Equivalent definition of independent increments of a stochastic process.

Let $$(X_t)_{t\geq0}$$ be a stochastic process on some probability sprace $$(\Omega, \mathcal{F}, P)$$. Then for $$s < t$$, we define the $$\textit{increment}$$ of the process, $$X_t - X_s$$ over the interval $$[s, t]$$. The process $$(X_t)_{t\geq0}$$ has $$\textit{independent increments}$$ if, for every set of real numbers $$0 \leq t_1 < t_2 < \ldots < t_n < \infty$$, the increments $$X_{t_2} - X_{t_1}, \ X_{t_3} - X_{t_2}, \ \ldots, \ X_{t_n} - X_{t_{n-1}}$$ are independent of eachother.

During our course in stochastic processes the lecturer mentioned an equivalent way to define independent incerements of a process. For every set of real numbers $$0 \leq t_1 < t_2 < \ldots < t_n < t_{n+1} < \infty$$ the increment $$X_{t_{n+1}}-X_{t_n}$$ is independent of the random vector $$(X_{t_1}, \ \ldots, X_{t_n})$$. I don't see a way how to prove the eqivalence. He said this equvalence withou writing it down so I could be missing something. Any help would be appreciated.

• One direction looks easier than the other. Can you prove the second condition implies the first? I also note that if we have a deterministic initial condition such as $X_0=0$ then the information $X_0,X_{t_1}, X_{t_2}, ..., X_{t_n}$ is equivalent to the information $X_{t_n}-X_{t_{n-1}}, X_{t_{n-1}}-X_{t_{n-2}}, ..., X_{t_2}-X_{t_1}, X_{t_1}-X_0$. Commented Apr 19 at 18:47
• We are studying point processes, so usually $X_0=0$. Commented Apr 20 at 7:08

The problem becomes easy if you assume $$X_0=0$$ surely, as you indicated in your comments. The main idea is to consider the vectors $$(X_0, X_{t_1}, ..., X_{t_n})$$ and $$(X_0, X_{t_1}-X_0, ..., X_{t_n}-X_{t_{n-1}})$$, and observe that any one of these two vectors can be used to obtain the other by simple addition or subtraction.

Fact 1: Fix positive integers $$n,k$$. Let $$Y, W_1, ..., W_n$$ be random variables. If $$Y$$ is independent of the random vector $$(W_1, ..., W_n)$$, then $$Y$$ is also independent of $$h(W_1, ..., W_n)$$ where $$h:\mathbb{R}^n\rightarrow\mathbb{R}^k$$ is some Borel measurable function.

Fact 2: Fix $$n\geq 2$$ as an integer. If $$(A_1, A_2, ..., A_n)$$ are random variables such that for each $$i \in \{2, ..., n\}$$, $$A_i$$ is independent of $$(A_1, ..., A_{i-1})$$, then $$A_1, ..., A_n$$ are mutually independent.

Forward direction: Suppose $$(X_t)_{t\geq 0}$$ has $$X_0=0$$ surely and also has the independent increments property. Fix $$n$$ as a postive integer and fix $$\{t_i\}_{i=1}^{n+1}$$ such that $$0\leq t_1\leq ... \leq t_{n+1}$$. We want to show $$X_{t_{n+1}}-X_{t_n}$$ is independent of $$(X_0,X_{t_1}, ..., X_{t_n})$$.

Since $$X_0=0$$ surely, we know $$X_{t_{n+1}}-X_{t_n}$$ is independent of $$(X_0, X_{t_1}-X_0, X_{t_2}-X_{t_1}, ..., X_{t_n}-X_{t_{n-1}})$$. So (by Fact 1) we know $$X_{t_{n+1}}-X_{t_n}$$ is independent of $$h(X_0, X_{t_1}-X_0, ..., X_{t_n}-X_{t_{n-1}})$$ for any measurable function $$h:\mathbb{R}^{n+1}\rightarrow\mathbb{R}^{n+1}$$. Observe that \begin{align} X_0 &= 0\\ X_{t_1}&=X_0+(X_{t_1}-X_0)\\ X_{t_2}&=X_0+(X_{t_1}-X_{0}) + (X_{t_2}-X_{t_1})\\ ...\\ X_{t_n} &= X_0+(X_{t_1}-X_{0}) + (X_{t_2}-X_{t_1}) + ... + (X_{t_n}-X_{t_{n-1}}) \end{align} So $$(X_0, X_{t_1}, ..., X_{t_n})=h(X_0, X_{t_1}-X_0, ..., X_{t_n}-X_{t_{n-1}})$$ for the measurable function $$h=(h_0, ..., h_n)$$ defined by \begin{align} &h_0(a_0, ..., a_n) = a_0\\ &h_1(a_0, ..., a_n)=a_0+a_1\\ &\cdots\\ &h_n(a_0,...,a_n)=a_0+a_1+...+a_n \end{align} $$\Box$$

Reverse direction: Suppose $$(X_t)_{t\geq 0}$$ satisfies for all positive integers $$n$$ and all times $$0\leq t_1\leq ...\leq t_{n+1}$$ that $$X_{t_{n+1}}-X_{t_n}$$ is independent of $$(X_0, ..., X_{t_n})$$. We want to show it has the independent increments property. Fix $$n$$. By Fact 1, we know $$X_{t_{n+1}}-X_{t_n}$$ is independent of $$h(X_0, ..., X_{t_n})$$ for any measurable function $$h$$. Clearly there is a measurable function that maps the pure times $$(X_0, ..., X_{t_n})$$ to their differences $$(X_1-X_0, ..., X_{t_n}-X_{t_{n-1}})$$. So $$X_{t_{n+1}}-X_{t_n}$$ is independent of $$(X_1-X_0, ..., X_{t_n}-X_{t_{n-1}})$$. By Fact 2, it holds that $$(X_t)_{t\geq 0}$$ has the independent increment property. $$\Box$$

The above proof of the forward direction uses $$X_0=0$$ surely, while the reverse direction does not.