In recent years, various notions of capacity and complexity have been proposed for
characterizing the generalization properties of stochastic gradient descent (SGD) in deep …

Dariusz Buraczewski Ewa Damek Thomas Mikosch The Equation X = AX + B Page 1
Springer Series in Operations Research and Financial Engineering Dariusz Buraczewski …

Let Z be a random variable with values in a proper closed convex cone, A a random
endomorphism of C and N a random integer. We assume that Z, A, N are independent …

We introduce the cluster index of a multivariate stationary sequence and characterize the
index in terms of the spectral tail process. This index plays a major role in limit theory for …

The theorem of Furstenberg and Kesten provides a strong law of large numbers for the norm
of a product of random matrices. This can be extended under various assumptions, covering …

In a variety of problems in pure and applied probability, it is relevant to study the large
exceedance probabilities of the perpetuity sequence Yn≔ B₁+ A₁ B₂+⋯+(A₁⋯ An-1) Bn …

Recent theoretical studies have shown that heavy-tails can emerge in stochastic
optimization due to 'multiplicative noise', even under surprisingly simple settings, such as …

Cyclic and randomized stepsizes are widely used in the deep learning practice and can
often outperform standard stepsize choices such as constant stepsize in SGD. Despite their …

We consider the random difference equations S= d (X+ S) Y and T= d X+ TY, where= d
denotes equality in distribution, X and Y are two nonnegative random variables, and S and T …

We consider the stochastic equation X= d AX+ B where A is a random diagonal matrix and
X, B are random vectors, X, A are independent and the equation is meant in law. We prove …