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Fix transpose typos in additive functional lecture #120

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May 7, 2020
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Fix transpose typos in additive functional lecture #120

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20 changes: 10 additions & 10 deletions source/rst/additive_functionals.rst
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Original file line number Diff line number Diff line change
Expand Up @@ -159,15 +159,15 @@ Next we construct a linear system
\begin{bmatrix}
1 & 0 & 0 \\
0 & A & 0 \\
\nu & D' & 1
\nu & D & 1
\end{bmatrix}
\begin{bmatrix}
1 \\
x_t \\
y_t
\end{bmatrix} +
\begin{bmatrix}
0 \\ B \\ F'
0 \\ B \\ F
\end{bmatrix}
z_{t+1}

Expand Down Expand Up @@ -763,9 +763,9 @@ functionals defined by :eq:`old1_additive_functionals` and :eq:`old2_additive_fu
.. math::

\begin{aligned}
H & := F + B'(I - A')^{-1} D
H & := F + D (I - A)^{-1} B
\\
g & := D' (I - A)^{-1}
g & := D (I - A)^{-1}
\end{aligned}


Expand Down Expand Up @@ -814,7 +814,7 @@ definitions just given,
1 & 0 & 0 & 0 & 0 \\
1 & 1 & 0 & 0 & 0 \\
0 & 0 & A & 0 & 0 \\
\nu & 0 & D' & 1 & 0 \\
\nu & 0 & D & 1 & 0 \\
0 & 0 & 0 & 0 & 1
\end{bmatrix}
\begin{bmatrix}
Expand All @@ -828,8 +828,8 @@ definitions just given,
0 \\
0 \\
B \\
F' \\
H'
F \\
H
\end{bmatrix}
z_{t+1}

Expand Down Expand Up @@ -934,7 +934,7 @@ Corresponding to the additive decomposition described above we have a multiplica
.. math::

\frac{M_t}{M_0}
= \exp (t \nu) \exp \Bigl(\sum_{j=1}^t H \cdot Z_j \Bigr) \exp \biggl( D'(I-A)^{-1} x_0 - D'(I-A)^{-1} x_t \biggr)
= \exp (t \nu) \exp \Bigl(\sum_{j=1}^t H \cdot Z_j \Bigr) \exp \biggl( D(I-A)^{-1} x_0 - D(I-A)^{-1} x_t \biggr)


or
Expand All @@ -957,7 +957,7 @@ and

.. math::

\tilde e(x) = \exp[g(x)] = \exp \bigl[ D' (I - A)^{-1} x \bigr]
\tilde e(x) = \exp[g(x)] = \exp \bigl[ D (I - A)^{-1} x \bigr]


An instance of class ``AMF_LSS_VAR`` (:ref:`above <amf_lss>`) includes this associated multiplicative functional as an attribute.
Expand Down Expand Up @@ -1041,7 +1041,7 @@ As we have seen, it has representation
\widetilde M_t = \exp \biggl( \sum_{j=1}^t \biggl(H \cdot z_j -\frac{ H \cdot H }{2} \biggr) \biggr), \quad \widetilde M_0 =1


where :math:`H = [F + B'(I-A')^{-1} D]`.
where :math:`H = [F + D(I-A)^{-1} B]`.

It follows that :math:`\log {\widetilde M}_t \sim {\mathcal N} ( -\frac{t H \cdot H}{2}, t H \cdot H )` and that consequently :math:`{\widetilde M}_t` is log normal.

Expand Down