FiniteStateMachine.asymptotic_moments: New method

[cheuberg]

Returns the main terms of expectation and variance of the sum of output labels and its covariance with the sum of input labels. This implements one of the results of the forthcoming paper: Clemens Heuberger, Sara Kropf and Stephan Wagner, Combinatorial Characterization of Independent Transducers via Functional Digraphs.

[cheuberg]

Updated reference HKW2014 to point to arxiv:1404.3680.

[cheuberg]

• merged current versions of all dependencies so that this merges cleanly with 6.2.beta8
• removed reference HKW2014 from FiniteStateMachine.final_components in order to avoid an error due to a duplicate reference
• replaced one new instance of "raise E, V" by "raise E(V)"

[dkrenn]

Made a couple of minor changes during the review.

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New commits:

​d22eea2

a couple of minor changes during review

[cheuberg]

reviewed your changes, those are fine for me.

Pushed a few commits:

• ​89ebf74 added two links in docstring
• ​40ecd46 added explanation to third example
• ​1447cd5 add examples to examples
• ​f13725d add explanation on PolynomialRing? vs SymbolicRing?
• ​878acff replaced parameter verbose by sage.misc.misc.verbose
• ​40da528 Explanations for >1 final components, periodic final components
• ​00e063e upgraded two tests to examples
• ​1908a5d unique initial states, all states are final

Finally, changed summary to reflect a name change in the method which occurred some time ago.

Please review those changes.

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Last 10 new commits:

​8e04e12	replaced one new instance of raise E, V by raise E(V) (cf. #15990)
​d22eea2	a couple of minor changes during review
​89ebf74	FiniteStateMachine.asymptotic_moments: added two links in docstring
​40ecd46	FiniteStateMachine.asymptotic_moments: added explanation to third example
​1447cd5	FiniteStateMachine.asymptotic_moments: add examples to examples
​f13725d	asymptotic_moments: add explanation on PolynomialRing vs SymbolicRing
​878acff	asymptotic_moments: replaced parameter verbose by sage.misc.misc.verbose
​40da528	Explanations for >1 final components, periodic final components
​00e063e	asymptotic_moments: upgraded two tests to examples
​1908a5d	asymptotic_moments: unique initial states, all states are final

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​78997f5

Corrected and improved version of commit 40da528

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​d56a769

Rewrote asymptotic_moments to output en+Order(1) etc.

[cheuberg]

I expect a rather trivial merge conflict with #16207 --- so whichever of the two tickets is reviewed earlier should be merged into the other.

[git]

Branch pushed to git repo; I updated commit sha1. New commits:

​24b6d69	explained the shifting in the Gray code-construction differently
​0b1200d	Merge branch 't/16143/fsm/example_gray_code' into fsm/asymptotic-moments

[cheuberg]

trac's automerge fails. Patch based on 6.2.beta8, all dependencies but #16143 are in 6.2.rc0, merges cleanly with 6.2.rc0.

[dkrenn]

Everything looks fine for me.

[dkrenn]

During review: very minor changes in docstring.

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New commits:

​b2aff73

very minor doctest changes: some spacings; added an 'a'

[cheuberg]

Replying to cheuberg:

I expect a rather trivial merge conflict with #16207 --- so whichever of the two tickets is reviewed earlier should be merged into the other.

#16207 now depends on this ticket, so merge conflict is taken care of.

Replying to dkrenn:

During review: very minor changes in docstring.

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New commits:

​b2aff73

very minor doctest changes: some spacings; added an 'a'

fine with me.

[dkrenn]

Ok, let's set this to positive...