# Brownian Motion

tejo
Posers and Puzzles 30 Sep '04 17:58
1. tejo
a unique loser
30 Sep '04 17:58
I could use some help with this exercise I just got.
Prove that with chance zero that limit for t-&gt;infinite (Brownian motion at time t)/(sqrt (t))=0.
I hope you understand what I mean and that you can help me.
Thanks,
Sander
2. 01 Oct '04 13:48
Originally posted by tejo
I could use some help with this exercise I just got.
Prove that with chance zero that limit for t->infinite (Brownian motion at time t)/(sqrt (t))=0.
I hope you understand what I mean and that you can help me.
Thanks,
Sander
Are you making us do ur homework?
3. tejo
a unique loser
01 Oct '04 14:00
Originally posted by howzzat
Are you making us do ur homework?
I am not making anyone to do anything. I am just asking for help, because I still haven't solved this one.
4. Acolyte
01 Oct '04 16:36
Originally posted by tejo
I could use some help with this exercise I just got.
Prove that with chance zero that limit for t->infinite (Brownian motion at time t)/(sqrt (t))=0.
I hope you understand what I mean and that you can help me.
Thanks,
Sander
Is time discrete or continuous? Could you define Brownian motion mathematically?
5. tejo
a unique loser
01 Oct '04 17:02
Originally posted by Acolyte
Is time discrete or continuous? Could you define Brownian motion mathematically?
I hope I will explain this well.

Bt:=integral from 0 to t Ns ds.

Ns is a completely random function of t. In other words, the continuous-time analogue of a sequence of independent identically distributed random variables.
1.Ns is independent of Nt for t != (not equal) s
2.The random variables Ns (s&gt;=0) all have the same probability distribution u
3.Expectancy of Ns = 0 (E(Nt)=0)

So Bt has the following requirements:
1.For any 0=t0&lt;=t1&lt;=...&lt;=tn the random vaiables Bt(j+1) - Bt(j) are independent (j=0,...,n-1)
2.Bt has stationary increments.
3. E(Bt)=0 for all t&gt;=0
4. E((B1)^2)=1 also true E((Bt)^2)=t
5. t-&gt;Bt is continuous a.s. with probability 1.

The Brownian Motion is also called Wiener process I believe.
6. Acolyte
01 Oct '04 23:061 edit
Originally posted by tejo
I hope I will explain this well.

Bt:=integral from 0 to t Ns ds.

Ns is a completely random function of t. In other words, the continuous-time analogue of a sequence of independent identically distributed random variables.
1.Ns is indep ...[text shortened]... 1.

The Brownian Motion is also called Wiener process I believe.
I understand the question now. I think the following will work: Bt (for t an integer) can be turned into a discrete summation; conditions 1. and 4. imply that the summands are iid with variance 1, and then a Central Limit Theorem does the rest.
7. tejo
a unique loser
04 Oct '04 11:51
Originally posted by Acolyte
I understand the question now. I think the following will work: Bt (for t an integer) can be turned into a discrete summation; conditions 1. and 4. imply that the summands are iid with variance 1, and then a Central Limit Theorem does the rest.
Thank you very much Acolyte. It worked really well.