Geodesic
Invariance principle in a~bilinear model with weak non-linearity
Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 8, Tome 320 (2004), pp. 97-105

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We consider a series of bilinear sequences $$ X_k^{(n)}=X_{k-1}^{(n)}+\varepsilon_k+b_n X_{k-1}^{(n)}\varepsilon_{k-1},\qquad k\ge 1, $$ with i.i.d. sequence $\varepsilon_k$, small bilinearity coefficients $b_n=\beta n^{-1/2}$ and show that the processes obtained from $X_k^{(n)}$ by usual scaling in time and space converge to a diffusion process $Y_\beta$. We provide an explicit form of $Y_\beta$, investigate the moments of $Y_\beta$ and study the limit behavior of some other quantities related to $X_k^{(n)}$ and important for statistical applications. 
@article{ZNSL_2004_320_a6,
     author = {M. A. Lifshits},
     title = {Invariance principle in a~bilinear model with weak non-linearity},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {97--105},
     publisher = {mathdoc},
     volume = {320},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2004_320_a6/}
}
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M. A. Lifshits. Invariance principle in a~bilinear model with weak non-linearity. Zapiski Nauchnykh Seminarov POMI, Probability and statistics. Part 8, Tome 320 (2004), pp. 97-105. http://geodesic.mathdoc.fr/item/ZNSL_2004_320_a6/