Given a finite irreducible Markov chain with transition matrix $P$ and stationary distribution $\pi$, how would you prove the following for any initial distribution vector $v$? $$\lim_{n \to \infty} \frac{v\sum\limits_{k=1}^n P^k}{n} = \pi$$ Inside the limit is the expected number of times we visit each state over the first $n$ time steps. This is a special case of the elementary renewal theorem (see ), but I'm hoping for a direct proof from the limit above. If I assume the limit exists, I can prove that it satisfies $\pi = \pi P$, but how can I prove the limit exists? A similar question was asked , but existence of the limit was not shown.
Consider an irreducible but possibly periodic Markov chain on a finite state space with transition matrix $P$. We know there exists a unique stationary distribution $\pi$. If the Markov chain were aperiodic, we would have $P^n_{ij} \to \pi(j)$ as $n \to \infty$. This fails if the chain is periodic, but we have convergence of the Cesaro averages: $$\frac{1}{n}\sum_{k=1}^n P^k_{ij} \to \pi(j) \text{ as } n \to \infty.$$ Can anyone point me to a reference that states this fact? Every reference I've seen only considers convergence for aperiodic chains, or "fixes" periodicity by considering a lazy version of the chain. Alternatively, is there a simple way to obtain this result using the result for aperiodic chains?
pgAdmin to display multiple result sets When writing my custom function I need to manually append node name to a taxonomy term link. To get the taxonomy term part done I use:    $link_uri = taxonomy_term_uri($term);   $path_alias = drupal_get_path_alias($link_uri['path']);   This works ok, now - I want to append node name ("/" . $node_name), but instead of raw string, I want the Pathauto transliterated output, as set in the URL alias configuration, for that node context.  What function should I use to mimic that output? I looked at hook_pathauto_alias_alter(&amp;$alias, array $context) but I'm not sure that's what I need. Realtek RTL8723be doesn't have good reception Only when I'm closer to the router, it shows any signal, but when I'm just little far from it, it is not showing wifi routers signals. Evaluate $\int^1_0 \log^2(1-x) \log^2(x) \, dx$ I have no idea where to even start. WolframAlpha cant compute it either.  $$\int^1_0 \log^2(1-x) \log^2(x) \, dx$$  I think it can be done with series, but I am not sure, can someone help a little so I can get a start??  Thanks! How do I catalog files on several external hard drives that I want to store off-line? I have quite a few external hard drives that I want to store off-line, and possibly off-site.  The contents is not saved in a structured manner, however; basically  there's lots of project data for lots of projects.  Is there any software for Windows XP that will catalog all the hard drives, so I can search the catalog, find the files, and get the ID of the disk that has the content? PythonTEX not working on current Overleaf compiler Is there a way to make PythonTEX compile as of today in Overleaf V2 ?  \documentclass[]{article} \usepackage[utf8]{inputenc} \usepackage{pythontex}  \begin{document} \begin{pyverbatim} def calc_ellipticity(theta, delta):     theta = np.deg2rad(theta)     psi = np.arcsin(np.sin(2*theta) * np.sin(delta))/2     return np.tan(psi) \end{pyverbatim} \end{document}   Compiling gives : ? PythonTEX ? Limit of a particular variety of infinite product/series Why do shadows from the sun join each other when near enough? I was laying on my bed, reading a book when the sun shone through the windows on my left. I happened to look at the wall on my right and noticed this very strange effect. The shadow of my elbow, when near the pages of the book, joined up with the shadow of the book even though I wasn't physically touching it.   I have posted :The video seems to be the wrong way up, but you still get the idea of what is happening.  What is causing this? Some sort of optical illusion where the light gets bent?  Coincidentally, I have been wondering about a similar effect recently where if you focus your eye on a nearby object, say, your finger, objects behind it in the distance seem to get curved/distorted around the edge of your finger. It seems awfully related...  EDIT: I could see the bulge with my bare eyes to the same extent as in the video! The room was well light and the wall was indeed quite bright. I wanted to create a WP plugin. So I requested and my request was approved by WordPress.org But I don't want to develop that plugin anymore. How can I delete the repository forever or request its deletion. Question: Best way to derive an ellipse formula when given bunch of points (say 60 points which when connected draws an ellipse)?  