Filtration in Markov Chains and stopping times Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Stopping times of Markov chainsMarkov chains and natural filtrationTime homogeneous Markov chain with random timesA question about the “stopping time with respect to a set” of a Markov chainContinuous time Markov Chain's Natural FiltrationGiven a list L of N elements uniformly sampled from a set A, what is the probability that L contains every element of A?Interchanging limit and expectation for irreducible Markov chainsIrreducible Markov Chain has finite stopping-time to a finite setDetermining whether the following are Markov ChainsStrong Markov property and time-homogeneity
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Filtration in Markov Chains and stopping times
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Stopping times of Markov chainsMarkov chains and natural filtrationTime homogeneous Markov chain with random timesA question about the “stopping time with respect to a set” of a Markov chainContinuous time Markov Chain's Natural FiltrationGiven a list L of N elements uniformly sampled from a set A, what is the probability that L contains every element of A?Interchanging limit and expectation for irreducible Markov chainsIrreducible Markov Chain has finite stopping-time to a finite setDetermining whether the following are Markov ChainsStrong Markov property and time-homogeneity
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I am trying to get a better understand of filtrations, and I can't seem to find any simple, concrete examples, so I will try to make one here.
Consider a discrete-time Markov chain with states $A,B$ that we run for 3 time steps. Is the following a correct representation of the corresponding filtration?
$$mathcalF_0 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_1 = emptyset, AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_2 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_3 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,
AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
If this is correct, what are some "stopping times" in this example? Suppose $tau_A$ is the first time at which the chain is in state $A$. Then for $tau_A$ to be a stopping time we must have $tau_A=t in mathcalF_t$ for $t=0,1,2,3$? What is the event $tau_A=t$ in the above filtration?
probability markov-chains filtrations
asked Mar 27 at 15:50
theQmantheQman
45538
$endgroup$
add a comment |
$begingroup$
I am trying to get a better understand of filtrations, and I can't seem to find any simple, concrete examples, so I will try to make one here.
Consider a discrete-time Markov chain with states $A,B$ that we run for 3 time steps. Is the following a correct representation of the corresponding filtration?
$$mathcalF_0 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_1 = emptyset, AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_2 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_3 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,
AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
If this is correct, what are some "stopping times" in this example? Suppose $tau_A$ is the first time at which the chain is in state $A$. Then for $tau_A$ to be a stopping time we must have $tau_A=t in mathcalF_t$ for $t=0,1,2,3$? What is the event $tau_A=t$ in the above filtration?
probability markov-chains filtrations
asked Mar 27 at 15:50
theQmantheQman
45538
$endgroup$
add a comment |
$begingroup$
I am trying to get a better understand of filtrations, and I can't seem to find any simple, concrete examples, so I will try to make one here.
Consider a discrete-time Markov chain with states $A,B$ that we run for 3 time steps. Is the following a correct representation of the corresponding filtration?
$$mathcalF_0 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_1 = emptyset, AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_2 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_3 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,
AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
If this is correct, what are some "stopping times" in this example? Suppose $tau_A$ is the first time at which the chain is in state $A$. Then for $tau_A$ to be a stopping time we must have $tau_A=t in mathcalF_t$ for $t=0,1,2,3$? What is the event $tau_A=t$ in the above filtration?
probability markov-chains filtrations
asked Mar 27 at 15:50
theQmantheQman
45538
$endgroup$
I am trying to get a better understand of filtrations, and I can't seem to find any simple, concrete examples, so I will try to make one here.
Consider a discrete-time Markov chain with states $A,B$ that we run for 3 time steps. Is the following a correct representation of the corresponding filtration?
$$mathcalF_0 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_1 = emptyset, AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_2 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
$$mathcalF_3 = emptyset, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,
AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB,AAA,AAB,ABA,ABB, BAA,BAB,BBA,BBB, AAA,AAB,ABA,ABB,BAA,BAB,BBA,BBB$$
If this is correct, what are some "stopping times" in this example? Suppose $tau_A$ is the first time at which the chain is in state $A$. Then for $tau_A$ to be a stopping time we must have $tau_A=t in mathcalF_t$ for $t=0,1,2,3$? What is the event $tau_A=t$ in the above filtration?
probability markov-chains filtrations
probability markov-chains filtrations
asked Mar 27 at 15:50
theQmantheQman
45538
asked Mar 27 at 15:50
theQmantheQman
45538
asked Mar 27 at 15:50
theQmantheQman
45538
asked Mar 27 at 15:50
theQmantheQman
45538
asked Mar 27 at 15:50
theQmantheQman
45538
45538
add a comment |
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1 Answer
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$begingroup$
This is correct.
The event $tau_A=1$ is exactly $AAA,AAB,ABA,ABBinmathcal F_1$. Indeed, this event means that the first step leads to $A$.
The event $tau_A=2$ is exactly $BAA,BABinmathcal F_2$. This event occures when the chain is at $B$ after the first step, and then at $A$ after the second step.
Try to find the event $tau_A=3$ in $mathcal F_3$.
answered Mar 27 at 16:37
NChNCh
7,1403825
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$begingroup$
This is correct.
The event $tau_A=1$ is exactly $AAA,AAB,ABA,ABBinmathcal F_1$. Indeed, this event means that the first step leads to $A$.
The event $tau_A=2$ is exactly $BAA,BABinmathcal F_2$. This event occures when the chain is at $B$ after the first step, and then at $A$ after the second step.
Try to find the event $tau_A=3$ in $mathcal F_3$.
answered Mar 27 at 16:37
NChNCh
7,1403825
$endgroup$
add a comment |
$begingroup$
This is correct.
The event $tau_A=1$ is exactly $AAA,AAB,ABA,ABBinmathcal F_1$. Indeed, this event means that the first step leads to $A$.
The event $tau_A=2$ is exactly $BAA,BABinmathcal F_2$. This event occures when the chain is at $B$ after the first step, and then at $A$ after the second step.
Try to find the event $tau_A=3$ in $mathcal F_3$.
answered Mar 27 at 16:37
NChNCh
7,1403825
$endgroup$
add a comment |
$begingroup$
This is correct.
The event $tau_A=1$ is exactly $AAA,AAB,ABA,ABBinmathcal F_1$. Indeed, this event means that the first step leads to $A$.
The event $tau_A=2$ is exactly $BAA,BABinmathcal F_2$. This event occures when the chain is at $B$ after the first step, and then at $A$ after the second step.
Try to find the event $tau_A=3$ in $mathcal F_3$.
answered Mar 27 at 16:37
NChNCh
7,1403825
$endgroup$
This is correct.
The event $tau_A=1$ is exactly $AAA,AAB,ABA,ABBinmathcal F_1$. Indeed, this event means that the first step leads to $A$.
The event $tau_A=2$ is exactly $BAA,BABinmathcal F_2$. This event occures when the chain is at $B$ after the first step, and then at $A$ after the second step.
Try to find the event $tau_A=3$ in $mathcal F_3$.
answered Mar 27 at 16:37
NChNCh
7,1403825
answered Mar 27 at 16:37
NChNCh
7,1403825
answered Mar 27 at 16:37
NChNCh
7,1403825
answered Mar 27 at 16:37
NChNCh
7,1403825
7,1403825
add a comment |
add a comment |
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