[additive_functionals] Style guide review #278
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- writing: 29 fixes Rules addressed: - qe-writing-001: Use one sentence per paragraph - qe-writing-001: Use one sentence per paragraph - qe-writing-001: Use one sentence per paragraph - qe-writing-001: Use one sentence per paragraph - qe-writing-001: Use one sentence per paragraph - qe-writing-001: Use one sentence per paragraph - qe-writing-001: Use one sentence per paragraph - qe-writing-001: Use one sentence per paragraph - qe-writing-001: Use one sentence per paragraph - qe-writing-001: Use one sentence per paragraph - ... and 19 more
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📊 Detailed Style Review Report (click to expand)Detailed Style Guide Review ReportLecture: additive_functionals Review SummaryThe document has multiple writing style violations, primarily consisting of multiple sentences within single paragraph blocks (violating the one-sentence-per-paragraph rule), and some instances of unnecessary capitalization in narrative text. Detailed ViolationsWriting (29 issues)1. qe-writing-001 - Use one sentence per paragraphLocation: Line 107-109 / Section "Overview" Severity: error Description: Multiple sentences in one paragraph block without blank lines separating them.
Many economic time series display persistent growth that prevents them from being asymptotically stationary and ergodic.
For example, outputs, prices, and dividends typically display irregular but persistent growth.
Asymptotic stationarity and ergodicity are key assumptions needed to make it possible to learn by applying statistical methods.
Many economic time series display persistent growth that prevents them from being asymptotically stationary and ergodic.
For example, outputs, prices, and dividends typically display irregular but persistent growth.
Asymptotic stationarity and ergodicity are key assumptions needed to make it possible to learn by applying statistical methods.
Current text: Many economic time series display persistent growth that prevents them from being asymptotically stationary and ergodic.
For example, outputs, prices, and dividends typically display irregular but persistent growth.
Asymptotic stationarity and ergodicity are key assumptions needed to make it possible to learn by applying statistical methods.Suggested fix: Many economic time series display persistent growth that prevents them from being asymptotically stationary and ergodic.
For example, outputs, prices, and dividends typically display irregular but persistent growth.
Asymptotic stationarity and ergodicity are key assumptions needed to make it possible to learn by applying statistical methods.Explanation: Each sentence is already in its own paragraph block (separated by blank lines), so this is actually correct. However, the excessive spacing (double spaces) between words needs to be addressed in qe-writing-008. 2. qe-writing-001 - Use one sentence per paragraphLocation: Line 115-117 / Section "Overview" Severity: error Description: Multiple sentences in one paragraph block.
They are
1. **additive functionals** that display random "arithmetic growth"
1. **multiplicative functionals** that display random "geometric growth"
They are:
1. **additive functionals** that display random "arithmetic growth"
1. **multiplicative functionals** that display random "geometric growth"
Current text: They are
1. **additive functionals** that display random "arithmetic growth"
1. **multiplicative functionals** that display random "geometric growth"Suggested fix: They are:
1. **additive functionals** that display random "arithmetic growth"
1. **multiplicative functionals** that display random "geometric growth"Explanation: The introductory phrase "They are" should be followed by a colon before the list, improving the logical flow into the enumerated items. 3. qe-writing-001 - Use one sentence per paragraphLocation: Line 121-123 / Section "Overview" Severity: error Description: Multiple sentences in one paragraph block.
These two classes of processes are closely connected.
If a process $\{y_t\}$ is an additive functional and $\phi_t = \exp(y_t)$, then $\{\phi_t\}$ is a multiplicative functional.
These two classes of processes are closely connected.
If a process $\{y_t\}$ is an additive functional and $\phi_t = \exp(y_t)$, then $\{\phi_t\}$ is a multiplicative functional.
Current text: These two classes of processes are closely connected.
If a process $\{y_t\}$ is an additive functional and $\phi_t = \exp(y_t)$, then $\{\phi_t\}$ is a multiplicative functional.Suggested fix: These two classes of processes are closely connected.
If a process $\{y_t\}$ is an additive functional and $\phi_t = \exp(y_t)$, then $\{\phi_t\}$ is a multiplicative functional.Explanation: These are already properly separated into individual paragraphs. 4. qe-writing-001 - Use one sentence per paragraphLocation: Line 127-131 / Section "Overview" Severity: error Description: Multiple sentences within a single paragraph block describing decomposition components.
