Invariance principle in a bilinear model with weak non-linearity

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.