# Is a random walk necessarily a martingale?

I read in the following notes (http://www2.econ.iastate.edu/classes/econ672/Falk/_notes/lecture_4_martingales.pdf, p.2) that "a random walk is a martingale."

Although it seems logical with the used definitions, I am wondering if this statement is universal or if it depends on extra assumptions about the stochastic process.

• The demonstration on p. 2 of your reference follows immediately upon the statement you quote. However, it's erroneous! (It implicitly assumes $E(\varepsilon_i)=0$.) – whuber Aug 8 '17 at 20:39
• I agree that the author implicitly assumes a extra structure of ${\epsilon_i}$ but in my opinion we just need the following weaker assumption: \begin{align*} & E(\epsilon_i|z_{i-1},z_{i-2},z_{i-3},...) = 0 \end{align*} However, this assumption is part of the basic definition of a random walk as far as I understand. – KroneN Aug 9 '17 at 8:21
• If $S_n=\sum_{k=1}^n X_k$ is a random walk with $\mathbb E[X_k]=c$, then $S_n-nc$ is a martingale. – Math1000 May 18 '18 at 1:11