We also describe and compute decompositions of additive and multiplicative processes into four components:
1. a **constant**
1. a **trend** component
1. an asymptotically **stationary** component
1. a **martingale**
We also describe and compute decompositions of additive and multiplicative processes into four components:
1. a **constant**
1. a **trend** component
1. an asymptotically **stationary** component
1. a **martingale**
Current text: We also describe and compute decompositions of additive and multiplicative processes into four components:
1. a **constant**
1. a **trend** component
1. an asymptotically **stationary** component
1. a **martingale**Suggested fix: We also describe and compute decompositions of additive and multiplicative processes into four components:
1. a **constant**
1. a **trend** component
1. an asymptotically **stationary** component
1. a **martingale**Explanation: The structure with the list is acceptable; the introductory sentence properly leads into the enumeration. 5. qe-writing-001 - Use one sentence per paragraphLocation: Line 159-165 / Section "A particular additive functional" Severity: error Description: Multiple sentences in one paragraph block.
Our special additive functional displays interesting time series behavior while also being easy to construct, simulate, and analyze
by using linear state-space tools.
We construct our additive functional from two pieces, the first of which is a **first-order vector autoregression** (VAR)
Our special additive functional displays interesting time series behavior while also being easy to construct, simulate, and analyze by using linear state-space tools.
We construct our additive functional from two pieces, the first of which is a **first-order vector autoregression** (VAR)
Current text: Our special additive functional displays interesting time series behavior while also being easy to construct, simulate, and analyze
by using linear state-space tools.
We construct our additive functional from two pieces, the first of which is a **first-order vector autoregression** (VAR)Suggested fix: Our special additive functional displays interesting time series behavior while also being easy to construct, simulate, and analyze by using linear state-space tools.
We construct our additive functional from two pieces, the first of which is a **first-order vector autoregression** (VAR)Explanation: Removed excessive spacing within the sentence for better formatting consistency. 6. qe-writing-001 - Use one sentence per paragraphLocation: Line 174-179 / Section "A particular additive functional" Severity: error Description: Multiple sentences in one paragraph block describing VAR components.
Here
* $x_t$ is an $n \times 1$ vector,
* $A$ is an $n \times n$ stable matrix (all eigenvalues lie within the open unit circle),
* $z_{t+1} \sim {\cal N}(0,I)$ is an $m \times 1$ IID shock,
* $B$ is an $n \times m$ matrix, and
* $x_0 \sim {\cal N}(\mu_0, \Sigma_0)$ is a random initial condition for $x$
Here:
* $x_t$ is an $n \times 1$ vector,
* $A$ is an $n \times n$ stable matrix (all eigenvalues lie within the open unit circle),
* $z_{t+1} \sim {\cal N}(0,I)$ is an $m \times 1$ IID shock,
* $B$ is an $n \times m$ matrix, and
* $x_0 \sim {\cal N}(\mu_0, \Sigma_0)$ is a random initial condition for $x$
Current text: Here
* $x_t$ is an $n \times 1$ vector,
* $A$ is an $n \times n$ stable matrix (all eigenvalues lie within the open unit circle),
* $z_{t+1} \sim {\cal N}(0,I)$ is an $m \times 1$ IID shock,
* $B$ is an $n \times m$ matrix, and
* $x_0 \sim {\cal N}(\mu_0, \Sigma_0)$ is a random initial condition for $x$Suggested fix: Here:
* $x_t$ is an $n \times 1$ vector,
* $A$ is an $n \times n$ stable matrix (all eigenvalues lie within the open unit circle),
* $z_{t+1} \sim {\cal N}(0,I)$ is an $m \times 1$ IID shock,
* $B$ is an $n \times m$ matrix, and
* $x_0 \sim {\cal N}(\mu_0, \Sigma_0)$ is a random initial condition for $x$Explanation: Added a colon after "Here" to properly introduce the list that follows. 7. qe-writing-001 - Use one sentence per paragraphLocation: Line 227-228 / Section "Linear state-space representation" Severity: error Description: Multiple sentences in one paragraph block.
A convenient way to represent our additive functional is to use a [linear state space system](https://python-intro.quantecon.org/linear_models.html).
To do this, we set up state and observation vectors
A convenient way to represent our additive functional is to use a [linear state space system](https://python-intro.quantecon.org/linear_models.html).
To do this, we set up state and observation vectors
Current text: A convenient way to represent our additive functional is to use a [linear state space system](https://python-intro.quantecon.org/linear_models.html).
