"To every event defined for the original random walk there corresponds an event of equal probability in the dual random walk, and in this way almost every probability relation has its dual." (William Feller, "An Introduction To Probability Theory And Its Applications", 1950)
"A random walk is a stochastic process, which is a general term for a chain of events which change in a random way with time. Sequential binomial sampling is another example of a stochastic. process, since each 'step' in the 'walk' represents the result of sampling one more item from the population to determine whether it is to be accepted or rejected. The decision to be reached by the process is to determine whether or not to accept a certain hypothesis about the population. Such a hypothesis might relate to the proportion of defective members, the proportion of members having a property within specified limits, or some similar statement." (David B MacNeil, "Modern Mathematics for the Practical Man", 1963)
"Random walk processes are significant, not only in many types of physical phenomena, but also in basic statistical methods, including decision making and the related sequential sampling." (David B MacNeil, "Modern Mathematics for the Practical Man", 1963)
"However, random walk theory also tells us that the chance that the balance never returns to zero - that is, that H stays in the lead for ever - is 0. This is the sense in which the 'law of averages' is true. If you wait long enough, then almost surely the numbers of heads and tails will even out. But this fact carries no implications about improving your chances of winning, if you're betting on whether H or T turns up. The probabilities are unchanged, and you don't know how long the 'long run' is going to be. Usually it is very long indeed." (Ian Stewart, "The Magical Maze: Seeing the world through mathematical eyes", 1997)
"In everyday language, a fair coin is called random, but not a coin that shows head more often than tail. A coin that keeps a memory of its own record of heads and tails is viewed as even less random. This mental picture is present in the term random walk, especially as used in finance." (Benoit B Mandelbrot, "Fractals and Scaling in Finance: Discontinuity, concentration, risk", 1997)
"A random walk is one in which future steps or directions cannot be predicted on the basis of past history. When the term is applied to the stock market, it means that short-run changes in stock prices are unpredictable. Investment advisory services, earnings forecasts, and chart patterns are useless. [...] What are often called 'persistent patterns' in the stock market occur no more frequently than the runs of luck in the fortunes of any gambler playing a game of chance. This is what economists mean when they say that stock prices behave very much like a random walk." (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)
"Perhaps the most common complaint about the weakness of the random-walk theory is based on a distrust of mathematics and a misconception of what the theory means. 'The market isn't random', the complaint goes, 'and no mathematician is going to convince me it is'. [...] But, even if markets were dominated during certain periods by irrational crowd behavior, the stock market might still well be approximated by a random walk. The original illustrative analogy of a random walk concerned a drunken man staggering around an empty field. He is not rational, but he's not predictable either." (Burton G Malkiel, "A Random Walk Down Wall Street", 1999)
"The concept of a random walk is simple but rich for its many applications, not only in finance but also in physics and the description of natural phenomena. It is arguably one of the most founding concepts in modern physics as well as in finance, as it underlies the theories of elementary particles, which are the building blocks of our universe, as well as those describing the complex organization of matter around us." (Didier Sornette, "Why Stock Markets Crash: Critical Events in Complex Systems", 2003)
"Monte Carlo is able to discover practical solutions to otherwise intractable problems because the most efficient search of an unmapped territory takes the form of a random walk. Today’s search engines, long descended from their ENIAC-era ancestors, still bear the imprint of their Monte Carlo origins: random search paths being accounted for, statistically, to accumulate increasingly accurate results. The genius of Monte Carlo - and its search-engine descendants - lies in the ability to extract meaningful solutions, in the face of overwhelming information, by recognizing that meaning resides less in the data at the end points and more in the intervening paths." (George B Dyson, "Turing's Cathedral: The Origins of the Digital Universe", 2012)
"A random walk, in general, is sometimes described as being similar to how a drunk person traverses a city. They know what direction or end point they want to reach but may take a very circuitous route to get there. The algorithm starts at one node and somewhat randomly follows one of the relationships forward or backward to a neighbor node. It then does the same from that node and so on, until it reaches the set path length. ('We say somewhat randomly because the number of relationships a node has, and its neighbors have, influences the probability a node will be walked through.)'" (Mark Needham & Amy E Hodler, "Graph Algorithms: Practical Examples in Apache Spark and Neo4j", 2019)
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