I got a question about the construction of Brownian motions. Just to be all on the same page, for me a Brownian motion on a certain probability space $(\Omega, \mathcal{A}, \mathbb{P})$ is a stochastic process $(B_t)_{t \in [0,\infty)}$, i.e. each $B_t$ is a real valued random variable on $\Omega$ such that
(i) $B_0 = 0$ a.s.
(ii) $B(\omega) : [0,\infty) \to \mathbb{R}$ is continuous for almost all $\omega \in \Omega$
(iii) The increments are independent, i.e. let $0\leq t_1< t_2 <...<t_n$, then $B_{t_n} - B_{t_{n-1}},\dots,B_{t_2}-B_{t_1}$ are independent
(iv) For all $0 \leq s < t$ we have $B_t - B_s \sim \mathcal{N}(0,t-s)$.
Now my question is: If I have a sequence of real valued random variables, say $(A_n)_{n \in \mathbb{N}_0}$, which satisfy (i) as well as (iii) and (iv) for all $t,s \in \mathbb{N}_0$, would it be possible to construct a Brownian motion $B$ such that
$B_n = A_n$ for all $n \in \mathbb{N}_0$.
My idea would be to interpolate linearly between $A_n$ and $A_{n+1}$ to obtain values for $B_t$ with $t \in (n, n+1)$. I.e.
$B_t = (1 - (\lceil t \rceil - t)) B_{\lfloor t \rfloor} + (\lceil t \rceil - t) B_{\lceil t \rceil}$
if we consider $\lceil n \rceil = n = \lfloor n \rfloor$ for integer values. Does that work or am I missing something here?