On a~functional version of the convergence of a~quadratic form in independent martingales to a~$\chi^2$ distribution

B. Cadre

Teoriâ veroâtnostej i ee primeneniâ, Tome 43 (1998) no. 1, pp. 82-96

Résumé

Let $(a_{ij})_{i,j\ge 1}$ be an infinite matrix of real numbers such that $a_{ii}=0$, $i\ge 1$ and let $(X^i)_{i\ge 1}$ be a sequence of independent martingales such that $\sup_{i\ge1}\mathsf{E}[(X_1^i)^4]\infty$ and for each $i\ge 1$ the predictable compensator of the quadratic variation of $X^i$ is the identity function. If for each $n\ge 1$, $\sigma_n^2=\sum^n_{i,j=1}a^2_{ij}$ we give a necessary and sufficient condition so that the process defined for each $n\ge 1$ and $t\ge 1$, by $\sigma_n^{-1}\sum_{i$ converges in law to $((2\sqrt{p})^{-1}\sum_{i=1}^p((B_t^i)^2-t))_{t\le1}$, where $p\ge 1$ and $B^1,\dots,B^p$ are $p$ independent standard Brownian motions. We then study the case where $(X^i)_{i\ge 1}$ is a sequence of independent solutions to the ‘`Structure Equation.’

Détail

Keywords: quadratic forms, $\chi^2$ distributions, functional limit theorems, stochastic calculus, Brownian motion.
Mots-clés : martingales