diff --git a/source/rst/heavy_tails.rst b/source/rst/heavy_tails.rst index 576fd346..01dee999 100644 --- a/source/rst/heavy_tails.rst +++ b/source/rst/heavy_tails.rst @@ -172,7 +172,7 @@ One impact of heavy tails is that sample averages can be poor estimators of the underlying mean of the distribution. To understand this point better, recall :doc:`our earlier discussion -` of the Law of Large Numbers, which considered iid :math:`X_1, +` of the Law of Large Numbers, which considered IID :math:`X_1, \ldots, X_n` with common distribution :math:`F` If :math:`\mathbb E |X_i|` is finite, then diff --git a/source/rst/kesten_processes.rst b/source/rst/kesten_processes.rst index 344b3157..4093ed6d 100644 --- a/source/rst/kesten_processes.rst +++ b/source/rst/kesten_processes.rst @@ -63,7 +63,7 @@ A **Kesten process** is a stochastic process of the form x_{t+1} = a_{t+1} x_t + \eta_{t+1} \quad \text{with } x_0 \text{ given} -where :math:`\{a_t\}_{t \geq 1}` and :math:`\{\eta_t\}_{t \geq 1}` are iid +where :math:`\{a_t\}_{t \geq 1}` and :math:`\{\eta_t\}_{t \geq 1}` are IID sequences. We will focus on the nonnegative scalar case, where :math:`x_t` takes values @@ -73,9 +73,9 @@ In particular, we will assume that * the initial condition :math:`x_0` is nonnegative, -* :math:`\{a_t\}_{t \geq 1}` is a nonnegative iid stochastic process and +* :math:`\{a_t\}_{t \geq 1}` is a nonnegative IID stochastic process and -* :math:`\{\eta_t\}_{t \geq 1}` is another nonnegative iid stochastic process, independent of the first. +* :math:`\{\eta_t\}_{t \geq 1}` is another nonnegative IID stochastic process, independent of the first. @@ -122,7 +122,7 @@ The GARCH(1, 1) volatility process takes the form \sigma_{t+1}^2 = \alpha_0 + \sigma_t^2 (\alpha_1 \xi_{t+1}^2 + \beta) -where :math:`\{\xi_t\}` is iid with :math:`\mathbb E \xi_t^2 = 1` and all parameters are positive. +where :math:`\{\xi_t\}` is IID with :math:`\mathbb E \xi_t^2 = 1` and all parameters are positive. Returns on a given asset are then modeled as @@ -131,7 +131,7 @@ Returns on a given asset are then modeled as r_t = \sigma_t \zeta_{t+1} -where :math:`\{\zeta_t\}` is again iid and independent of :math:`\{\xi_t\}`. +where :math:`\{\zeta_t\}` is again IID and independent of :math:`\{\xi_t\}`. Notice that the volatility sequence :math:`\{\sigma_t\}`, which drives the dynamics, is a Kesten process. @@ -152,7 +152,7 @@ Wealth then evolves according to where :math:`\{R_t\}` is the gross rate of return on assets. -If :math:`\{R_t\}` and :math:`\{y_t\}` are both iid, then :eq:`wealth_dynam` +If :math:`\{R_t\}` and :math:`\{y_t\}` are both IID, then :eq:`wealth_dynam` is a Kesten process. @@ -386,7 +386,7 @@ We can express this idea by stating that a suitably defined measure \frac{s_{t+1}}{s_t} = a_{t+1} -for some positive iid sequence :math:`\{a_t\}`. +for some positive IID sequence :math:`\{a_t\}`. One implication of Gibrat's law is that the growth rate of individual firms does not depend on their size. @@ -413,7 +413,7 @@ to s_{t+1} = a_{t+1} s_t + b_{t+1} -where :math:`\{a_t\}` and :math:`\{b_t\}` are both iid and independent of each +where :math:`\{a_t\}` and :math:`\{b_t\}` are both IID and independent of each other. In the exercises you are asked to show that :eq:`firm_dynam` is more