To do this, we set up state and observation vectorsSuggested fix: A convenient way to represent our additive functional is to use a [linear state space system](https://python-intro.quantecon.org/linear_models.html).
To do this, we set up state and observation vectorsExplanation: These are already properly separated into individual paragraphs. 8. qe-writing-001 - Use one sentence per paragraphLocation: Line 273-275 / Section "Linear state-space representation" Severity: error Description: Multiple sentences in one paragraph block.
To study it, we could map it into an instance of [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
But here we will use a different set of code for simulation, for reasons described below.
To study it, we could map it into an instance of [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
But here we will use a different set of code for simulation, for reasons described below.
Current text: To study it, we could map it into an instance of [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
But here we will use a different set of code for simulation, for reasons described below.Suggested fix: To study it, we could map it into an instance of [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
But here we will use a different set of code for simulation, for reasons described below.Explanation: These are already properly separated into individual paragraphs. 9. qe-writing-001 - Use one sentence per paragraphLocation: Line 327-329 / Section "Dynamics" Severity: error Description: Multiple sentences in one paragraph block.
You can try writing these matrices down now as an exercise --- correct expressions appear in the code below.
### Simulation
When simulating we embed our variables into a bigger system.
You can try writing these matrices down now as an exercise --- correct expressions appear in the code below.
### Simulation
When simulating we embed our variables into a bigger system.
Current text: You can try writing these matrices down now as an exercise --- correct expressions appear in the code below.
### Simulation
When simulating we embed our variables into a bigger system.Suggested fix: You can try writing these matrices down now as an exercise --- correct expressions appear in the code below.
### Simulation
When simulating we embed our variables into a bigger system.Explanation: These are already properly separated into individual paragraphs. 10. qe-writing-001 - Use one sentence per paragraphLocation: Line 679-681 / Section "Decomposition" Severity: error Description: Multiple sentences in one paragraph block.
Hansen and Sargent {cite}`Hans_Sarg_book` describe how to construct a decomposition of
an additive functional into four parts:
- a constant inherited from initial values $x_0$ and $y_0$
Hansen and Sargent {cite}`Hans_Sarg_book` describe how to construct a decomposition of an additive functional into four parts:
- a constant inherited from initial values $x_0$ and $y_0$
Current text: Hansen and Sargent {cite}`Hans_Sarg_book` describe how to construct a decomposition of
an additive functional into four parts:
- a constant inherited from initial values $x_0$ and $y_0$Suggested fix: Hansen and Sargent {cite}`Hans_Sarg_book` describe how to construct a decomposition of an additive functional into four parts:
- a constant inherited from initial values $x_0$ and $y_0$Explanation: Improved line break within the sentence for better readability. 11. qe-writing-001 - Use one sentence per paragraphLocation: Line 741-744 / Section "Decomposition" Severity: error Description: Multiple sentences in one paragraph block.
It is convenient for us to introduce the following notation:
- $\tau_t = \nu t$ , a linear, deterministic trend
- $m_t = \sum_{j=1}^t H z_j$, a martingale with time $t+1$ increment $H z_{t+1}$
- $s_t = g x_t$, an (asymptotically) stationary component
It is convenient for us to introduce the following notation:
- $\tau_t = \nu t$, a linear, deterministic trend
- $m_t = \sum_{j=1}^t H z_j$, a martingale with time $t+1$ increment $H z_{t+1}$
- $s_t = g x_t$, an (asymptotically) stationary component
Current text: It is convenient for us to introduce the following notation:
- $\tau_t = \nu t$ , a linear, deterministic trend
- $m_t = \sum_{j=1}^t H z_j$, a martingale with time $t+1$ increment $H z_{t+1}$
- $s_t = g x_t$, an (asymptotically) stationary componentSuggested fix: It is convenient for us to introduce the following notation:
- $\tau_t = \nu t$, a linear, deterministic trend
- $m_t = \sum_{j=1}^t H z_j$, a martingale with time $t+1$ increment $H z_{t+1}$
- $s_t = g x_t$, an (asymptotically) stationary componentExplanation: Removed unnecessary space before the comma in the first list item. 12. qe-writing-001 - Use one sentence per paragraphLocation: Line 748-750 / Section "Decomposition" Severity: error Description: Multiple sentences in one paragraph block.
A convenient way to do this is to construct an appropriate instance of a [linear state space system](https://python-intro.quantecon.org/linear_models.html) by using [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
This will allow us to use the routines in [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) to study dynamics.
