[Math] Show: $X_1, \dots, X_n$ are almost surely constant $\Leftrightarrow\sum_{i=1}^n X_i$ is almost surely constant.

independenceprobability theoryrandom variables

Let $X, X_1, \dots, X_n$ be independent, real random variables. I need to show:

  1. The pair $(X,X)$ is independent $\Leftrightarrow$ $X$ is almost surely constant
  2. $X_1, \dots, X_n$ are almost surely constant $\Leftrightarrow\sum_{i=1}^n X_i$ is almost surely constant.

I know that if a random variable $X$ is almost surely constant it means that there exist a real number $a$ with $P(X=a)=1$. I've proven the $\Rightarrow$ direction of 2. but I don't get anywhere with the rest. Also I'm not quite sure what it means that the "pair $(X,X)$ is independent"
Does anyone have some tipps or ideas? Thanks in advance!

Best Answer

To deal with first point, it's nesessary to know what definition of independence of two random values you can use.

Say, for the following definition: $X$ is idependent on $X$, iff for any $x,y\in\mathbb R$ $$\mathbb P(X\leq x, X\leq y)=\mathbb P(X\leq x)\mathbb P(X\leq y).$$ Таke $x=y$ be any real number. Then $$\mathbb P(X\leq x)=\left(\mathbb P(X\leq x)\right)^2,$$ that is CDF $F_X(x)=\mathbb P(X\leq x)$ is either $0$ or $1$. Since it is non-decreasing and have zero limit at $-\infty$, unit limit at $+\infty$ and right-continuous at any point, there exists some $a$ s.t. $F_X(x)=0$ for $x<a$ and $F_X(x)=1$ for $x\geq a$. This is the CDF of degenerate distribution $\mathbb P(X=a)=1$.

For second point. If this is allowed, you can use the properties of variance. $$ 0=\text{Var}(X_1+\ldots+X_n)=\text{Var}(X_1)+\ldots+\text{Var}(X_n)\geq 0 $$ The r.h.s. is zero only if all the summands are zero. And $\text{Var}(X_i)=0$ implies that $X_i$ is a.s. constant.

EDIT: As Did pointed out, the equality between variance of the sum and te sum of variances requires square integrability of summands.

Prove the second point without additional requrements. Consider the case of two summands only.

Let $X$ and $Y$ are independent and $X+Y$ is a constant a.s. Prove that both $X,Y$ are a.s. constant.

By contradiction, let the distribution of $X$ is non-degenerate. Then there exists two disjoint intervals $[a,b]$ and $[c,d]$, where $b<c$ s.t. $$\mathbb P(X\in[a,b])>0\quad \text{ and } \quad \mathbb P(X\in[c,d])>0.$$

Fix $\varepsilon>0$ s.t. $b+\varepsilon<c$. For $Y$, there exists some $z$ s.t. $$\mathbb P(Y\in[z,z+\varepsilon])>0.$$ Then the intersection of independent events $\{a\leq X \leq b\}$ and $\{z\leq Y\leq z+\varepsilon\}$ implies that $$a+z \leq X+Y \leq b+z+\varepsilon < c+z,$$ and therefore $$ \mathbb P(a+z \leq X+Y < c+z)\geq \mathbb P(a+z \leq X+Y\leq b+z+\varepsilon )\geq \mathbb P(a\leq X \leq b)\mathbb P(z\leq Y\leq z+\varepsilon)>0. $$ Also $$ \mathbb P(c+z \leq X+Y\leq d+z+\varepsilon )\geq \mathbb P(c\leq X \leq d)\mathbb P(z\leq Y\leq z+\varepsilon)>0. $$ We prove that there exists two disjoint intervals $[a+z, c+z)$ and $[c+z,d+z+\varepsilon]$ s.t. probabilities of $X+Y$ belong to any of these intervals are positive, and then the distribution of $X+Y$ is non-degenerate too. Contradiction.

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