## 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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