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Is the records a Markov chain?


Prove Markov Chain by definitionConvergence of the number of visits in a Markov ChainTransition probability matrix of Markov chainDifficult to comprehend markov chain and its characteristicsProve that $W_n := (X_n,Y_n)$ is a Markov chain and determine the transition probabilities.Symmetric states of a Markov Chainprove homogeneous markov chainHow to know this is Markov Chain or not?Markov chain transition kernel inferenceMarkov Chain: Ehrenfest













0












$begingroup$


Let $X_1, X_2, dots$ be independent random variables such that $PX_i = j = alpha_j, j geq 0$. Say that a record occurs at time $n$ if $X_n > max(X_1, dots, X_n-1)$, where $X_0 = -infty$, and if a record does occur at time $n$ call $X_n$ the record value. Let $R_i$ denote the ith record value.



(a) Argue that $R_i, i geq 1$ is a Markov chain and compute its transition probabilities.



(b) Let $T_i$ denote the time between the ith and $(i + 1)$st record. Is $T_i, i geq 1$ a Markov chain? What about $(R_i, T_i), i geq 1$? Compute transition probabilities where appropriate.



(c) Let $S_n = sum_i=1^n T_i, n geq 1$. Argue that $S_n, n geq 1$ is a Markov chain and find its transition probabilities.



The Problem was from Chapter 4 of "Stochastic Processes" by M. Ross, I've solved the first question, which is $
P_ij = left{
beginarrayll
0 quad i geq j \
alpha_j/sum_k=i+1^infty alpha_k quad i < j \
endarray
right. $



I think the $T_i$ are independent from each other(thus a trivial Markov chain), whose transition probability is its probability. But I don't know how solve the last two question exactly. Thx for help.










share|cite|improve this question







New contributor




charmpeach is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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$endgroup$
















    0












    $begingroup$


    Let $X_1, X_2, dots$ be independent random variables such that $PX_i = j = alpha_j, j geq 0$. Say that a record occurs at time $n$ if $X_n > max(X_1, dots, X_n-1)$, where $X_0 = -infty$, and if a record does occur at time $n$ call $X_n$ the record value. Let $R_i$ denote the ith record value.



    (a) Argue that $R_i, i geq 1$ is a Markov chain and compute its transition probabilities.



    (b) Let $T_i$ denote the time between the ith and $(i + 1)$st record. Is $T_i, i geq 1$ a Markov chain? What about $(R_i, T_i), i geq 1$? Compute transition probabilities where appropriate.



    (c) Let $S_n = sum_i=1^n T_i, n geq 1$. Argue that $S_n, n geq 1$ is a Markov chain and find its transition probabilities.



    The Problem was from Chapter 4 of "Stochastic Processes" by M. Ross, I've solved the first question, which is $
    P_ij = left{
    beginarrayll
    0 quad i geq j \
    alpha_j/sum_k=i+1^infty alpha_k quad i < j \
    endarray
    right. $



    I think the $T_i$ are independent from each other(thus a trivial Markov chain), whose transition probability is its probability. But I don't know how solve the last two question exactly. Thx for help.










    share|cite|improve this question







    New contributor




    charmpeach is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$














      0












      0








      0





      $begingroup$


      Let $X_1, X_2, dots$ be independent random variables such that $PX_i = j = alpha_j, j geq 0$. Say that a record occurs at time $n$ if $X_n > max(X_1, dots, X_n-1)$, where $X_0 = -infty$, and if a record does occur at time $n$ call $X_n$ the record value. Let $R_i$ denote the ith record value.



      (a) Argue that $R_i, i geq 1$ is a Markov chain and compute its transition probabilities.



      (b) Let $T_i$ denote the time between the ith and $(i + 1)$st record. Is $T_i, i geq 1$ a Markov chain? What about $(R_i, T_i), i geq 1$? Compute transition probabilities where appropriate.



      (c) Let $S_n = sum_i=1^n T_i, n geq 1$. Argue that $S_n, n geq 1$ is a Markov chain and find its transition probabilities.



      The Problem was from Chapter 4 of "Stochastic Processes" by M. Ross, I've solved the first question, which is $
      P_ij = left{
      beginarrayll
      0 quad i geq j \
      alpha_j/sum_k=i+1^infty alpha_k quad i < j \
      endarray
      right. $



      I think the $T_i$ are independent from each other(thus a trivial Markov chain), whose transition probability is its probability. But I don't know how solve the last two question exactly. Thx for help.










      share|cite|improve this question







      New contributor




      charmpeach is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      Let $X_1, X_2, dots$ be independent random variables such that $PX_i = j = alpha_j, j geq 0$. Say that a record occurs at time $n$ if $X_n > max(X_1, dots, X_n-1)$, where $X_0 = -infty$, and if a record does occur at time $n$ call $X_n$ the record value. Let $R_i$ denote the ith record value.



      (a) Argue that $R_i, i geq 1$ is a Markov chain and compute its transition probabilities.



      (b) Let $T_i$ denote the time between the ith and $(i + 1)$st record. Is $T_i, i geq 1$ a Markov chain? What about $(R_i, T_i), i geq 1$? Compute transition probabilities where appropriate.



      (c) Let $S_n = sum_i=1^n T_i, n geq 1$. Argue that $S_n, n geq 1$ is a Markov chain and find its transition probabilities.



      The Problem was from Chapter 4 of "Stochastic Processes" by M. Ross, I've solved the first question, which is $
      P_ij = left{
      beginarrayll
      0 quad i geq j \
      alpha_j/sum_k=i+1^infty alpha_k quad i < j \
      endarray
      right. $



      I think the $T_i$ are independent from each other(thus a trivial Markov chain), whose transition probability is its probability. But I don't know how solve the last two question exactly. Thx for help.







      markov-chains






      share|cite|improve this question







      New contributor




      charmpeach is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question







      New contributor




      charmpeach is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question






      New contributor




      charmpeach is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      asked Mar 13 at 8:41









      charmpeachcharmpeach

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