Theory of probability and random processes

By Leonid Koralov, Y. Sinai. 2007

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On Lipschitz Bounds of General Convolutional Neural Networks (2018, background): “X ( t ) is said to be strict-sense-stationary (SSS) (see, for instance, [22], Chapter 16) if all of its ﬁnite-order moments are time-invariant (its cumulative distribution does not change with time).”
Some covariance models based on normal scale mixtures (2011, background): “√ |f(x) + f(y)| , x, y ∈R, in which the values of f :R → R are strictly positive definite matrices, see also Paciorek (2003) and Porcu et al. (2009). Here, f(x)+ f(y) is not a variogram in general, but the proof of Theorem 5 is still applicable if Ĉ in Equation (9) is replaced by Ĉ(ω,x, y) = exp(−ω(”
Large deviations for Markovian nonlinear Hawkes processes (2011, background): “Now, by Jensen’s inequality, we have lim inf Therefore, by the usual ergodic theorem, (for a reference, see Chapter 16.4 of Ko-ralov and Sinai [14]), for any (ˆ λ, ˆ π ) ∈ Q e , we have, Hence, we have lim inf Recall that (3.19) We claim that It is easy to see that for (ˆ λ, ˆ π ) ∈ Q e , g C 1…”
Stochastic Processes and Applications: Diffusion Processes, the Fokker-Planck and Langevin Equations (2014, background): “Well known textbooks on probability th eory are [4, 14, 15, 44, 45, 37, 71].”