More typos

probability/martingales.qmd

@@ -228,7 +228,7 @@ $$
 Therefore, $\{W_t\}_{t \ge 1}$ is a supermartingale, but we cannot immediately apply the supermartingale convergence theorem because we don't know that $W_t$ is bounded from below. 
 
 
-For an arbitrary $a > 0$, define the stopping time $τ = \inf \bigl\{ t : \sum_{s=1}^{t} Y_s > a \bigr\}.$ Then, the stopped sequence $\{W_{τ \wedge t}\}_{t \ge 1}$ is also a supermartingale because
+For an arbitrary $a > 0$, define the stopping time $τ = \inf \bigl\{ t : \sum_{s=1}^{t} Y'_s > a \bigr\}.$ Then, the stopped sequence $\{W_{τ \wedge t}\}_{t \ge 1}$ is also a supermartingale because
 $$
 \EXP[ W_{τ \wedge (t+1)} \mid \mathcal F_t ]
 = W_t \IND\{ τ \le t \} + \EXP[ W_{t+1} \mid \mathcal F_t ] \IND\{ τ > t\}

rl/stochastic-approximation.qmd

@@ -272,7 +272,7 @@ It is worthwhile to compare the conditions of @thm-borkar-meyn and @thm-vidyasag
 
 :::{.callout-note collapse="true"} 
 #### Proof
-We first start by establishing a bound on $\EXP[V(θ_{t+1} \mid \ALPHABET  F_t]$. To do so, observe that by Taylor series, we have
+We first start by establishing a bound on $\EXP[V(θ_{t+1}) \mid \ALPHABET  F_t]$. To do so, observe that by Taylor series, we have
 $$
   V(θ + η) = V(θ) + \langle \GRAD V(θ), η \rangle
   + \frac 12 \langle η, \GRAD^2 V(θ + λη)η \rangle
@@ -308,7 +308,7 @@ Subsituting in the above bound, we get:
 \begin{equation}\label{eq:vidyasagar-1-pf-step-1}
 \EXP[V(θ_{t+1}) \mid \ALPHABET F_t] \le V(θ_t) 
 + α_t \dot V(θ_t)
-+ α_t M \bigl[ σ^2 + (σ^2 + L^2)\NORM{θ_t - θ^*}_2^2 \bigr].
++ α_t^2 M \bigl[ σ^2 + (σ^2 + L^2)\NORM{θ_t - θ^*}_2^2 \bigr].
 \end{equation}
 
 ### Proof of part 1. {-}