Question

1.the distribution of the processes are different though they have same finite dimensional distributions give a example

Let {Xn : n ≥ 0} denote the random walk on 9 cycle. Express it as a random walk on a group (G, ·) with transition probabilities given by pxy = µ(y · x −1 ) for an appropriate distribution µ on G.

Answer 1

Answer:-

Given That:-

1.the distribution of the processes are different though they have same finite dimensional distributions give a example?

Let U be a uniform [0,1]R.variable defined on a prob space Define for

and   for all and  

then

both have same finite distribution .

Now

we look at events

then

Here for

i.e.,

the distribution of the process are difference but they have same finite dimensional distributions

Let {Xn : n ≥ 0} denote the random walk on 9 cycle. Express it as a random walk on a group (G, ·) with transition probabilities given by pxy = µ(y · x −1 ) for an appropriate distribution µ on G.?

g cycle group is   

addition module g

  

as we can move one step forward or one step backward.

if then s.t.

x=2,