I made some typos below capital and lowercase - all I means I and all J means j....

X is a discrete-time continuous-state random process that satisfies the condition

fX(I+1)X(1),X(2),...,X(i)(xi+1x1,x2,...,xI)=fX(I+1)X(i)(xi+1xI)

for every i integer with I>= 1 and for all real numbers x1,x2,...,xI+1 . Intuitively, this property

say that for this random process, the state it is at time i+1 depends on the past through only

the state it was at time i.

a) Show that for every j such that 1<= j < i and real numbers x1,x2,...,xi .

fX(i)X(j),X(j+1),...,X(i1)(xixj,xj+1,...,xi1)=fX(i)X(i1)(xixi1)

b) Show that for such a process "the past and the future are independent given the present", i.e. for all i and real numbers xi1,xi,xi+1 it holds

fX(i+1),X(i1)X(i)(xi+1,xi1xI)=fX(I+1)X(i)(xi+1xI)fX(i1)X(i)(xi1xI)

c) Prove that for every j such that 1<= j < i and real numbers xj,xj+1,xj+2,...,xi it holds

fX(j)X(j+1),X(j+2),...,X(i)(xjxj+1,xj+2,...,xi)=fX(j)X(j+1)(xjxj+1)