Convergence in Probability and Convergence with Probability 1


Definations:

For a random vector X and a sequence of random vectors X_{1},\cdots X_{n},\cdots defined on a probability space (\Omega,\mathcal{F},P).

1. Convergence in Probability or Convergence of probability measures, is kind of convergence in measure, in concept of the measure theory.
X_{n}\stackrel{P}{\to}X: P\{||X_{n}-X||>\epsilon\}\to0 for any fixed \epsilon>0.

2. Convergence with Probability 1, is kind of pointwise convergence in real analysis, it is also called almost surely convergence in the measure theory.
X_{n}\stackrel{a.s.}{\to}X: if P\{X_{n}\to X\}=1 or P\{X_{n}\nrightarrow X\}=0.

Convergence in Probability is weaker than Convergence with Probability 1, as stated below:

Theorem: X_{n}\stackrel{a.s.}{\to}X\Rightarrow X_{n}\stackrel{P}{\to}X

Proof: \{X_{n}\to X\}={\omega: for each \epsilon>0 there exists an N(\omega)>0 such that ||X_{k}(\omega)-X(\omega)||\leq\epsilon for all k>N(\omega)}

={\cap_{\epsilon>0}{\omega: there exists an N(\omega)>0 such that ||X_{k}(\omega)-X(\omega)||\leq\epsilon for all k>N(\omega)}}

={\cap_{\epsilon>0}\cup_{n=1}^{\infty}{\omega: ||X_{k}(\omega)-X(\omega)||\leq\epsilon for all k>N(\omega)}}

={\cap_{\epsilon>0}\cup_{n=1}^{\infty}\cap_{k>n}{\omega: ||X_{k}(\omega)-X(\omega)||\leq\epsilon}}

So X_{n}\stackrel{a.s.}{\to}X\Longleftrightarrow P\{X_{n}\to X\}=1

\Longleftrightarrow P\{\cap_{\epsilon>0}\cup_{n=1}^{\infty}\cap_{k>n}\{\omega:||X_{k}(\omega)-X(\omega)||\leq\epsilon\}\}=1

\Longleftrightarrow P\{\cup_{n=1}^{\infty}\cap_{k>n}\{\omega:||X_{k}(\omega)-X(\omega)||\leq\epsilon\}\}=1 for all \epsilon>0

\Longleftrightarrow\lim_{n\to\infty}P\{\cap_{k>n}\{\omega:||X_{k}(\omega)-X(\omega)||\leq\epsilon\}\}=1 for all \epsilon>0

\Longleftrightarrow\lim_{n\to\infty}P\{\cup_{k>n}\{\omega:||X_{k}(\omega)-X(\omega)||>\epsilon\}\}=0 for all \epsilon>0

\Rightarrow\lim_{n\to\infty}P\{||X_{n}-X||>\epsilon\}=0 for all \epsilon>0, or P\{||X_{n}-X||>\epsilon\}\to0 for any fixed \epsilon>0, that is X_{n}\stackrel{P}{\to}X.

if X_{n}\stackrel{a.s.}{\to}X, let the set of unconvergence points as U_{a.s.}=\{\omega:X_{n}(\omega)\nrightarrow X(\omega)\} then P\{U_{a.s.}\}=0.
if X_{n}\stackrel{P}{\to}X, let the set of unconvergence points as U_{P}=\{\omega:X_{n}(\omega)\nrightarrow X(\omega)\} then we donot have P\{U_{P}\}=0.

Convergence in Probability do not require P\{U_{P}\}=0 but release the condition to \lim_{n\to\infty}P\{U_{P}(n)\}=0, where U_{P}(n) is a set which will go smaller as n\to\infty.

From the proof above we can see that U_{P}=\{\cup_{\epsilon}\cap_{n=1}^{\infty}\cup_{k}\{\omega:||X_{k}(\omega)-X(\omega)||>\epsilon\}\} and U_{P}(n)=\{\omega:||X_{n}-X||>\epsilon\}, that is to say U_{P} is bigger than U_{a.s.}

References:
A Course in Large Sample Theory(Lecture notes), Xianyi Wu
A Course in Large Sample Theory, Thomas S. Ferguson

Advertisements
This entry was posted in analysis, convergence, probability. Bookmark the permalink.

24 Responses to Convergence in Probability and Convergence with Probability 1

  1. Emilio says:

    Remarkable issues here. I’m very glad to see your article. Thank you so much and I’m taking a look forward to contact you.

    Will you kindly drop me a mail?

