Weiner process

Consider a particle which is undergoing Brownian motion performes a random walk such that as time changes from  to , the position of the particle also changes from  to . One should be aware that the total displacement of the particle in time  is . Also suppose that the random variable  denotes the length of the  step taken by the particle in the time interval of , such that  and  and  is independent of both  and .

Now these  are i.i.d. and assume you divide the interval length into  equal subintervals each of length , such that . The total displacement is given by

Hence:

such that

such that

Now assume as ,  then

and

(3.6)

Moreover consider

	(3.7)
	(3.8)

If (3.6), (3.7) and (3.8) are true it would mean that

,  and

(3.9)