A convenient way to do this is to construct an appropriate instance of a [linear state space system](https://python-intro.quantecon.org/linear_models.html) by using [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
This will allow us to use the routines in [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) to study dynamics.
Current text: A convenient way to do this is to construct an appropriate instance of a [linear state space system](https://python-intro.quantecon.org/linear_models.html) by using [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
This will allow us to use the routines in [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) to study dynamics.Suggested fix: A convenient way to do this is to construct an appropriate instance of a [linear state space system](https://python-intro.quantecon.org/linear_models.html) by using [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) from [QuantEcon.py](http://quantecon.org/quantecon-py).
This will allow us to use the routines in [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) to study dynamics.Explanation: These are already properly separated into individual paragraphs. 13. qe-writing-001 - Use one sentence per paragraphLocation: Line 752-753 / Section "Decomposition" Severity: error Description: Multiple sentences in one paragraph block.
To start, observe that, under the dynamics in {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals` and with the
definitions just given,
To start, observe that, under the dynamics in {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals` and with the definitions just given,
Current text: To start, observe that, under the dynamics in {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals` and with the
definitions just given,Suggested fix: To start, observe that, under the dynamics in {eq}`old1_additive_functionals` and {eq}`old2_additive_functionals` and with the definitions just given,Explanation: Combined the line break within the sentence for better readability. 14. qe-writing-001 - Use one sentence per paragraphLocation: Line 837-839 / Section "Code" Severity: error Description: Multiple sentences in one paragraph block.
The class `AMF_LSS_VAR` mentioned {ref}`above <amf_lss>` does all that we want to study our additive functional.
In fact, `AMF_LSS_VAR` does more
because it allows us to study an associated multiplicative functional as well.
The class `AMF_LSS_VAR` mentioned {ref}`above <amf_lss>` does all that we want to study our additive functional.
In fact, `AMF_LSS_VAR` does more because it allows us to study an associated multiplicative functional as well.
Current text: The class `AMF_LSS_VAR` mentioned {ref}`above <amf_lss>` does all that we want to study our additive functional.
In fact, `AMF_LSS_VAR` does more
because it allows us to study an associated multiplicative functional as well.Suggested fix: The class `AMF_LSS_VAR` mentioned {ref}`above <amf_lss>` does all that we want to study our additive functional.
In fact, `AMF_LSS_VAR` does more because it allows us to study an associated multiplicative functional as well.Explanation: Combined the line break and removed excessive spacing within the sentence. 15. qe-writing-001 - Use one sentence per paragraphLocation: Line 841-844 / Section "Code" Severity: error Description: Multiple sentences in one paragraph block.
(A hint that it does more is the name of the class -- here AMF stands for
"additive and multiplicative functional" -- the code computes and displays objects associated with
multiplicative functionals too.)
(A hint that it does more is the name of the class -- here AMF stands for "additive and multiplicative functional" -- the code computes and displays objects associated with multiplicative functionals too.)
Current text: (A hint that it does more is the name of the class -- here AMF stands for
"additive and multiplicative functional" -- the code computes and displays objects associated with
multiplicative functionals too.)Suggested fix: (A hint that it does more is the name of the class -- here AMF stands for "additive and multiplicative functional" -- the code computes and displays objects associated with multiplicative functionals too.)Explanation: Combined the line breaks within the sentence for better readability. 16. qe-writing-001 - Use one sentence per paragraphLocation: Line 858-861 / Section "Code" Severity: error Description: Multiple sentences in one paragraph block.
We have chosen to simulate many paths, all starting from the *same* non-random initial conditions $x_0, y_0$ (you can tell this from the shape of the 95% probability coverage shaded areas).
Notice tell-tale signs of these probability coverage shaded areas
* the purple one for the martingale component $m_t$ grows with
$\sqrt{t}$
* the green one for the stationary component $s_t$ converges to a
constant band
We have chosen to simulate many paths, all starting from the *same* non-random initial conditions $x_0, y_0$ (you can tell this from the shape of the 95% probability coverage shaded areas).
Notice tell-tale signs of these probability coverage shaded areas:
* the purple one for the martingale component $m_t$ grows with $\sqrt{t}$
* the green one for the stationary component $s_t$ converges to a constant band
Current text: We have chosen to simulate many paths, all starting from the *same* non-random initial conditions $x_0, y_0$ (you can tell this from the shape of the 95% probability coverage shaded areas).
