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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. 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.2011-05-27
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    @Didier: Thanks! The notation was copied from Chung's book and at first confused me.2011-05-27
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    Typo: In my comment, replace *Likewise for $g(y^-)$ and $y* by *Likewise for $g(x^-)$ and $y*. // 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 $x$.2011-05-28

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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.2011-05-15
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    @Tim: Might want to change the title too.2011-05-15
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    @Theo: I think you should add that if the limit is non-zero then it is an odd function + a scalar.2011-05-15
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    @Asaf: Thanks, done.2011-05-15
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    @Theo: Thanks! Nice to know its relation to odd function.2011-05-15
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    @Tim: Unfortunately, I can't give you a good answer to your supplemented question (asking about symmetric probability distributions) and I suspect that $F(-x-)$ is a typo for $F(-x)$. But wait a while, someone among the strong probabilists we have here can certainly give you a better founded answer than I can.2011-05-15
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    @Theo: Thanks! I now 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.2011-05-15
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    @Tim: That makes sense, yes.2011-05-15
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    @Theo: Definitely not a typo, see my comment.2011-05-27