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Why $ \lim_{t\to\infty} \frac{W_t}{t} = 0 $

I tried to write it as Wiener process is defined which is: $ \lim_{t\to\infty} \frac{W_t}{t} = \lim_{t\to\infty}\lim_{N\to\infty} \frac{\sum_{k=1}^{\lfloor Nt \rfloor} \varepsilon_{k}}{\sqrt{N}*t}*\frac{\lfloor Nt \rfloor}{\lfloor Nt \rfloor} $

and then use a Law of large numbers but I got expression $ 0*\infty $

any hints or solution would be appreciated.

Pablo
  • 461

1 Answers1

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Ok, I tried different approach, we know that, $$ Z_t=t*W_\frac{1}{t} $$ is also a wiener process, then wanted limit is equiwalent to $$ \lim_{t\to 0 } \frac{Z_\frac{1}{t}}{\frac{1}{t}}=\lim_{t\to 0} t*Z_\frac{1}{t} $$ which is 0 , is it true ?

Pablo
  • 461
  • This makes no sens e at all. The question has been answered by user Did at https://math.stackexchange.com/questions/1099655/law-of-large-numbers-for-brownian-motion-direct-proof-using-l2-convergence – Kavi Rama Murthy Apr 11 '19 at 23:12