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I was wondering if there is a name for function $F: \mathbb{R} \rightarrow \mathbb{R}$ with property $$F(x)+F(-x)=\lim_{y\rightarrow\infty}F(y), \forall x \in \mathbb{R}?$$

An example is when $F$ is the probability distribution function of a symmetric density function. A random variable with such probability distribution function is called symmetric.

I would like to know if properties of such kind of functions have been studied both generally and especially for probability theory i.e. when $F$ is a probability distribution function.

Added: Also I wonder in Kai Lai Chung's A Course in Probability Theory where he wrote for the probability distribution function of a symmetric distribution $$F(x)=1-F(-x-), \forall x \in \mathbb{R},$$ what is the meaning of $-x-$? I guess that because a probability distribution function is a right-continuous function, maybe Chung wanted to emphasize that it is the left limit that is used to define symmetric distribution.

Thanks and regards!

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Yes, the notation $g(x^+)$ is routinely used as a shorthand for the limit of $g(y)$ when $y\to x$ restricted to $y>x$. Likewise for $g(y^-)$ and $y<x$. However, usually the $+$ and $-$ signs are written as exponents and not like in your post. Finally, note that one cannot replace the condition $F(x)=1-F((-x)^-)$ by $F(x)=1-F(-x)$, otherwise the distributions of symmetric Bernoulli random variables would not be... well, symmetric. –  Did May 27 '11 at 20:51
    
@Didier: Thanks! The notation was copied from Chung's book and at first confused me. –  Tim May 27 '11 at 21:32
    
Typo: In my comment, replace *Likewise for $g(y^-)$ and $y<x$* by *Likewise for $g(x^-)$ and $y<x$*. // Remark: Chung **defines** the notations $g(x+)$ and $g(x-)$ on page 2 of Chapter 1 Section 1. // Chung uses $g(x+)$ and $g(x-)$ for the right- and left-limits of $g$ at $x$, probably to avoid a conflict of notations with $x^+$ and $x^-$ the positive and negative parts of the real number $x$. –  Did May 28 '11 at 8:01

1 Answer 1

up vote 5 down vote accepted

If we assume that the limit on the right hand side is zero, then $F$ is called antisymmetric or odd, that is, it satisfies $F(x) = -F(-x)$.

Now let's assume that the limit $c = \lim_{x \to \infty} F(x) + F(-x)$ exists. Then $G(x) = F(x) - \frac{c}{2}$ satisfies $G(x) + G(-x) = F(x) + F(-x) - c$, hence $\lim_{y \to \infty} G(y) + G(-y) = 0$, so $G$ is odd by the previous reasoning, hence $F(x) = G(x) + c/2$ is the sum of an odd function and the scalar $c/2$. Odd functions arise quite frequently and a bunch of their properties can be found on the Wikipedia page linked to above.

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Thanks! Sorry about some confusion and I just made some change. –  Tim May 15 '11 at 18:08
    
@Tim: Might want to change the title too. –  Ben Alpert May 15 '11 at 18:09
    
@Theo: I think you should add that if the limit is non-zero then it is an odd function + a scalar. –  Asaf Karagila May 15 '11 at 18:14
    
@Asaf: Thanks, done. –  t.b. May 15 '11 at 18:22
    
@Theo: Thanks! Nice to know its relation to odd function. –  Tim May 15 '11 at 18:41

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