How to prove that $P(Z_1 = j| Z_0 = i) = P(Z_n+1 = j| Z_n = i)?$

The name of the pictureThe name of the pictureThe name of the pictureClash Royale CLAN TAG#URR8PPP











up vote
0
down vote

favorite












Currently I am reading this Markov Chains' notes.
At page $112,$ section $4.2$ Transition matrix, I fail to understand how author obtains the following.




As seen above, the random evolution of a Markov chain $(Z_n)_nin mathbbN$ is determined
by the data of
$$P_i,j := P(Z_1 = j | Z_0 = i), i, j in mathbbS,$$
which coincides with the probability $P(Z_n+1 = j | Z_n = i)$ which is independent of $n in mathbbN.$




I do not understand how author obtains the following equality
$$P(Z_1 = j| Z_0 = i) = P(Z_n+1 = j| Z_n = i).$$
I think Markov property might be applied here, but I do not know how.



Any hint is appreciated.










share|cite|improve this question

























    up vote
    0
    down vote

    favorite












    Currently I am reading this Markov Chains' notes.
    At page $112,$ section $4.2$ Transition matrix, I fail to understand how author obtains the following.




    As seen above, the random evolution of a Markov chain $(Z_n)_nin mathbbN$ is determined
    by the data of
    $$P_i,j := P(Z_1 = j | Z_0 = i), i, j in mathbbS,$$
    which coincides with the probability $P(Z_n+1 = j | Z_n = i)$ which is independent of $n in mathbbN.$




    I do not understand how author obtains the following equality
    $$P(Z_1 = j| Z_0 = i) = P(Z_n+1 = j| Z_n = i).$$
    I think Markov property might be applied here, but I do not know how.



    Any hint is appreciated.










    share|cite|improve this question























      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Currently I am reading this Markov Chains' notes.
      At page $112,$ section $4.2$ Transition matrix, I fail to understand how author obtains the following.




      As seen above, the random evolution of a Markov chain $(Z_n)_nin mathbbN$ is determined
      by the data of
      $$P_i,j := P(Z_1 = j | Z_0 = i), i, j in mathbbS,$$
      which coincides with the probability $P(Z_n+1 = j | Z_n = i)$ which is independent of $n in mathbbN.$




      I do not understand how author obtains the following equality
      $$P(Z_1 = j| Z_0 = i) = P(Z_n+1 = j| Z_n = i).$$
      I think Markov property might be applied here, but I do not know how.



      Any hint is appreciated.










      share|cite|improve this question













      Currently I am reading this Markov Chains' notes.
      At page $112,$ section $4.2$ Transition matrix, I fail to understand how author obtains the following.




      As seen above, the random evolution of a Markov chain $(Z_n)_nin mathbbN$ is determined
      by the data of
      $$P_i,j := P(Z_1 = j | Z_0 = i), i, j in mathbbS,$$
      which coincides with the probability $P(Z_n+1 = j | Z_n = i)$ which is independent of $n in mathbbN.$




      I do not understand how author obtains the following equality
      $$P(Z_1 = j| Z_0 = i) = P(Z_n+1 = j| Z_n = i).$$
      I think Markov property might be applied here, but I do not know how.



      Any hint is appreciated.







      probability markov-chains markov-process






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Sep 5 at 7:46









      Idonknow

      3,115643109




      3,115643109




















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote













          You cannot prove this. Many books consider what are called homogeneous Markov chain. For these what you are trying to prove is part of the definition. There are non-homogeneous MC's so Markov property does not imply this property.






          share|cite|improve this answer




















            Your Answer




            StackExchange.ifUsing("editor", function ()
            return StackExchange.using("mathjaxEditing", function ()
            StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
            StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
            );
            );
            , "mathjax-editing");

            StackExchange.ready(function()
            var channelOptions =
            tags: "".split(" "),
            id: "69"
            ;
            initTagRenderer("".split(" "), "".split(" "), channelOptions);

            StackExchange.using("externalEditor", function()
            // Have to fire editor after snippets, if snippets enabled
            if (StackExchange.settings.snippets.snippetsEnabled)
            StackExchange.using("snippets", function()
            createEditor();
            );

            else
            createEditor();

            );

            function createEditor()
            StackExchange.prepareEditor(
            heartbeatType: 'answer',
            convertImagesToLinks: true,
            noModals: false,
            showLowRepImageUploadWarning: true,
            reputationToPostImages: 10,
            bindNavPrevention: true,
            postfix: "",
            noCode: true, onDemand: true,
            discardSelector: ".discard-answer"
            ,immediatelyShowMarkdownHelp:true
            );



            );













             

            draft saved


            draft discarded


















            StackExchange.ready(
            function ()
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2905986%2fhow-to-prove-that-pz-1-j-z-0-i-pz-n1-j-z-n-i%23new-answer', 'question_page');

            );

            Post as a guest






























            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote













            You cannot prove this. Many books consider what are called homogeneous Markov chain. For these what you are trying to prove is part of the definition. There are non-homogeneous MC's so Markov property does not imply this property.






            share|cite|improve this answer
























              up vote
              1
              down vote













              You cannot prove this. Many books consider what are called homogeneous Markov chain. For these what you are trying to prove is part of the definition. There are non-homogeneous MC's so Markov property does not imply this property.






              share|cite|improve this answer






















                up vote
                1
                down vote










                up vote
                1
                down vote









                You cannot prove this. Many books consider what are called homogeneous Markov chain. For these what you are trying to prove is part of the definition. There are non-homogeneous MC's so Markov property does not imply this property.






                share|cite|improve this answer












                You cannot prove this. Many books consider what are called homogeneous Markov chain. For these what you are trying to prove is part of the definition. There are non-homogeneous MC's so Markov property does not imply this property.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Sep 5 at 8:05









                Kavi Rama Murthy

                26.3k31437




                26.3k31437



























                     

                    draft saved


                    draft discarded















































                     


                    draft saved


                    draft discarded














                    StackExchange.ready(
                    function ()
                    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2905986%2fhow-to-prove-that-pz-1-j-z-0-i-pz-n1-j-z-n-i%23new-answer', 'question_page');

                    );

                    Post as a guest













































































                    這個網誌中的熱門文章

                    How to combine Bézier curves to a surface?

                    Carbon dioxide

                    Why am i infinitely getting the same tweet with the Twitter Search API?