Functional asymptotic behavior of some random multilinear forms (Q1382549)

For each \(n \geq 1, \) let \(\left (\chi _k^n\right )_{k\geq 0}\) be a real-valued sequence of martingale differences defined on a filtered space and \((a_{i_1,\dots ,i_d}^n)_{i_1,\dots ,i_d \geq 0} \) be a sequence of arrays with real coefficients. Let \[ S_n = \left (\sum _{i_1,\dots ,i_d = 1}^{[nt]} a_{i_1,\dots ,i_d}^n \chi _{i_1}^n\dots \chi _{i_d}^n\right )_{t\leq 1} \] denote a multilinear form in random variables \(\chi _k^n\). Asymptotic distribution of \(S_n\) is established; the proving method is based on asymptotic representation of \(S_n\) by means of iterated stochastic integrals.