Pairwise independence. Let X1, . .. Xn be a sequence of random variables, suppose that Xi Xi and Xj are independent for every pair (i, j) with 1 < i="">< j="">< n.="" does="" this="" imply="">1, .. . Xn are independent? Sketch a proof or counterexample.
Sequential independence. Let X1, . . . Xn be a sequence of random variables. Suppose that for every 1 < m="">< n="" -1="" the="" random="" sequence="">1, . .. Xm) is independent of the next random variable X m+1. Does this imply X1, . . . Xn are independent? Sketch a proof or give a counterexample.