3. Consider the syrmnetric simple random walk described above. {a} If we take a di'erent scaling for the simple random walk, namely, the space scale by El and the time scale by i, then what happens to the characteristic function of the scaled symmetric simple random walk, $13.1: [H] as It goes to innity? b Secondl what ha if on kee the same a ace scale i but different time scale ( 3', spells :r P p n \"15, in other 1ili'iords, What happens to tlg aim} as 11 goes to innity? b) The process 3n = 22:1 xk is known as the symmetric random walla Scaled symmetric random walk is attained by stepping down time by :andalso scaling doum the step sizeiofthe smmetricrandomwallrsm The characteristicmctionofsn isgr'lsenhj.r agtw) = spams] = Esq eiopm which is mm)" For t 3: D and as n > o:- the distribution of the scaled random walls: W\"(t} converges to anormal distributionwith expected mean zero and 1variance t while making the central limit assumption Additionally, this will form a standard Brownian motion that has continuous paths that are sanilar_1_;o the length of intervals dened in the half line- The characteristic rnotion for this scaled symmetrical random walk will be H- @15- HIE) : E-[Eix] = Z Eixkp(x) For t .2 {II and as u > (no this function will he a rigorous Brownian motion of the process W\"(t} and it will be nowhere dierentiahle