Some new results on local limit theorems with an addendum: On the positions of the maximum of the simple symmetric random walk

The topic is two-fold. We first recall the notion of local limit theorems and show how a multinomial local limit theorem generalizes the Black-Scholes formula. Secondly we introduce the concept of almost sure local limit theorems and prove such a result for Bernoulli processes, thus extending the deMoivre-Laplace theorem to the category of a.s. limit theorems. An application yields new methods of quantile estimation in statistics. In the second part of the talk we will consider $X_n$ - i.i.d. random variables with $P(X_1=1)=P(X_1=-1)=1/2$ and $S_n$ -- the partial sums form the random walk mentioned in the title. We derive some (apparently new) observations concerning the positions of the maximum of the random walk up to time $n$.