Can a stochastic process with independent random variables be a stochastic process with independent increments? We know that if a stochastic process is a stochastic process with independent increments, we must have:
For any natural number $n$ and all real $0≤α_1<β_1≤α_2<β_2≤\cdots ≤α_n<β_n$, the increments
$X(β_1)−X(α_1),\cdots,X(β_n)−X(α_n)$ are mutually-independent random variables.
So my question is: Can a stochastic process with independent random variables be a stochastic process with independent increments?

I will answer the question when the index set is $\mathbb N$ but the idea of the proof works for other index sets like $[0,\infty)$ too.
If $(X_n)$ is independent and has independent increments then  $X_{n+1}-X_{n-1}=[X_{n+1}-X_{n}]+[X_{n}-X_{n-1}]$, a sum of two indepndent r.v.'s. Denoting by $h_n$ the characteristic function of $X_n$ we get $$h_{n+1}(t)h_{n-1}(-t)=[h_{n+1}(t)h_{n}(-t)][h_{n}(t)h_{n-1}(-t)].$$ This implies that $|h_n(t)|\equiv 1$ in a neighborhood of $0$. Hence, $X_n$ is a.s  constant.
