Skorokhod
In probability theory and stochastic processes, "Skorokhod" can refer to several concepts, most commonly to Anatoliy Volodymyrovych Skorokhod (1930-2011), a Ukrainian mathematician who made significant contributions to the field. As such, Skorokhod is associated with several mathematical concepts and theorems named after him.
Skorokhod's Representation Theorem: This theorem provides conditions under which weak convergence of probability measures implies the existence of random variables defined on a common probability space that converge almost surely. More formally, if a sequence of probability measures μn on a Polish space converges weakly to a probability measure μ, then there exist random variables Xn and X, all defined on the same probability space, such that Xn has distribution μn, X has distribution μ, and Xn converges to X almost surely. This theorem is a powerful tool for transferring results about convergence of distributions to results about almost sure convergence, which is often easier to work with.
Skorokhod Space (D[0,1]): This is the space of càdlàg functions (French: continu à droite, limité à gauche), meaning functions that are right-continuous and have left limits, defined on the interval [0, 1]. It is frequently used as the state space for stochastic processes, particularly those with jumps, like Lévy processes and certain types of Markov processes. Unlike the space of continuous functions, which is complete under the uniform norm, the Skorokhod space requires a different notion of distance to ensure completeness and to allow weak convergence to capture the behavior of processes with discontinuities.
Skorokhod Metric: Several metrics can be defined on the Skorokhod space D[0,1] that induce a topology under which the space is Polish (a complete, separable metric space). These metrics are designed to allow for "time changes" in the functions being compared, making them suitable for analyzing stochastic processes where the timing of events may be subject to small perturbations. One common Skorokhod metric involves reparameterizations of the time axis.
Other Contributions: Beyond the representation theorem and the Skorokhod space, Anatoliy Skorokhod made fundamental contributions to the theory of stochastic differential equations, limit theorems for stochastic processes, and the study of random evolutions. His work significantly advanced the understanding and application of probability theory in various fields, including physics, engineering, and finance.