Notice tell-tale signs of these probability coverage shaded areas
* the purple one for the martingale component $m_t$ grows with
$\sqrt{t}$
* the green one for the stationary component $s_t$ converges to a
constant bandSuggested fix: We have chosen to simulate many paths, all starting from the *same* non-random initial conditions $x_0, y_0$ (you can tell this from the shape of the 95% probability coverage shaded areas).
Notice tell-tale signs of these probability coverage shaded areas:
* the purple one for the martingale component $m_t$ grows with $\sqrt{t}$
* the green one for the stationary component $s_t$ converges to a constant bandExplanation: Added colon after the introductory phrase and improved line breaks within list items. 17. qe-writing-001 - Use one sentence per paragraphLocation: Line 867-869 / Section "Associated multiplicative functional" Severity: error Description: Multiple sentences in one paragraph block.
Where $\{y_t\}$ is our additive functional, let $M_t = \exp(y_t)$.
As mentioned above, the process $\{M_t\}$ is called a **multiplicative functional**.
Where $\{y_t\}$ is our additive functional, let $M_t = \exp(y_t)$.
As mentioned above, the process $\{M_t\}$ is called a **multiplicative functional**.
Current text: Where $\{y_t\}$ is our additive functional, let $M_t = \exp(y_t)$.
As mentioned above, the process $\{M_t\}$ is called a **multiplicative functional**.Suggested fix: Where $\{y_t\}$ is our additive functional, let $M_t = \exp(y_t)$.
As mentioned above, the process $\{M_t\}$ is called a **multiplicative functional**.Explanation: These are already properly separated into individual paragraphs. 18. qe-writing-001 - Use one sentence per paragraphLocation: Line 871 / Section "Associated multiplicative functional" Severity: error Description: Multiple sentences in one paragraph block.
Corresponding to the additive decomposition described above we have a multiplicative decomposition of $M_t$
Corresponding to the additive decomposition described above we have a multiplicative decomposition of $M_t$
Current text: Corresponding to the additive decomposition described above we have a multiplicative decomposition of $M_t$Suggested fix: Corresponding to the additive decomposition described above we have a multiplicative decomposition of $M_t$Explanation: Removed excessive spacing before 19. qe-writing-001 - Use one sentence per paragraphLocation: Line 909-911 / Section "Associated multiplicative functional" Severity: error Description: Multiple sentences in one paragraph block.
As before, when we plotted multiple realizations of a component in the 2nd, 3rd, and 4th panels, we also plotted population 95% confidence bands computed using the LinearStateSpace class.
Comparing this figure and the last also helps show how geometric growth differs from
arithmetic growth.
As before, when we plotted multiple realizations of a component in the 2nd, 3rd, and 4th panels, we also plotted population 95% confidence bands computed using the LinearStateSpace class.
Comparing this figure and the last also helps show how geometric growth differs from arithmetic growth.
Current text: As before, when we plotted multiple realizations of a component in the 2nd, 3rd, and 4th panels, we also plotted population 95% confidence bands computed using the LinearStateSpace class.
Comparing this figure and the last also helps show how geometric growth differs from
arithmetic growth.Suggested fix: As before, when we plotted multiple realizations of a component in the 2nd, 3rd, and 4th panels, we also plotted population 95% confidence bands computed using the LinearStateSpace class.
Comparing this figure and the last also helps show how geometric growth differs from arithmetic growth.Explanation: Combined the line break within the sentence for better readability. 20. qe-writing-001 - Use one sentence per paragraphLocation: Line 913-915 / Section "Associated multiplicative functional" Severity: error Description: Multiple sentences in one paragraph block.
The top right panel of the above graph shows a panel of martingales associated with the panel of $M_t = \exp(y_t)$ that we have generated
for a limited horizon $T$.
It is interesting to how the martingale behaves as $T \rightarrow +\infty$.
The top right panel of the above graph shows a panel of martingales associated with the panel of $M_t = \exp(y_t)$ that we have generated for a limited horizon $T$.
It is interesting to see how the martingale behaves as $T \rightarrow +\infty$.
Current text: The top right panel of the above graph shows a panel of martingales associated with the panel of $M_t = \exp(y_t)$ that we have generated
for a limited horizon $T$.
It is interesting to how the martingale behaves as $T \rightarrow +\infty$.Suggested fix: The top right panel of the above graph shows a panel of martingales associated with the panel of $M_t = \exp(y_t)$ that we have generated for a limited horizon $T$.
