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I have recently been studying the convergence of a sequence of random variables. However, Let $\left\{ X_{n}\right\} _{1}^{\infty}$ be a sequence of random variables defined on $\left(\Omega,F,P\right)$ where the range of each term $X_{n}$ is the singleton set $\left\{ 1+\frac{1}{n}\right\} .$ First, I wish to be able to find $\left\{ F_{X_{n}}\right\} _{1}^{\infty}$ and $\left\{ f_{X_{n}}\right\} _{1}^{\infty}$. Secondly, I'd like to find out whether or not

a. $\left\{ X_{n}\right\} _{1}^{\infty}$ converge in distribution

b. $\left\{ X_{n}\right\} _{1}^{\infty}$ converge in probability

c. $\left\{ X_{n}\right\} _{1}^{\infty}$ converge in almost sure sense

d. $\left\{ X_{n}\right\} _{1}^{\infty}$ converge in $p$-th moment for $p\geq1$.

I know the definitions of convergence in distribution, probability and almost sure convergence. However, I do not know what convergence in $p$-th moment mean.
Please, any help on how to begin will be appreciated. Thanks.

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girdav: thanks for the comment – Mike Jun 1 '11 at 19:11
up vote 1 down vote accepted

You have the sequence of constant variables $X_n = 1+1/n$ ($X_n (\omega) = 1+1/n$ for any $\omega \in \Omega$). One should expect that it converges to $1$ as $n \to \infty$, in all modes. For a) you need to show that $$ \mathop {\lim }\limits_{n \to \infty } P(1 + 1/n \le x) = P(1 \le x), $$ for any $x \neq 1$. For b) you need to show that $$ \mathop {\lim }\limits_{n \to \infty } P(|(1 + 1/n) - 1| > \varepsilon ) = 0, $$ for any $\varepsilon > 0$. For c) you need to show that $$ P(\lim _{n \to \infty } (1 + 1/n) = 1) = 1. $$ For d) you need to show that $$ \mathop {\lim }\limits_{n \to \infty } {\rm E}|(1 + 1/n) - 1|^p = 0. $$

EDIT: Concerning the sequence of distribution functions $(F_{X_n})$, note that $$ F_{X_n } (x): = P(X_n \le x) = P(1 + 1/n \le x). $$ The distribution function of the limit $X=1$ is given by $F_X (x)=P(1 \leq x)$, $x \in \mathbb{R}$. It is important to remember that $X_n$ converges in distribution to $X$ if and only if $F_{X_n}(x) \to F_X (x)$ (as $n \to \infty$) for any $x \in \mathbb{R}$ which is a continuity point of $F_X$. (Note that, in our case, $x$ is a continuity point of $F_X$ if and only if $x \neq 1$.)

Finally, $X_n$ (like any other discrete random variable) does not have a probability density function; hence, $f_{X_n}$ does not exist.

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Thanks for your answer. How do I find $\left\{ F_{X_{n}}\right\} _{1}^{\infty}$ and $\left\{ f_{X_{n}}\right\} _{1}^{\infty}$ – Mike Jun 1 '11 at 18:39
Mike: I'll add details concerning this question. – Shai Covo Jun 1 '11 at 18:42
Details added... – Shai Covo Jun 1 '11 at 19:07
Thanks very much. I really aprreciate your help. – Mike Jun 1 '11 at 19:16
Mike: Thanks, glad to help. – Shai Covo Jun 1 '11 at 19:19

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