Markov Chains¶
Norris, J. R. (1997). Markov Chains. Cambridge University Press.
Cited by¶
5 citations across 5 artifacts.
Each citation links to the sentence it supports in the citing article.
Primes¶
- Markov Process
- The structural claim is sharp and almost paradoxically strong: a single, sufficiently-rich present state is a complete summary of all history relevant to the future, so prediction requires only the state now and a transition rule, not the trajectory that produced it.
This sourceCanonical modern textbook: develops the Markov property as a conditional-independence statement (present state screens off the past), the state–transition-kernel structure, sufficient-statistic framing, and long-run behavior (stationary distributions, mixing, recurrence/absorption) determined by the transition structure alone.
Supported in partVerified against the publisher's abstract
The textbook abstract shows Markov chains applied across genetics, queues and other domains, backing the substrate-independence frame but not the screening-off mechanism itself.
“There are applications to simulation, economics, optimal control, genetics, queues and many other topics, and exercises and examples drawn both from theory and practice.”
- The structural claim is sharp and almost paradoxically strong: a single, sufficiently-rich present state is a complete summary of all history relevant to the future, so prediction requires only the state now and a transition rule, not the trajectory that produced it.
- Stochastic Process
- The Markov toolkit — transition kernels, stationary distributions, mixing times, absorbing-state analysis — transfers from statistical physics (the Metropolis dynamics) to NLP (hidden Markov models) to operations (queue occupancy) to PageRank, because all are Markov processes and the present-screens-off-the-past structure licenses the same eigenvalue and steady-state machinery.
This sourceDevelops transition kernels, stationary distributions, mixing, and absorbing-state analysis for the Markov subclass of stochastic processes.
Supported in partVerified against the publisher's abstract
Norris supplies the Markov-chain machinery the claim's frame rests on, and names queues among the book's applications, but the abstract documents none of the Metropolis/HMM/PageRank transfers.
“Markov chains are central to the understanding of random processes. This is not only because they pervade the applications of random processes, but also because one can calculate explicitly many quantities of interest.”
- The Markov toolkit — transition kernels, stationary distributions, mixing times, absorbing-state analysis — transfers from statistical physics (the Metropolis dynamics) to NLP (hidden Markov models) to operations (queue occupancy) to PageRank, because all are Markov processes and the present-screens-off-the-past structure licenses the same eigenvalue and steady-state machinery.
Domain-specific¶
Mechanisms¶
- Markov Chain Process Model
- If the future depends on how long you have been in a state, or on the route you took, first-order transition probabilities are simply wrong, and long-horizon extrapolations compound the error step by step.
This sourceDefines the first-order Markov property as future evolution depending on the present state rather than elapsed duration or the prior route.
- If the future depends on how long you have been in a state, or on the route you took, first-order transition probabilities are simply wrong, and long-horizon extrapolations compound the error step by step.
- State-Transition Kernel
- Because multi-step forecasts come from repeated composition, small errors in the one-step entries compound over the horizon through the Chapman-Kolmogorov
This sourceDerives multi-step transition probabilities by repeated Chapman–Kolmogorov composition of the transition kernel.
- Because multi-step forecasts come from repeated composition, small errors in the one-step entries compound over the horizon through the Chapman-Kolmogorov
Verification¶
Does it exist? Confirmed. This work's DOI resolves to a registered record, which fixes its identity. That is all it fixes.
Does it back the claim? Read against the text for 2 of 5 citations: 2 supported in part. Each verdict is shown under its citation below, with what in the work backs the sentence.
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Registry ID ref:fa47114b619a · see in the full table