On asymptotic of the hitting times of bounded sets for random walks

Consider the probability $p_n$ that a centred random walk does not hit a fixed bounded set at the first $n$ steps. Kesten and Spitzer (1963) found the asymptotic of $p_n$ for integer-valued random walks but their technique does not apply for a general case. We obtain the asymptotic of $p_n$ for any centred random walks with finite variance, and prove a conditional limit theorem for a typical trajectory of the walk. Our initial interest to the problem was motivated by the particular case that the set was an interval. Here the asymptotic of $p_n$ implies that, under the stated assumptions, the size of the largest gap within the range of the random walk is of a constant order.