  Let B_t^H be a d-dimensional fractional Brownian motion with Hurst parameter
H\in(0,1). Assume d\geq2. We prove that the renormalized self-intersection
local time\ell=\int_0^T\int_0^t\delta(B_t^H-B_s^H) ds dt
-E\biggl(\int_0^T\int_0^t\delta (B_t^H-B_s^H) ds dt\biggr) exists in L^2 if and
only if H<3/(2d), which generalizes the Varadhan renormalization theorem to any
dimension and with any Hurst parameter. Motivated by a result of Yor, we show
that in the case 3/4>H\geq\frac{3}{2d}, r(\epsilon)\ell_{\epsilon} converges in
distribution to a normal law N(0,T\sigma^2), as \epsilon tends to zero, where
\ell_{\epsilon} is an approximation of \ell, defined through (2), and
r(\epsilon)=|\log\epsilon|^{-1} if H=3/(2d), and
r(\epsilon)=\epsilon^{d-3/(2H)} if 3/(2d)<H.
