractional Brownian motion. Our results reveal the existence of a new class of random interfaces which properties appear to be robust. The mechanism that generates correlations at large distance is identified and it explains the appearance of such correlations in several situations of interest such as the physics of earthquakes or the propagation of energy through a diffusive medium.