Fluctuations of Discrete-Time Random Walks

These notes are devoted to fluctuations of one-dimensional random walks. We discuss various approaches to first-passage times and to the corresponding conditional distributions. After discussion of some classical methods, such as reflection principle for simple random walks and Wiener-Hopf factorisation, we proceed to the universality approach, which has been developed in recent past. Considering one-dimensional case allows us to avoid some technical obstacles and to present the core of this method in a more transparent way. It turns out that the universality method is much more robust than the Wiener-Hopf factorisation and allows one to consider walks with non-identically distributed or even dependent increments.