# Distributions of increments of a stochastic process

1. Do distributions of increments of a stochastic process $\{X_t, t \geq 0\}$ mean joint distributions of finitely many increments over non-overlapping time intervals, i.e. $\{L(X_{t_n} - X_{t_{n-1}}, \dots, X_{t_2} - X_{t_1}), \forall n \in \mathbb{N}, 0\leq t_1 < t_2 < \dots < t_n\}$, where $L()$ means law of random variables?
2. Can joint distributions of finitely many increments and finite dimensional distributions ($\{L(X_{t_n}, \dots, X_{t_1}), \forall n \in \mathbb{N}, 0\leq t_1 < t_2 < \dots < t_n\}$) be derived from each other? The former can be from the latter, but the reverse may not be true I think.

• If $X_0$ can be considered as an increment, then finite dimensional distributions can be derived from joint distributions of finitely many increments? The question is: is $X_0$ considered an increment? (I don't think it is)
• If $X_0$ is a given deterministic value $c$ instead of a random variable, then finite dimensional distributions can be derived from joint distributions of finitely many increments. But is it still true if $X_0$ is a random variable but not a deterministic value? (I don't think it is)

For example, I am not clear about why it can be proved that independent increments implies Markov property by assuming $X_0 = 0$ here?

Thanks and regards!

-