As in the vonjd's answer martingale property makes some sense only if considered with the risk premium and risk-free rate ("stochastic discount factor" they say). Discounted stock price process is assumed to be a martingale in many studies.
The root of H02's "evil" is Fama's Efficient Market Hypothesis Survey. It is the most clear and comprehensive survey about the efficient markets hypothesis in 1970s. Both in the paper that you referred to and Fama's paper H01 is not highly regarded.
Concisely he divides the hypothesis into three: Weak form, semi strong form and strong form. Weak form deals only with the past price information, semi strong form adds market reaction to public information (i.e. earnings, dividends, etc.) and the strong form adds the information homogeneity (i.e. no part of the market has private information that can be used to obtain extra gains). To the impatient reader, he defends weak form and semi strong form but he admits deficiency to some extent in the strong form (though he still defends it is enough for him to deem the markets efficient).
Most of what he did is to survey previous studies (including his and his colleagues) and draw conclusions by compare and contrast. Weak form is the most interesting since most of the modern mathematical models has at least a notion of it, including option pricing models. If the memory serves, first a term "fair game" is coined. Then skepticism towards random walk models are discussed and random walk theories are deemed not so solid and not even necessary for a market to be efficient. There are also discussions about the Gaussian assumption (in a negative way).
Semi strong form says that the market adjusts for publicly available information and the market is unbiased towards these information. For stock splits, dividends and earnings he cites studies that even if there is extra movement caused by these kind of information it is confined to a very small subset of assets.
Fama says there is limited evidence against strong form, indicating specialists (aka market makers) have access to privileged information (i.e. order limits). But it was not much of an interest to me so I urge you to read from the source.
Final word. You should only take these as an idea and search for more. Well, it is a model and "some" representation of the market. Academics are usually lazy enough to take those for granted and have 'reasons' to do so ("the guy and followers of the theory earned many Nobel prizes").
ps. I love Taleb's remarks (although with some reservation due to my far less than full grasp about the field) about Nobel prizes related to asset price theoreticians. See some here.