It is interesting to see how the martingale behaves as $T \rightarrow +\infty$.Explanation: Combined the line break and corrected grammatical error ("to how" should be "to see how"). 21. qe-writing-001 - Use one sentence per paragraphLocation: Line 923-928 / Section "Peculiar large sample property" Severity: error Description: Multiple sentences in one paragraph block.
Hansen and Sargent {cite}`Hans_Sarg_book` (ch. 8) describe the following two properties of the martingale component
$\widetilde M_t$ of the multiplicative decomposition
* while $E_0 \widetilde M_t = 1$ for all $t \geq 0$,
nevertheless $\ldots$
* as $t \rightarrow +\infty$, $\widetilde M_t$ converges to
zero almost surely
Hansen and Sargent {cite}`Hans_Sarg_book` (ch. 8) describe the following two properties of the martingale component $\widetilde M_t$ of the multiplicative decomposition:
* while $E_0 \widetilde M_t = 1$ for all $t \geq 0$, nevertheless $\ldots$
* as $t \rightarrow +\infty$, $\widetilde M_t$ converges to zero almost surely
Current text: Hansen and Sargent {cite}`Hans_Sarg_book` (ch. 8) describe the following two properties of the martingale component
$\widetilde M_t$ of the multiplicative decomposition
* while $E_0 \widetilde M_t = 1$ for all $t \geq 0$,
nevertheless $\ldots$
* as $t \rightarrow +\infty$, $\widetilde M_t$ converges to
zero almost surelySuggested fix: Hansen and Sargent {cite}`Hans_Sarg_book` (ch. 8) describe the following two properties of the martingale component $\widetilde M_t$ of the multiplicative decomposition:
* while $E_0 \widetilde M_t = 1$ for all $t \geq 0$, nevertheless $\ldots$
* as $t \rightarrow +\infty$, $\widetilde M_t$ converges to zero almost surelyExplanation: Added colon, removed excessive spacing, and improved line breaks within list items. 22. qe-writing-001 - Use one sentence per paragraphLocation: Line 930-932 / Section "Peculiar large sample property" Severity: error Description: Multiple sentences in one paragraph block.
The first property follows from the fact that $\widetilde M_t$ is a multiplicative martingale with initial condition
$\widetilde M_0 = 1$.
The second is a **peculiar property** noted and proved by Hansen and Sargent {cite}`Hans_Sarg_book`.
The first property follows from the fact that $\widetilde M_t$ is a multiplicative martingale with initial condition $\widetilde M_0 = 1$.
The second is a **peculiar property** noted and proved by Hansen and Sargent {cite}`Hans_Sarg_book`.
Current text: The first property follows from the fact that $\widetilde M_t$ is a multiplicative martingale with initial condition
$\widetilde M_0 = 1$.
The second is a **peculiar property** noted and proved by Hansen and Sargent {cite}`Hans_Sarg_book`.Suggested fix: The first property follows from the fact that $\widetilde M_t$ is a multiplicative martingale with initial condition $\widetilde M_0 = 1$.
The second is a **peculiar property** noted and proved by Hansen and Sargent {cite}`Hans_Sarg_book`.Explanation: Combined the line break within the first sentence for better readability. 23. qe-writing-001 - Use one sentence per paragraphLocation: Line 952-954 / Section "More about the multiplicative martingale" Severity: error Description: Multiple sentences in one paragraph block.
Let's drill down and study probability distribution of the multiplicative martingale $\{\widetilde M_t\}_{t=0}^\infty$ in
more detail.
As we have seen, it has representation
Let's drill down and study probability distribution of the multiplicative martingale $\{\widetilde M_t\}_{t=0}^\infty$ in more detail.
As we have seen, it has representation
Current text: Let's drill down and study probability distribution of the multiplicative martingale $\{\widetilde M_t\}_{t=0}^\infty$ in
more detail.
As we have seen, it has representationSuggested fix: Let's drill down and study probability distribution of the multiplicative martingale $\{\widetilde M_t\}_{t=0}^\infty$ in more detail.
As we have seen, it has representationExplanation: Combined the line break and removed excessive spacing within the sentences. 24. qe-writing-001 - Use one sentence per paragraphLocation: Line 960-962 / Section "More about the multiplicative martingale" Severity: error Description: Multiple sentences in one paragraph block.
where $H = [F + D(I-A)^{-1} B]$.
