Exercise is from the book "Wstęp do rachunku prawdopodobieństwa" Jakubowski, Sztencel
I need to show that family of random variables given by $$ Y_t = tW_{1/t},\,t>0 $$ And $Y_0=0$ where $W_t$ is a Wiener process is also a Wiener process.
I'm stuck with showing that the trajectory is continuous, i. e. that $$ \lim_{t\to 0^+} Y_t = 0 $$ How can I show that the trajectory is continuous? If possible, I'd like an advice rather than full answer.
My professor said something about the law of iterated logarithm (which is unkown to me). Thank you
The law of iterated logarithm for Brownian motion says $$ \limsup_{t\to\infty}\frac{B_t}{\sqrt{2t\log\log t}}=+1 \text{ a.s.} $$ (and symmetry about $0$ gives liminf is -1 a.s.). You might have seen this for the sum of zero-mean unit-variance independent random variables, sitting between the law of large numbers $\mathbb{P}(S_n/n\to0)=1$ and the central limit theorem $S_n/\sqrt{n}\sim N(0,1)$. Can you finish it off from here?