Wilks's theorem: $-2\log\lambda\overset{d}{\longrightarrow}\chi^2_1$ under $H_0$

Consider the problem of testing $H_0:\theta=\theta_0$ against $H_1:\theta\ne\theta_0$, where $\theta_0$ is a specified value of $\theta$, the parameter of a one-parameter Cram'er family. If $\lambda(x)$ is the likelihood ratio statistic based on sample of size $n$,