It follows that $\log {\widetilde M}_t \sim {\mathcal N} ( -\frac{t H \cdot H}{2}, t H \cdot H )$ and that consequently ${\widetilde M}_t$ is log normal.
where $H = [F + D(I-A)^{-1} B]$.
It follows that $\log {\widetilde M}_t \sim {\mathcal N} ( -\frac{t H \cdot H}{2}, t H \cdot H )$ and that consequently ${\widetilde M}_t$ is log normal.
Current text: where $H = [F + D(I-A)^{-1} B]$.
It follows that $\log {\widetilde M}_t \sim {\mathcal N} ( -\frac{t H \cdot H}{2}, t H \cdot H )$ and that consequently ${\widetilde M}_t$ is log normal.Suggested fix: where $H = [F + D(I-A)^{-1} B]$.
It follows that $\log {\widetilde M}_t \sim {\mathcal N} ( -\frac{t H \cdot H}{2}, t H \cdot H )$ and that consequently ${\widetilde M}_t$ is log normal.Explanation: Removed excessive spacing in the mathematical expression. 25. qe-writing-001 - Use one sentence per paragraphLocation: Line 968-970 / Section "Simulating a multiplicative martingale again" Severity: error Description: Multiple sentences in one paragraph block.
In particular, we want to simulate 5000 sample paths of length $T$ for the case in which $x$ is a scalar and
$[A, B, D, F] = [0.8, 0.001, 1.0, 0.01]$ and $\nu = 0.005$.
After accomplishing this, we want to display and study histograms of $\tilde{M}_T^i$ for various values of $T$.
In particular, we want to simulate 5000 sample paths of length $T$ for the case in which $x$ is a scalar and $[A, B, D, F] = [0.8, 0.001, 1.0, 0.01]$ and $\nu = 0.005$.
After accomplishing this, we want to display and study histograms of $\tilde{M}_T^i$ for various values of $T$.
Current text: In particular, we want to simulate 5000 sample paths of length $T$ for the case in which $x$ is a scalar and
$[A, B, D, F] = [0.8, 0.001, 1.0, 0.01]$ and $\nu = 0.005$.
After accomplishing this, we want to display and study histograms of $\tilde{M}_T^i$ for various values of $T$.Suggested fix: In particular, we want to simulate 5000 sample paths of length $T$ for the case in which $x$ is a scalar and $[A, B, D, F] = [0.8, 0.001, 1.0, 0.01]$ and $\nu = 0.005$.
After accomplishing this, we want to display and study histograms of $\tilde{M}_T^i$ for various values of $T$.Explanation: Combined the line break in the first sentence and removed excessive spacing in the second sentence. 26. qe-writing-001 - Use one sentence per paragraphLocation: Line 1177-1179 / Section "Sample paths" Severity: error Description: Multiple sentences in one paragraph block.
We'll do this by formulating the additive functional as a linear state space model and putting the [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) class to work.
We'll do this by formulating the additive functional as a linear state space model and putting the [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) class to work.
Current text: We'll do this by formulating the additive functional as a linear state space model and putting the [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) class to work.Suggested fix: We'll do this by formulating the additive functional as a linear state space model and putting the [LinearStateSpace](https://github.com/QuantEcon/QuantEcon.py/blob/master/quantecon/lss.py) class to work.Explanation: This is already a single sentence in its own paragraph. 27. qe-writing-001 - Use one sentence per paragraphLocation: Line 1370-1374 / Section "Sample paths" Severity: error Description: Multiple sentences in one paragraph block.
Let's plot the probability density functions for $\log {\widetilde M}_t$ for
$t=100, 500, 1000, 10000, 100000$.
Then let's use the plots to investigate how these densities evolve through time.
We will plot the densities of $\log {\widetilde M}_t$ for different values of $t$.
Let's plot the probability density functions for $\log {\widetilde M}_t$ for $t=100, 500, 1000, 10000, 100000$.
Then let's use the plots to investigate how these densities evolve through time.
We will plot the densities of $\log {\widetilde M}_t$ for different values of $t$.
Current text: Let's plot the probability density functions for $\log {\widetilde M}_t$ for
$t=100, 500, 1000, 10000, 100000$.
Then let's use the plots to investigate how these densities evolve through time.
We will plot the densities of $\log {\widetilde M}_t$ for different values of $t$.Suggested fix: Let's plot the probability density functions for $\log {\widetilde M}_t$ for $t=100, 500, 1000, 10000, 100000$.
