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The asymptotic behavior of locally square integrable martingales. (English) Zbl 0831.60053

Let \(M= \{M_t, t\geq 0\}\) be a locally square integrable martingale with predictable quadratic variation \(\langle M\rangle\) and the jump process \(|\Delta M|\). The author gives various rates of increase of \(M_t\) as \(t\to \infty\) under the different restrictions on \(|\Delta M|\) and \(\langle M\rangle\). One of the results: let \(|\Delta M|\leq H(\langle M\rangle/\text{LLg}\langle M\rangle)^{1/2}\) a.s., where \(H\) is a predictable process. Then \[ \{\langle M\rangle_\infty= \infty\}\subset \Biggl\{\limsup_{t\to \infty} {|M_t|\over \sqrt{2\langle M\rangle_t\text{LLg}\langle M\rangle_t}}\leq a(K)\Biggr\}\quad\text{a.s.}, \] where \(\text{LLg } x= \log(\log(x\vee e^e))\), \(K= \limsup_{t\to\infty} H(t)\), \(a(K)\) is the unique solution of the equation \(a^2\psi(\sqrt 2 aK)= 1\), \(\psi (x)= {2(1+ x)\log(1+ x)- 2x\over x^2}\), \(x> 0\), and \(a(\infty)= \infty\). The asymptotic behavior of stochastic integrals \(X= B\cdot M\) for a predictable process \(B\) is investigated in terms of \(|\Delta M|\) and the rates of increase of \(B\). In the last chapter the author gives some examples how to get the convergence rates of different estimators in the statistics of stochastic processes, such as \(\text{AR}(1)\) model, Poisson process, gamma process.

MSC:

60G44 Martingales with continuous parameter
60F15 Strong limit theorems
60H05 Stochastic integrals
62M09 Non-Markovian processes: estimation
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