# Sums and differences with three independent random variables

Suppose I have three mutually independent random variables $$X_1$$, $$X_2$$, $$X_3$$. Let $$Y_1 = X_1-X_3$$, and let $$Y_2 = X_2-X_3$$. Are $$Y_1$$ and $$Y_2$$ are dependent or independent?

Edit 1: If this is not true in general, what extra assumptions would need to be added to guarantee independence of $$Y_1$$ and $$Y_2$$?

Edit 2: Are there generalizations of this concept to a collection of $$n$$ mutually independent random variables? By this I mean the following: take $$X_1, \dots, X_n$$ mutually independent random variables $$(n\in \mathbb{N}, n>3)$$, and let $$Y_i = X_i - X_n$$, $$i\in S = \{1,\dots,n-1\}$$. Under which conditions would the $$\{Y_i\}_S$$ be mutually independent?

They may be dependent. Put $$X_1 = X_2 = 0$$, $$X_3$$ - Bernoulli random Variable.

There's a notion "n independent random variables", which may be found in every book on probability.

Addition: if $$Y_1$$ and $$Y_2$$ are independent then the characteristic function of $$(X_1 - X_3, X_2 - X_3)$$ is

$$Ee^{i \bigl( a(X_1-X_3) + b(X_2-X_3) \bigr) } = Ee^{i a(X_1-X_3)} \cdot Ee^{i b(X_2-X_3)} \ \ \ (\star)$$ As $$X_1, X_2, X_3$$ are independent left-hand side of ($$\star$$) is $$Ee^{i \bigl( a X_1 + b X_2 -(a+b) X_3 \bigr) }=Ee^{i a X_1} Ee^{i b X_2} Ee^{i ( -(a+b) X_3)}$$ and right-hand side of $$(\star)$$ is $$Ee^{i a X_1 } Ee^{i (-a)X_3} Ee^{i b X_2} Ee^{i (-b)X_3}.$$ Thus $$Ee^{i a X_1} Ee^{i b X_2} Ee^{i ( -(a+b) X_3)} = Ee^{i a X_1 } Ee^{i (-a)X_3} Ee^{i b X_2} Ee^{i (-b)X_3}.$$ Suppose that $$Ee^{i a X_1} = 0$$ only for $$a$$ from a set of measure $$0$$. Also suppose that $$Ee^{i b X_2} = 0$$ only for $$b$$ from a set of measure $$0$$. These assumptions holds true for all popular distributions. We have $$Ee^{i ( -(a+b) X_3)} = Ee^{i (-a)X_3} Ee^{i (-b)X_3}$$ for almost all $$a$$ and $$b$$. Characteristic functions are continious, hence $$Ee^{i ( -(a+b) X_3)} = Ee^{i (-a)X_3} Ee^{i (-b)X_3}$$ for all $$a$$ and $$b$$. Put $$t = -a, s = -b$$. Thus $$Ee^{i (t+s) X_3} = Ee^{i t X_3} Ee^{i s X_3}$$ for all $$t$$ and $$s$$. We got that the characteristic function of a vector $$(X_3, X_3)$$ is equal to the product of characteristic functions of it's components. Hence $$X_3$$ and $$X_3$$ are independent and hence $$X_3 = const$$.

If $$X_3 = const$$ then $$Y_1$$ and $$Y_2$$ are obviously independent.

So the condition that you need is the condition $$X_n = const$$.

Remark: there are characteristic functions which are equal to $$0$$ on the set of positive measure, e.g. $$(1-|t|)I_{|t| < 1}$$. It follows from Polya theorem about characteristic functions.

• thank you for your response. please see edits to the question that clarify its intent. Oct 15, 2021 at 20:12