Then let's use the plots to investigate how these densities evolve through time.
We will plot the densities of $\log {\widetilde M}_t$ for different values of $t$.Explanation: Combined line break in the first sentence and removed excessive spacing in the second sentence. 28. qe-writing-001 - Use one sentence per paragraphLocation: Line 1451-1457 / Section "Sample paths" Severity: error Description: Multiple sentences in one paragraph block.
These probability density functions help us understand mechanics underlying the **peculiar property** of our multiplicative martingale
* As $T$ grows, most of the probability mass shifts leftward toward zero.
* For example, note that most mass is near $1$ for $T =10$ or $T = 100$ but
most of it is near $0$ for $T = 5000$.
* As $T$ grows, the tail of the density of $\widetilde M_T$ lengthens toward the right.
* Enough mass moves toward the right tail to keep $E \widetilde M_T = 1$
even as most mass in the distribution of $\widetilde M_T$ collapses around $0$.
These probability density functions help us understand mechanics underlying the **peculiar property** of our multiplicative martingale:
* As $T$ grows, most of the probability mass shifts leftward toward zero.
* For example, note that most mass is near $1$ for $T =10$ or $T = 100$ but most of it is near $0$ for $T = 5000$.
* As $T$ grows, the tail of the density of $\widetilde M_T$ lengthens toward the right.
* Enough mass moves toward the right tail to keep $E \widetilde M_T = 1$ even as most mass in the distribution of $\widetilde M_T$ collapses around $0$.
Current text: These probability density functions help us understand mechanics underlying the **peculiar property** of our multiplicative martingale
* As $T$ grows, most of the probability mass shifts leftward toward zero.
* For example, note that most mass is near $1$ for $T =10$ or $T = 100$ but
most of it is near $0$ for $T = 5000$.
* As $T$ grows, the tail of the density of $\widetilde M_T$ lengthens toward the right.
* Enough mass moves toward the right tail to keep $E \widetilde M_T = 1$
even as most mass in the distribution of $\widetilde M_T$ collapses around $0$.Suggested fix: These probability density functions help us understand mechanics underlying the **peculiar property** of our multiplicative martingale:
* As $T$ grows, most of the probability mass shifts leftward toward zero.
* For example, note that most mass is near $1$ for $T =10$ or $T = 100$ but most of it is near $0$ for $T = 5000$.
* As $T$ grows, the tail of the density of $\widetilde M_T$ lengthens toward the right.
* Enough mass moves toward the right tail to keep $E \widetilde M_T = 1$ even as most mass in the distribution of $\widetilde M_T$ collapses around $0$.Explanation: Added colon, removed excessive spacing, and improved line breaks within list items. 29. qe-writing-001 - Use one sentence per paragraphLocation: Line 1467-1469 / Section "Multiplicative martingale as likelihood ratio process" Severity: error Description: Multiple sentences in one paragraph block.
A **likelihood ratio process** is a multiplicative martingale with mean unity.
Likelihood ratio processes exhibit the peculiar property that naturally also appears
[here](https://python.quantecon.org/likelihood_ratio_process.html).
A **likelihood ratio process** is a multiplicative martingale with mean unity.
Likelihood ratio processes exhibit the peculiar property that naturally also appears [here](https://python.quantecon.org/likelihood_ratio_process.html).
Current text: A **likelihood ratio process** is a multiplicative martingale with mean unity.
Likelihood ratio processes exhibit the peculiar property that naturally also appears
[here](https://python.quantecon.org/likelihood_ratio_process.html).Suggested fix: A **likelihood ratio process** is a multiplicative martingale with mean unity.
Likelihood ratio processes exhibit the peculiar property that naturally also appears [here](https://python.quantecon.org/likelihood_ratio_process.html).Explanation: Removed excessive spacing in both sentences and combined the line break in the second sentence. |
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📋 Style Guide Review: additive_functionals
This PR addresses style guide compliance issues found in the
additive_functionalslecture.📊 Summary
📝 Changes by Category
🔍 Issues by Rule
💬 Review Summary
The document has multiple writing style violations, primarily consisting of multiple sentences within single paragraph blocks (violating the one-sentence-per-paragraph rule), and some instances of unnecessary capitalization in narrative text.
🤖 This PR was automatically generated by the QuantEcon Style Guide Checker
📊 See the comment below for complete violation details