Is the independence of increments in Brownian motion a trivial consequence of other properties?

Question

Standard Brownian motion $W(t)$ is usually defined with 4 following statements:

1. $W(t) =0.$
2. For any $t < T$ the random variable $W(T) - W(t)\sim N(0, T-t)$, e.g. it's normal
3. For any $0<t_{1}<...<t_{n}$ increments $W(t_{i}) - W(t_{i-1})$ are independent random variables.
4. the function $W(t)$ is continuous.

Surely condition 3) is redundant as it's a consequence of 2) ? We can consider 3 arbitrary times $t_{1}<t_{2}<t_{3}<t{_n}$ and observe that:

$W(t_{3}) - W(t_{2})\sim N(0, t_{3} - t_{2})$ and $W(t_{2}) - W(t_{1})\sim N(0, t_{2} - t_{1})$.

They are both normal variables with zero mean and variance which only depends on the "end points" of the time increments, not the intermediate values, hence trivially the distribution of $W(t_{3}) - W(t_{2})$ has no relevance to distribution of $W(t_{2}) - W(t_{1})$...

And surely this immediately extends to Brownian motion with drift too?

So why do we require 3) ? Or am I misunderstanding something?

Answer 1

By the argument you noted $$W(t_3) - W(t_2) \sim N(0 , t_3 - t_2)$$ and $$W(t_2) - W(t_1) \sim N(0, t_2 - t_1)$$. So we have two normally distributed random variables. We can note that their sum is distributed normal with variance parameter equal to sum of the variances of each term by 2) condition. (Meaning that $t_3 - t_1 = t_3 - t_2 + t_2 - t_1$). Which means that their correlation ($\rho (W(t_3) - W(t_2), W(t_2) - W(t_1))$) should be zero. (As sum of two normals is also normal but with the variance including correlation).

So they are uncorrelated.

But uncorrelatedness (even for normally distributed random variables), does not imply independence.

https://en.wikipedia.org/wiki/Normally_distributed_and_uncorrelated_does_not_imply_independent

But if we had that $(W(t_3) - W(t_2), W(t_2) - W(t_1))$ were jointly normal (meaning that every linear combination of them were normal), the uncorrelatedness would imply independence.

So, not straightly, but the third condition implies that $(W(t_3) - W(t_2), W(t_2) - W(t_1))$ are jointly Gaussian.

Which is a desirable condition as it is the same as finite dimensional distribution is normal.

I hope I was helpful. (Sorry for posting this as answer. I just always write in informal manner).

Answer 2

It is a part of the definition not a consequence. You are confusing the following two statements: (a). the laws of $W_{t_3}-W_{t_2}$ and $W_{t_2}-W_{t_1}$ have laws that depend only on the time increment, i.e. $t_3-t_2$ and $t_2-t_1$, respectively and (b). $W_{t_3}-W_{t_2}$ and $W_{t_2}-W_{t_1}$ are independent as random variables.

The first is statement about laws and the second has to do with their joint laws; the two properties have almost nothing do with each other. Two random variables having the same or related laws does not mean they are independent.