Suppose that $(X_m)_{m=1}^k,(Y_m)_{m=1}^k$ are sequences of independent and uniform on $[0,1]$ random variables. I am trying to find a coupling of the sequences such that:
$$\sum_{m=1}^nX_m<1\iff (Y_m)_{m=1}^n \text{ is monotone}$$
Well, I am struggling with it, I thought of determining the random variables based of those who came before them in the sequence - I mean, after knowing the value of $X_1,Y_1$ we determine the values of $X_2,Y_2$ etc...
Hints will be appreciated
