Re: Questions: the central limit theorem when n is a random variable.
- From: hrubin@xxxxxxxxxxxxxxxxxxxx (Herman Rubin)
- Date: 26 Nov 2005 12:35:33 -0500
In article <3625480.1133021540101.JavaMail.jakarta@xxxxxxxxxxxxxxxxxxxxxx>,
zql <zqlict@xxxxxxxxxxx> wrote:
>Dear miss/sir:
> I feel very gloomy when I was faced with the following questions:
> The central limit theorem:
> Assume that Xi (i=1,2...) is i.i.d., and E(xi) and Var(xi) exist, then Sn=(X1+...Xn) appoaches normal distribution when n approaches inf.
> Noting that n is a variable in this theorem, My Question is the following:
> Assume that
> (1) X(i) is i.i.d., and all moments of X(i) exist.
> (2) N(t) is a random variable.
> (3) for any i, N(t) is independent of X(i).
> (4) N(t) goes to inf. if t goes to inf..
> (5) S[N(t)]=(x(1)+...+x(N(t)))
> Question 1:
> Does S[N(t)]approaches Normal distribution?
Not necessarily. This is easily seen; if N(t) = m with
probability 1/2, and 2m with probability 1/2, the
distribution is clearly not normal.
> Question 2:
> if (2)and (3) become (2'): N(t) is a stopping time of the sequence {x(i)},
> does S[N(t)]approaches Normal distribution?
A similar argument holds. The above is a stopping time,
but also N(t) can be defined by X(1) to have the properties
above.
> I am very grateful if you can give me any suggestions or recommend some books about the questions to me!
If the means are 0, the important property is that N(t)
is "close" to a constant. If not, it will itself have
to be not only close to a constant, but approximately
normally distributed itself.
> Thank for your attention very much!
>Best wishes.
>Zhao Qinglin
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@xxxxxxxxxxxxxxx Phone: (765)494-6054 FAX: (765)494-0558
.
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