Hi I didn't get the first line on Pg 9 of Ch-4.If the processes are independent so how can we generalise and conclude that the increments are independent as well? Kindly explain. Thank you
Hello If process A is independent of process B then changes in process A are independent of changes in process B. ie if the values of process A have no impact on / relationship with process B and vice versa, then the changes in value of process A have no impact on / relationship with the changes of value for process B. Say we have discrete valued processes then: \( P(A_{s_1} - A_{t_1} = a, B_{s_2} - B_{t_2} = b) = \Sigma_{v_1} \Sigma_{v_2} P(A_{t_1} = v_1, A_{s_1} = v_1 + a, B_{t_2} = v_2, B_{s_2} = v_2 + b) \) \( = \Sigma_{v_1} \Sigma_{v_2} P(A_{t_1} = v_1, A_{s_1} = v_1 + a) * P(B_{t_2} = v_2, B_{s_2} = v_2 + b) \) by independence of \( \{A_t\} \) and \( \{B_t\} \). \( = \Sigma_{v_1} P(A_{t_1} = v_1, A_{s_1} = v_1 + a) * \Sigma_{v_2} P(B_{t_2} = v_2, B_{s_2} = v_2 + b) \) \( = P(A_{s_1} - A_{t_1} = a) * P(B_{s_2} - B_{t_2} = b) \) therefore they are independent. Hope this helps Andy
