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Let $X$ denote a real-valued random variable with distribution function $F$ and characteristic function $\phi$. Suppose that $\phi$ satisfies the following condition:

$$\lim_{T\to\infty}\int_{-T}^{T}\phi(t) ~dt = 2.$$

What can be said about the distribution?

Attempt:

The distribution is symmetric across the $x = 0$ axis (?)

The distribution is absolutely continuous.

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statsguyz
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1 Answers1

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Remember that $$ F_X(a)-\lim_{x\uparrow a}F_X(x) = \lim_{T\to\infty} \frac{1}{2T} \int_{-T}^T e^{-ita}\varphi_X(t)\,dt \, . \qquad (*) $$

If you make $a=0$ in this formula, what happens? The product rule for limits will be useful. Can you use this to say something about the distribution function at the origin?

Zen
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    What I can see is that if a = 0, then the $e^{-ita}$ becomes 1. As T approaches infinity, the $\frac{1}{2T}$ term becomes zero. I must be missing something. – statsguyz Jun 28 '14 at 07:54
  • If $\lim_{x\to\infty}g(x)=0$ and $\lim_{x\to\infty}h(x)=2$, what can you say about $\lim_{x\to\infty}g(x)h(x)$? – Zen Jun 28 '14 at 12:35
  • Also, since $F_X$ is right continuous (check this: http://stats.stackexchange.com/questions/25238/how-can-i-prove-that-the-cumulative-distribution-function-is-right-continuous), the LHS of $(*)$ is the size of the jump of $F_X$ at $a$. – Zen Jun 28 '14 at 12:40
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    Suppose that $$\lim_{x\to c} f(x) =L$$ and $$\lim_{x\to c} g(x) =M$$. Then $$ \lim_{x\to c} [f(x) g(x)] = \lim_{x\to c} f(x) \lim_{x\to c} g(x) = L M $$ This is what I have for the product rule of limits. So then the equation should equal zero if we simply have L = 2 and M = 0. – statsguyz Jun 29 '14 at 02:06
  • And you conclude that $F_X(0)-\lim_{x\uparrow 0}F_X(x)=0$, meaning that $F_X$ is left continuous at the origin, but since $F_X$ is always righ continuous, you have that $F_X$ is continuous at the origin, which is equivalent to $\Pr(X=0)=0$. – Zen Jun 29 '14 at 02:19
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    Ah ok I think I understand. So basically we know that this function (like all distribution functions) is right continuous, but we are able to ascertain that $Pr(X=0)=0$ from the given information. – statsguyz Jun 29 '14 at 02:47
  • Exactly. Perfect. – Zen Aug 15 '14 at 17:59