Let's model the motion of a jumping frog as a continuous-time stochastic process. The model for this process depends on a parameter $V_\infty$, the "true" average velocity of the frog. We want to infer $V_\infty$ from a finite set of observations.
The process: The frog jumps a distance $J_i \in \mathbb{R}^+$ at times $t_i \in \mathbb{R}$, for $i=1,...,N$. The average velocity of the frog over long time intervals converges to $V_\infty$. The jumps are instantaneous and all in the same direction. The underlying stochastic process is stationary, at least in a wide-sense.
Possible estimator of $V_\infty$: An estimator of $V_\infty$ could be the ratio between the empirical mean of the jumps over the empirical mean of the "waiting time" between consecutive jumps: $$ v_N = \left( \frac{t_N-t_1}{N-1} \right)^{-1} \left( \frac{1}{N}\sum_{i=1}^N J_i \right) $$ Assuming to know the details of the process, is there a way to tell how much $v_N$ is an accurate estimate of $V_\infty$? Are there better (possibly unbiased) estimators for $V_\infty$?
Question: given the observations $( t_i \, , \, J_i )_{i=1..N}$, how to theoretically infer $V_\infty$? I am OK with partial answers: I hope that someone can help me find the correct mathematical framework for this kind of question. Is there any known kind of stochastic process with real and positive amplitudes that occur at random times? It seems like a generalization of the Poisson process (maybe this process is a subordinator?). Any good reference?
It seems not easy: We have data only for a "single and partial history" of this stochastic frog, not different possible realizations of its motion (we do not have an ensemble of frogs all with the same $V_\infty$). Is it possible to quantify how much the estimate of $V_\infty$ given by $v_N$ is reliable? Partial answers pointing out the difficulties or which hypotheses could improve the feasibility of the task are well appreciated.
Related practical question and partial answer: this problem may be regarded as the "theoretical counterpart" of How to fit a cumulative time series?: to practically estimate $V_\infty$, we could fit the cumulative time series constructed from the observed data $(t_i, J_i)$. After some research, I concluded that a satisfying practical answer can be found in Fitting a Straight Line to Certain Types of Cumulative Data (1957), Parameter Estimation with Cumulative Errors (1974) and the methods described in Statistical estimates of the pulsar glitch activity (2021).