Limit theorem for large deviations probabilities of certain forms
Consider the quadratic forms \(S_n=\sum^n_{k=1} \sum^{k-1}_{j=1} \beta_n^{k-j} \xi_k\xi_j\), \(n\geq 1\), where \(\{\beta_n\}\) is a real sequence such that \(\beta_n\to 1\) and \((\xi_k, {\mathcal F}_k, k\geq 1)\) is a uniformly bounded, square-integrable martingale difference sequence such that \(E(\xi^2_k \mid {\mathcal F}_{k-1}) =1\) a.s., \(k\geq 1\). Assume that \(n(1- \beta^2_n) \to\gamma \geq 0\). The main results present explicit real sequences \(\{x_n\}\) such that \[ \sup_{0\leq x\leq x_n} \left | {P(n^{-1} a_nS_n> x)\over 1-G (x)}-1 \right| \to 0 \text{ as } n\to \infty, \] where \(\{a_n\}\) is a real sequence and \(G\) is a distribution function (d.f.). This result holds, in particular, when \(\gamma= +\infty\) if \(a_n= (1-\beta^2_n)^{1/2}\) and \(G\) is the standard normal d.f. and, in general, with \(a_n=1\) and \(G(x)= P[\int^1_0 Y(t)dW(t)\leq x]\), where \(W(t)\) is a standard Wiener process and \(Y(t)\) is an Ornstein-Uhlenbeck process.
scientific article; zbMATH DE number 1296187
scientific article