Независимые случайные величины, многочлены Уика и негауссовский стохастический интеграл: теория вероятностей, системы Радемахера, плотность произведений, экспонента Уика

This book provides a self‑contained construction of independent sequences of random variables for various families of measures (discrete, diffuse, and with a single jump) by transporting generalized Rademacher sequences via partition functions. The first part develops Wick polynomials and their generating exponential, establishes recurrence formulas and differential equations, and gives explicit expressions for several classical laws (normal, gamma, exponential, Poisson, binomial). The second part studies the density of products of independent random variables, proves the density of polynomial algebras under exponential moment conditions, and applies these results to a Hilbert‑space valued mapping \(\Phi\). The third part constructs a non‑Gaussian stochastic integral \(\displaystyle\int \Phi(h)_s\,d\Phi(g)_s\) using chaos operators \(\varphi^\) and \(\varphi^\), proves an integration‑by‑parts formula, computes the norm, and establishes the existence of continuous modifications as well as the quadratic variation of the integral process. The text is intended for Master’s students and researchers who wish to master advanced tools of stochastic analysis beyond the Gaussian framework.