<p>I suppose the answer to my question follows from the proof of the <a href="http://en.wikipedia.org/wiki/L%C3%A9vy%27s_continuity_theorem">Lévy’s continuity theorem</a>,
but probably one could suggest me a direct reference to the corresponding answer.</p>

<p>The question is as follows. Suppose the sequence of characteristic functions (c.f.) $\phi_n$ of random variables $X_n$ converge point-wise to some c.f. $\phi$ of random variable $X$ with convergence rate $\alpha(x,n)$, i.e. $\phi(x) = \phi_n(x) + O(\alpha(x,n))$ where $\alpha(x,n)$ tends to $0$ while $n$ tends to infinity. Does this implies that $P(X_n \leq x) = P(X \leq x) + O(\alpha(x,n))$?</p>

CC BY-SA 3.0
2013-06-22T11:45:00.357Z
You probably already now that convergence in distribution of $X_n$ to $X$ is equivalent to the convergence of cumulative distribution function at the points of continuity of $F_X$ (cdf of $X$). So here, convergence may fail when we consider $X$ with a non continuous cdf (for example, $X_n=1/n$ and $X=0$).
Davide Giraudo
CC BY-SA 3.0
2013-06-23T10:20:48.353Z
Thank you for your remark, but I am only interested in the case when cdfs of $X_n$ and $X$ are continious.
Mikhail Berlinkov
CC BY-SA 3.0
2013-06-23T10:21:50.010Z
Yes, I guessed it was you had in mind. I will think on it.
Davide Giraudo
CC BY-SA 3.0
2013-07-05T16:29:56.490Z
I think you might get a faster answer to this on [math.se] or MO. Let me know if you want to migrate this to one of them. I can migrate it to [math.se] if you want, however, for MO we have to first check with them that the migration is fine.
Kaveh
CC BY-SA 3.0
2013-07-10T07:30:32.590Z
@Kaveh: Yes, I suppose the question is more appropriate for Mathematics
Mikhail Berlinkov
CC BY-SA 3.0
2013-07-10T07:40:05.180Z
So would you like the question migrated to [math.se]?
Kaveh
CC BY-SA 3.0
2013-09-05T08:11:20.067Z
@Kaveh: Yes, I would.
Mikhail Berlinkov
<reference-request><pr.probability>
Convergence speed in Lévy’s continuity theorem
Mikhail Berlinkov
# Convergence speed in Lévy’s continuity theorem

(Asked by: username_0 on 2013-05-30)

I suppose the answer to my question follows from the proof of the [Lévy’s continuity theorem](http://en.wikipedia.org/wiki/L%C3%A9vy%27s_continuity_theorem),
but probably one could suggest me a direct reference to the corresponding answer.

The question is as follows. Suppose the sequence of characteristic functions (c.f.) $\phi\_n$ of random variables $X\_n$ converge point-wise to some c.f. $\phi$ of random variable $X$ with convergence rate $\alpha(x,n)$, i.e. $\phi(x) = \phi\_n(x) + O(\alpha(x,n))$ where $\alpha(x,n)$ tends to $0$ while $n$ tends to infinity. Does this implies that $P(X\_n \leq x) = P(X \leq x) + O(\alpha(x,n))$?

**Tags:** reference-request, pr.probability

> **Comments:**
> **username_1**: You probably already now that convergence in distribution of $X_n$ to $X$ is equivalent to the convergence of cumulative distribution function at the points of continuity of $F_X$ (cdf of $X$). So here, convergence may fail when we consider $X$ with a non continuous cdf (for example, $X_n=1/n$ and $X=0$).  
> **username_0**: Thank you for your remark, but I am only interested in the case when cdfs of $X_n$ and $X$ are continious.  
> **username_1**: Yes, I guessed it was you had in mind. I will think on it.  
> **username_2**: I think you might get a faster answer to this on [math.se] or MO. Let me know if you want to migrate this to one of them. I can migrate it to [math.se] if you want, however, for MO we have to first check with them that the migration is fine.  
> **username_0**: @username_2: Yes, I suppose the question is more appropriate for Mathematics  
> **username_2**: So would you like the question migrated to [math.se]?  
> **username_0**: @username_2: Yes, I would.  

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## 0 answers

