Suppose that $N=\{N_t\}_{t\in \mathbb{R}^+}$ is a counting process that counts the number of events by time $t$, $T=\{T_n\}_{n\in \mathbb{N}_0}$ is an arrival time process, i.e., the time at which an event happens, and $W=\{W_n\}_{n\in \mathbb{N}_0}$ is an interarrival process, i.e., the time between consecutively arriving events. Concerning the event that exactly $n$ events have occurred by time $t$, we know that
\begin{equation} \{N_t= n\} = \{T_n \leq t\} \cap \{W_{n+1}> t- T_{n}\} \end{equation}
Now, I would like to know how we can represent the event that between $n_1$ and $n_2$ events have occurred by time $t$, i.e., $\{n_1 \leq N_t \leq n_2\}$, based on events constructed by process $T$ and $W$. More precisely, if it is possible to find a representation like the foregoing one in this case?
\begin{equation} \{n_1\leq N_t \leq n_2\} = ? \end{equation}
$$\{ {n_1 \leq N_t \leq n_2} \} = \{ T_{n_1} \leq t \} \cap \{ \sum_{i = 1}^{n2-n1+1} W_{n_1 + i} \geq t-T_{n_1}\}$$ Reduces to the standard form if $n_2 = n_1$.