Background: Using R Momocs library function  which returns confidence ellipse coordinates. Would be nice to abstract those points into a formula.  You can see my ellipse points in the image below;    Any suggestions much appreciated, Thanks \pdfendlink ended up in different nesting level than \pdfstartlink How to remove conference information from the ACM 2017 SIGCONF template? I am doing an analysis with 2 groups of patients (control and disease) whose blood pressures have been measured continuously for 24 hours. My first goal is to test the effect of GROUP on BP. In the beginning, I built the mixed effect model with TIME as random effect variable and group as fixed effect model. The model I used is listed below:  mod &lt;- lme(BP ~ factor(GROUP), random=~1+ TIME| ID, data=gdata)   However, the experimenter wants to know the effects of GROUP on the slope of blood pressure changing per hour. In another words, we need to have blood pressure slope, which is the change of blood pressure by time, as response variable in the model. And I generated another model as bellow:  mod &lt;- lme(BP ~ factor(GROUP)*TIME, random=~1+ TIME| ID , data=gdata)   However, in the new model, the time works both as random effect variable and fixed effect variable. I am not sure if this is allowed in the mixed effect model. Alt attribute inserted as empty what's the type of movement when the electron jump from level to another when gaining or losing a quantum of energy is it actually jump or move in a specific movement? I have an Asus R500VD laptop which came with the preinstalled Windows 8. As actually this computer will be used by my grandmother (and she's only using win 7) I'd like to downgrade it.  Thanks to other thread here, I was able to successfully boot computer from Windows 7 USB stick, but unfortunately during install I got a message:  A required CD/DVD drive device driver is missing. (...)   and I can't get pass it... This is what I've done:   I've downloaded all drivers from Asus Website (), unzip and copied to visible location for windows 7 installation - I've picked each one of them, none has helped. As I belive it's some kind of problem with SATA/RAID/AHCI drivers I went to Intel site and downloaded various "Intel Rapid Storage Technology" drivers - a lot of them were recognized by win7, but none of them helped.   Does anyone have an idea what can I do next? Any help would be really appreciated. \bordermatrix with brackets [ ] instead of parentheses ( ) I have the following code:  \documentclass{article} \begin{document} $ M = \bordermatrix{~ &amp; x &amp; y \cr               A &amp; 1 &amp; 0 \cr               B &amp; 0 &amp; 1 \cr} $ \end{document}   But this code makes a matrix with () and i would like it to be with [] How do i do this? What are these notices above my post? My post has been marked with an informational notice in a box above it.   What should I do about this? Who attaches notices to posts? What kinds of notices are available? What should I do if I think a notice should be added to a post? What should I do if I think a notice should be removed from a post? How to get status start_date and end_date from status_track table? I have two tables:   things:  +------+----------------+------------+ | id   | current_status | created_at | +------+----------------+------------+ | 7770 | active         | 2016-08-09 | +------+----------------+------------+  thing_status_tract:  +----------+-------------+-----------+---------------------+ | thing_id | status_from | status_to | changed_at          | +----------+-------------+-----------+---------------------+ |     7770 | incomplete  | inactive  | 2016-08-09 16:26:22 | |     7770 | inactive    | active    | 2016-08-10 12:31:04 | +----------+-------------+-----------+---------------------+    I need data in the following form. This table has a status and its corresponding start and end timestamp for a particular thing_id:  +----------+-------------+---------------------+---------------------+ | thing_id | status      | status_start_date   | status_end_date     | +----------+-------------+---------------------+---------------------+ |     7770 | incomplete  | 2016-08-09 00:00:00 | 2016-08-09 16:26:22 | |     7770 | inactive    | 2016-08-09 16:26:22 | 2016-08-10 12:31:04 | |     7770 | active      | 2016-08-10 12:31:04 | now()               | +----------+-------------+---------------------+---------------------+   How to do that with a SQL query? If $0<a<b$, prove that $a<\sqrt{ab}<\frac{a+b}{2}<b$