Independence of the indicator functions of record values for Multivariate independent data

We consider a sequence of random vectors on $\mathbb{R}^d, \ d\geq 1$. We consider the record values based on the simultaneous strict inequality of the coordinates. The indicator record variable (irv) of the j-th observation is the function that assigns the value 1 (one) if that observation is a record value and the null value otherwise. Here, we give a detailed and a thorough proof that the indicator functions are independent, whenever the data are themselves independent, not necessarily <i>iid</i>, in $\mathbb{R}^d, \ d\geq 1$. We compare that proof with available proofs in dimension one. Indeed, in seminal works on records, in particular in Ahsanullah(2024), Nevzorov(2001), Resnick (1987), Ahsanullah and Nevzorov (2015), etc., the independence of record indicator functions is usually validated based on logical reasoning, and so, is not rigorously proved. This allows us to undertake a detailed and thorough proof for independent data, not necessity <i>iid</i>, in $\mathbb{R}^d, \ d\geq 1$. The proof includes leads for not even independent data.