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Expand Up @@ -19,12 +19,13 @@
\draw[important line, thick] (Z1) -- (O) node[right] {};
\draw[important line, thick] (Py) -- (Z2) node[right] {$S$};
\draw[important line, blue,->] (O) -- (y) node[right] {$y$};
\draw[important line, blue,->] (O) -- (Py') node[anchor = north west, text width=5em] {$P y'$};
\draw[dotted line] (0.54,0.27) -- (0.51,0.33);
\draw[dotted line] (0.57,0.36) -- (0.51,0.33);
\draw[dotted line] (-0.22,-0.11) -- (-0.25,-0.05);
\draw[dotted line] (-0.31,-0.08) -- (-0.25,-0.05);
\draw[dashed line, black] (y) -- (Py);
\draw[dashed line, black] (y') -- (Py') node[anchor = north west, text width=5em] {$P y'$};
\draw[dashed line, black] (y') -- (Py');
\end{tikzpicture}

\end{document}
16 changes: 8 additions & 8 deletions lectures/additive_functionals.md
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Expand Up @@ -77,7 +77,7 @@ import matplotlib.pyplot as plt
from scipy.stats import norm, lognorm
```

## A Particular Additive Functional
## A particular additive functional

{cite}`Hansen_2012_Eca` describes a general class of additive functionals.

Expand Down Expand Up @@ -123,7 +123,7 @@ initial condition for $y$.
The nonstationary random process $\{y_t\}_{t=0}^\infty$ displays
systematic but random *arithmetic growth*.

### Linear State-Space Representation
### Linear state-space representation

A convenient way to represent our additive functional is to use a [linear state space system](https://python-intro.quantecon.org/linear_models.html).

Expand Down Expand Up @@ -872,7 +872,7 @@ Notice tell-tale signs of these probability coverage shaded areas
* the green one for the stationary component $s_t$ converges to a
constant band

### Associated Multiplicative Functional
### Associated multiplicative functional

Where $\{y_t\}$ is our additive functional, let $M_t = \exp(y_t)$.

Expand Down Expand Up @@ -929,7 +929,7 @@ It is interesting to how the martingale behaves as $T \rightarrow +\infty$.

Let's see what happens when we set $T = 12000$ instead of $150$.

### Peculiar Large Sample Property
### Peculiar large sample property

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
Expand Down Expand Up @@ -958,7 +958,7 @@ It remains constant at unity, illustrating the first property.

The purple 95 percent frequency coverage interval collapses around zero, illustrating the second property.

## More About the Multiplicative Martingale
## More about the multiplicative martingale

Let's drill down and study probability distribution of the multiplicative martingale $\{\widetilde M_t\}_{t=0}^\infty$ in
more detail.
Expand All @@ -973,7 +973,7 @@ 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.

### Simulating a Multiplicative Martingale Again
### Simulating a multiplicative martingale again

Next, we want a program to simulate the likelihood ratio process $\{ \tilde{M}_t \}_{t=0}^\infty$.

Expand All @@ -984,7 +984,7 @@ After accomplishing this, we want to display and study histograms of $\tilde{M}_

Here is code that accomplishes these tasks.

### Sample Paths
### Sample paths

Let's write a program to simulate sample paths of $\{ x_t, y_{t} \}_{t=0}^{\infty}$.

Expand Down Expand Up @@ -1257,7 +1257,7 @@ These probability density functions help us understand mechanics underlying the
* 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 $0$.

### Multiplicative Martingale as Likelihood Ratio Process
### Multiplicative martingale as likelihood ratio process

[This lecture](https://python.quantecon.org/likelihood_ratio_process.html) studies **likelihood processes**
and **likelihood ratio processes**.
Expand Down
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