The Random Walk Hypothesis is a financial econometric hypothesis that stocks have the same distribution and independent of each other: that stocks are a random variable and not predictable in a macro space.

To set up the random walk hypothesis, let’s begin with some time \(t\), an asset return \(r_t\), some time elapsed \(k\), and some future asset return \(r_{t+k}\).

We will create two random variables \(f(r_t)\) and \(g(r_{t+k})\), which \(f\) and \(g\) are arbitrary functions we applied to analyze the return at that time.

The number of alpha vectors needed to perform PBVI is one for each of your belief sample. Which is a bad idea. Perseus is essentially PBVI, where this idea is explored slightly.

The preamble is the same as PBVI:

we keep track of a bunch of alpha vectors and belief samples (which we get from point selection):

\begin{equation} \Gamma = \{\alpha_{1}, \dots, \alpha_{m}\} \end{equation}

and

\begin{equation} B = \{b_1, \dots, b_{m}\} \end{equation}

To preserve the lower-boundedness of these alpha vectors, one should seed the alpha vectors via something like blind lower bound

a randomized turing machine is a turing machine with functions \(\delta_{0}\), \(\delta_{1}\). During computation, we take either \(\delta_{0}\) or \(\delta_{1}\) each with probability \(\frac{1}{2}\).

See also randomized algorithm.

decision

a randomized TM decides a particular language \(L\) if, \(\forall x \in \Sigma^{*}\), we have that:

\begin{align} &x \in L \implies \text{Pr}\qty [M\text{ accepts } x] \geq \frac{2}{3} \\ &x \not\in L \implies \text{Pr}\qty [M\text{ accepts } x] \leq \frac{1}{3} \end{align}

NOTE! we have to prove this for all \(x\). “most \(x\)” is not good enough.