Rate of convergence in the strong law of large numbers for martingales (Q760707)

Let \(\{(S_ n,{\mathcal F}_ n)\), \(n\geq 0\}\) be a martingale with \(S_ 0=0\) a.s., \(S_ n=\sum^{n}_{k=1}X_ k\), \(n\geq 1\). Let us put \(\sigma^ 2_ k=EX^ 2_ k\), \(s^ 2_ k=\sum^{k}_{i=1}\sigma^ 2_ i\), \(k\geq 1\), \(S_{N_ n}=\sum^{N_ n}_{k=1}X_ k\), \(M^ 2_ n=\sum^{N_ n}_{k=1}\sigma^ 2_ k\), where \(\{N_ n\), \(n\geq 1\}\) is a sequence of positive integer-valued random variables not necessarily independent of \(\{X_ n\), \(n\geq 1\}\). Write \[ Z_{\infty}(t,\alpha)=\sum^{\infty}_{n=1}P(| S_{N_ n}| \geq tM_ n^{1+2\alpha}),\quad H(t)=\sum^{\infty}_{n=1}P(| M^ 2_ n-\lambda s^ 2_ n| \geq ts^ 2_ n) \] where \(t>0\), \(\alpha >0\) are constants and \(\lambda\) is a positive random variable such that \(P(a\leq \lambda \leq b)=1\) for some constants \(0<a\leq b<\infty\). In this paper we present sufficient conditions under which \[ \lim \inf_{t\to 0^+}t^{1/\alpha}[Z_{\infty}(t,\alpha)+H(t)]\geq \lim \inf_{t\to 0^+} F(\alpha,t,b)t^{1/\alpha}\quad and \] \[ \limsup_{t\to 0^+}t^{1/\alpha}[Z_{\infty}(t,\alpha)-H(t)]\leq \limsup_{t\to 0^+} F(\alpha,t,a)t^{1/\alpha}, \] where \(F(\alpha,t,x)=2\sum^{\infty}_{n=1}\Phi (-tx^{\alpha}s_ n^{2\alpha})\) and \(\Phi\) is the standard normal distribution function.