s23-probability-theory/inputs/a_1_counterexamples.tex

\section{(Counter)examples}

Consistent families and inconsistent families

Notions of convergence

Exercise 4.3
10.2


\begin{example}[Martingale not converging in $L^1$]
    Let $\Omega = [0,1]$, $\bP = \lambda\upharpoonright [0,1]$.
    Define $X_n \coloneqq 2^n \cdot \One_{[0,2^n]}$,
    and let $(\cF_n)_n$ be the canonical filtration.
    Then $(X_n)_{n}$ is a Martingale
    with $\bE[X_0] = 1$,
    but $X_n \xrightarrow{a.s.} 0$.
\end{example}


Stopping times

\begin{example}[{Martingale such that $\bE[X_T] \neq \bE[X_0]$}]
    Consider the simple random walk and $T = \inf \{n : X_n \ge 1\}$.
    Obviously $X_T = 1$.
\end{example}