  2. Vimax Diet says:

    Does your site have a contact page? I’m having trouble locating it but, I’d like to send you an email.
    I’ve got some ideas for your blog you might be interested in hearing. Either way, great site and I look forward to seeing it develop over time.

  3. Hi, its good post on the topic of media print, we all be familiar with media is a great source of information.

  4. Today, I went to the beach with my children. I found a sea shell and gave it to my 4 year old daughter and said
    “You can hear the ocean if you put this to your ear.” She placed the shell to her ear and screamed.
    There was a hermit crab inside and it pinched her ear.
    She never wants to go back! LoL I know this is entirely off topic
    but I had to tell someone!

  5. Attractive component to content. I just stumbled upon your website and in accession capital to claim that I get in fact loved account your weblog posts.
    Any way I will be subscribing on your augment or even I achievement you get admission to consistently fast.

  6. Fabulous, what a web site it is! This blog gives helpful information
    to us, keep it up.

  7. hi!,I love your writing so so much! percentage we
    keep up a correspondence extra about your post on AOL?

    I need an expert on this house to solve my problem.
    Maybe that’s you! Looking ahead to see you.

  8. Howdy! Do you know if they make any plugins to protect against hackers?
    I’m kinda paranoid about losing everything I’ve
    worked hard on. Any suggestions?

  9. Hey There. I discovered your weblog the use of msn.
    This is a very neatly written article. I will make sure to bookmark it and return to read extra
    of your useful information. Thanks for the post. I’ll definitely return.

  10. I usually do not write a leave a response, however I glanced through some remarks on Convergence in Probability and Convergence with Probability
    1 | Bruce Zhou on Statistics. I do have a few questions for you if you tend not to mind.

    Could it be simply me or do some of these responses look like they are
    left by brain dead visitors? 😛 And, if you are writing at other sites, I
    would like to follow you. Could you list of the complete urls of all your public
    pages like your Facebook page, twitter feed, or linkedin profile?

  11. reubendominquez says:

    I’m not that much oof a internet reader to be honest but you sites really nice, keep it up!
    I’ll go ahead and bookmark your website to come back later.
    All the best

  12. cliffordy50ai says:

    This is a topic that is near to my heart… Best wishes!
    Exactly where are your contact details though?

  13. I loved as much as you’ll receive carried out right here.
    The sketch is tasteful, your authored material stylish. nonetheless, you command get got an
    edginess over that you wish be delivering the following.

    unwell unquestionably come more formerly again as exactly the same nearly very often inside case you shield
    this increase.

  14. thyroid cure says:

    What you published was actually very logical.
    However, think about this, suppose you were to write a killer headline?
    I ain’t saying your content isn’t solid, however what if you added a headline to maybe grab a person’s attention?
    I mean Convergence in Probability and Convergence with Probability 1 | Bruce Zhou on Statistics is kinda vanilla.
    You could look at Yahoo’s front page and see how they
    create news headlines to get viewers to click.

    You might add a related video or a picture or two to grab people excited about what you’ve written.

    In my opinion, it would bring your posts a little livelier.

  15. henna artist says:

    This post is actually a nice one it assists
    new web people, who are wishing in favor of blogging.

  16. Hi, the whole thing is going sound here and ofcourse every one is sharing data, that’s really good,
    keep up writing.

  17. Jetta says:

    Fantastic items from you, man. I have bear in mind your stuff previous to and you’re simply too
    magnificent. I really like what you’ve acquired here, really like what you’re stating and the best
    way in which you are saying it. You are making it enjoyable and you continue to take care of to stay it sensible.

    I can not wait to read much more from you.
    That is actually a great web site.

  18. It is like the real guitar greats who spent 12 to16 hours a day, day in day out for years and who attained legendary status.
    For more information about how to find a good iron visit.
    A great way to identify the most valuable home improvement jobs is to chat with
    a realtor in your area.

  19. mobile games says:

    Hiya very nice website!! Guy .. Beautiful .. Amazing ..

    I’ll bookmark your blog and take the feeds additionally?
    I’m glad to seek out so many useful information here within the put up, we
    want develop extra strategies in this regard, thank you
    for sharing. . . . . .

  20. You ought to take part in a contest for one of the finest blogs on the internet.
    I most certainly will recommend this web site!

  21. Magnificent website. A lot of helpful information here.
    I’m sending it to some friends ans additionally sharing in delicious.

    And of course, thanks to your effort!

  22. This is a gokd tip particularly tto those new to the blogosphere.
    Short but very precise info… Thanks for sharing this one.
    A must resad post!

  23. gta 5 says:

    Its not my first time to visit this website, i am visiting
    this website dailly and get good data from here everyday.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s