Ordered random walks with heavy tails ∗

Quantum Field Theory of Black-Swan Events

Free and weakly interacting particles are described by a secondquantized nonlinear Schrödinger equation, or relativistic versions of it. They describe Gaussian random walks with collisions. By contrast, the fields of strongly interacting particles are governed by effective actions, whose extremum yields fractional field equations. Their particle orbits perform universal Lévy walks with heavy ta...

Heavy-Tailed Analogues of the Covariance Matrix for ICA

Independent Component Analysis (ICA) is the problem of learning a square matrix A, given samples of X = AS, where S is a random vector with independent coordinates. Most existing algorithms are provably efficient only when each Si has finite and moderately valued fourth moment. However, there are practical applications where this assumption need not be true, such as speech and finance. Algorith...

E¢ cient Rare-event Simulation for the Maximum of Heavy-tailed Random Walks

Let (Xn : n 0) be a sequence of iid rv’s with negative mean. Set S0 = 0 and de…ne Sn = X1 + :::+Xn. We propose an importance sampling algorithm to estimate the tail of M = maxfSn : n 0g that is strongly e¢ cient for both light and heavy-tailed increment distributions. A key feature of our algorithm is that it is state-dependent. In the presence of light tails, our procedure leads to Siegmund’s ...

Upper Tails for Intersection Local times of Random Walks in Supercritical Dimensions

We determine the precise asymptotics of the logarithmic upper tail probability of the total intersection local time of p independent random walks in Z under the assumption p(d − 2) > d. Our approach allows a direct treatment of the infinite time horizon. Mathematics Subject Classification (2000): 60F10, 60G50, 60K35.

Upper Tails of Self-intersection Local times of Random Walks: Survey of Proof Techniques