Transmission of Poisson process properties: $(N_t :t \geq 0 )$ = P.process $\implies \frac{1}{t+1}N_{t+1}$, $\frac{1}{2}N_{4t}$, (...) = P.processes?

Question

$(N_t :t \geq 0 )$ is a Poisson process with variable $\lambda$.

• $A_t = \frac{1}{t+1}N_{t+1}$
• $B_t = \frac{1}{2}N_{4t}$
• $C_t = \frac{1}{2}N_{2t}$
• $D_t = \frac{5}{2}N_{t+1}$

Is that true that $A_t, B_t, C_t, D_t$ are also Poisson processes?

I think it should be true since all of them look like scaling of values $N_{t}$ with avoiding some of them. However I have seen that only specific transformations transmission Wiener proces properties. Therefore I would like to know how it works with Poisson processes.

Answer 1

Unless you use a different definition, none of these are Poisson processes.

A Poisson process can be interpreted as a counting process (i.e. counting the number of events that have happened). Scaling by $\frac{1}{2}$ or $\frac{5}{2}$ is therefore not allowed, as it wouldn't increment by one anymore.