I would like to create confidence intervals for martingale estimators.
For the processes $$B_{t}(a,k)=\int_0^ta_s(k)\,\mathbb 1_{\{\mathbb E[\lambda_{s}(k)]>0\}}\,\mathrm ds$$ we have estimators $$\hat{B}^n_t(a,k)=\int_0^t\big(\lambda^n_s(k)\big)^{-1}\,\mathrm dN^n_s(k).$$ These estimators are consistent and asymptotically normal with mean=$0$. $N^n$ is a sequence of point processes with stochastic intensity $\lambda^n$.
We observe i.i.d. copies $(N_t^i,\lambda_t^i)$, $1\le i\le n$ of pairs $(N,\lambda)$. For the variance, we have the estimator $$\hat{C}_t(a,k)=n\int_0^t\frac1{[\lambda^n_s(k)]^{2}}\,\mathrm dN^n_s(k).$$ How do I now create a confidence interval?