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I was doing some revision and had an admittedly elementary question. My lecture notes say, the following are properties of Brownian Motion {$B_t$}

  1. (Normal or Gaussian increments) For all $s < t, B_t − B_s$ has $\text{N} (0, t − s)$ distribution, i.e Normal distribution with mean $0$ and variance $t − s$.

  2. (Independent increments) $B_t − B_s$ is independent of the past, that is, of $B_u, 0 ≤ u ≤ s$.

My question is that; In practice, what exactly does Property 2 tell us, which property 1 does not?

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Property $1$ says nothing about independence. It only tells you about the distribution of the increments $B_t-B_s$. If not for property $2$, you may have that $B_t-B_s$ is dependent on $B_s-B_u$ for some $u<s$.

Property $1$ only tells you that $B_t-B_s$ has the distribution $N(0,t-s)$, and that $B_s-B_u$ has the distribution $N(0,s-u)$. Given property $1$, they may or may not be independent, but property $2$ guarantees independence.

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