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Is the records a Markov chain?

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0

$begingroup$

Let $X_1, X_2, dots$ be independent random variables such that $P{X_i = j} = alpha_j, j geq 0$. 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{
begin{array}{ll}
0 quad i geq j \
alpha_j/sum_{k=i+1}^{infty} alpha_k quad i < j \
end{array}
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

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asked Mar 13 at 8:41

charmpeachcharmpeach

12

$endgroup$

add a comment | 

0

$begingroup$

Let $X_1, X_2, dots$ be independent random variables such that $P{X_i = j} = alpha_j, j geq 0$. 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{
begin{array}{ll}
0 quad i geq j \
alpha_j/sum_{k=i+1}^{infty} alpha_k quad i < j \
end{array}
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

asked Mar 13 at 8:41

charmpeachcharmpeach

12

$endgroup$

add a comment | 

$begingroup$

Let $X_1, X_2, dots$ be independent random variables such that $P{X_i = j} = alpha_j, j geq 0$. 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{
begin{array}{ll}
0 quad i geq j \
alpha_j/sum_{k=i+1}^{infty} alpha_k quad i < j \
end{array}
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

asked Mar 13 at 8:41

charmpeachcharmpeach

12

$endgroup$

share|cite|improve this question

asked Mar 13 at 8:41

charmpeachcharmpeach

12

share|cite|improve this question

asked Mar 13 at 8:41

charmpeachcharmpeach

12

asked Mar 13 at 8:41

charmpeachcharmpeach

12

asked Mar 13 at 8:41

charmpeachcharmpeach

12