# Likelihood ratio testing on infinite dimensional vector

I am performing a likelihood ratio test on a vector containing observations $\textbf{a}$. The number of observation is a random variable $\in \{1,2,\ldots\}$, so the vector of observations may have infinite length. Is that OK? The likelihood ratio between $H_0$ and $H_1$ is: \begin{align} \nonumber \Lambda(\textbf{a}) &= \frac{\mathbb{P}_{\mathbf{A}\mid H_1}(\textbf{a}\mid H_1)}{\mathbb{P}_{\mathbf{A}\mid H_0}\left(\textbf{a}\mid H_0\right) } \\ \nonumber &= \frac{\sum_{k} \mathbb{P}_{\mathbf{A}\mid H_1, N_1}(\mathbf{a}\mid H_1, k) \mathbb{P}_{N_1}(k) }{\sum_k \mathbb{P}_{\mathbf{A}\mid H_0, N_0}(\mathbf{a}\mid H_0, k) \mathbb{P}_{N_0}(k) } \end{align} where $\mathbb{P}_{N_0}(k) = \mathbb{P}(N_0=k)$ is the probability mass function (pmf) for the number of observations under $H_0$, and $\mathbb{P}_{N_1}(k)= \mathbb{P}(N_1=k)$ is the pmf for the number of observations under $H_1$.

• Can you explain how is it possible that you have an infinite number of observations? – kjetil b halvorsen Aug 6 '17 at 21:01