$W(t)$ is a Brownian motion and $c>0$. I need to verify that

1)$X(t)=W(c+t)-W(c)$ and $X(t)=cW(t/c^2)$ are Brownian motions.

2) $Z(t)=tW(1/t)$ can be showm as $lim_{t-\rightarrow\infty} Z(t)=0$. shoe that Z(t) - Brownian motion

I assumed that it is necessary to check the following properties for $X(t)=W(c+t)-W(c)$ and $X(t)=cW(t/c^2)$

1) We have 1. $W(0)=0$ 2. for all $0=t_0, t_1, t_2, \dots,t_m$ the increments $W(t_1)-W(t_2), W(t_2)-W(t_m), \dots, W(t_m)-W(t_m-1)$ are independent 3. the increments are normally distributed with $E[W(t_k)-W(t_k-1)]=0$ and $Var[W(t_k)-W(t_k-1)]=t_k-t_k-1$.

$X(t) = 0$ almost surely, but I am stuck with checking 2) and 3).

2) Here I assumed, that for $tW(1/t)$,t>0, so if for all $0=t_0<t_1<t_2< \dots<t_m$ the RVs $W(t_1), W(t_2)\dots W(t_m)$ are normally jointly distributed with 0 mean and covariance matrix then $Z(t)$ is Brownian motion$ but I don't really know how to start 0 mean and covariance matrix checking.

I wil really appriciate help.


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