The title already reveals my question. I was wondering how specific the characterisitics of a random walk are defined and if every time series that is not predictable belongs to the class of random walks.

Edit: unpredictable in a sense that the past observations provide no information which makes predictions of the future feasible.

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    $\begingroup$ Please explain what you mean by "not predictable:" either this has a clear quantitative definition, which is needed to answer your question; or it does not, in which case your question is unanswerable. $\endgroup$
    – whuber
    Jun 7, 2021 at 15:54
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    $\begingroup$ It also depends on what you mean by a random walk $\endgroup$
    – Henry
    Jun 7, 2021 at 16:32
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    $\begingroup$ Also, it is not true that you cannot predict a random walk. You just cannot do better than saying that it will stay where it currently is, which, in view of large variance of the random walk, then leads to large prediction errors. $\endgroup$ Jun 8, 2021 at 7:36
  • $\begingroup$ Your definition of "unpredictable" is a fine one, but it leads to a trivial answer: unpredictable processes have independent marginal distributions, by definition. That's not true of any "random walk" in the conventional sense. There is a relationship: random walks in a general sense (of solutions of stochastic differential equations) are built out of processes that have independent identically distributed marginals (or, in even greater generality, out of Martingale processes). $\endgroup$
    – whuber
    Jun 8, 2021 at 16:17


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