Positivity of integrated random walks

Consider the sequence of partial sums of a centered random walk with finite variance. We study asymptotics of the probability that the first $n$ terms of this sequence are positive, as $n \to \infty$. The first result here is due to Ya. Sinai (1992) who came to the problem considering solutions of the Burgers equation with random initial data. The speaker's original motivation emerged as these probabilities appeared in his study of certain properties of sticky particle systems with random initial positions. The new results are significantly sharper than those presented at the seminar a year ago.