Malkiel first publishes the classic A Random Walk Down Wall Street.
Economists and believers in the random walk hypothesis apply this to the stock market.
Random Walk Theory Definition  Investopedia
Bachelier’s original version of the random walk hypothesis was crude by today’s standards. It had prices follow an arithmetic random walk with zero drift. The modern version developed out of the work of multiple researchers, including those already mentioned, as well as Osborne (1959), Moore (1960), Alexander (1961. 1964) and Granger and Morgenstern (1963). It states that the log returns follow an arithmetic random walk with a drift reflecting the longterm return from equity investment. Stated another way, prices follow a geometric random walk with drift.
The random walk hypothesis is more an empirical observation than a theoretical result. Fundamentally, it is an empirical observation that price series are well modeled with a random walk. However, researchers did offer a theoretical explanation for why prices should follow a random walk. They noted that prices change in response to news items—earnings reports, releases of economic indicators, merger announcements, etc. If news items are assumed to arise independently (the relative probabilities of upcoming news being good or bad is unaffected by whether recent news has been good or bad) then price changes should be independent. Also, the volume of news items affecting a price is sufficiently large that the central limit theorem applies, and price changes over any discernible period should be approximately normal. Somewhat after the fact, Samuelson (1965) and Mandelbrot (1966) rigorously formalized this theoretical justification of the random walk hypothesis.
The squarerootoftime pattern in its confidence bands for longtermforecasts is of profound importance in finance (it is the basis of the theoryof options pricing), and the random walk model often provides a good benchmarkagainst which to judge the performance of more complicated models.
Financial Concepts: Random Walk Theory  …
he idea behind the random walk theory or as it is often called – the efficient market hypothesis, is that information is unpredictable and random and accordingly stock prices also move unpredictably. Let us for a moment assume that some formula with great confidence predicts that the share price of Company X which is currently at 100, will rise dramatically in three days and reach 110.
And because the standard deviation of the forecasterror is the square root of its variance, it follows that the standard error of a kstepahead forecastis larger than that of the 1stepahead forecast by a factor ofsquarerootofk. This is the socalled "square root of time"rule for the errors of random walk forecasts, and it explains thesidewaysparabola shape of the confidence bands for longterm forecasts: that'sthe shape of the graph of Y=SQRT(X).
Random Walk Model  Duke University
In the early 1960s, the literature of the random walk hypothesis took two new directions. The first would extend results as an efficient market hypothesis. The other looked for flaws in the random walk model. Two obvious flaws, which were noted early on, were the fact that log price changes were
leptokurtic, rendering the normal distribution an imperfect representation, and
heteroskedastic.
The first, in particular, was noted by Osborne (1959) and Alexander (1961). Mandlebrot (1963) proposed that leptokurtosis be addressed by replacing the normal distribution of a random walk with a stable Paretian distribution.
Neither flaw affords technicians trading opportunities, but researchers sifted through massive volumes of historical price data, looking for flaws that might. Their published results formed the literature on market anomalies which ultimately contributed to the dubious field of behavioral finance.
Again, though, the mean value of the steps in a finite sample of randomwalkdata generally does not provide a good estimate of the current rate of drift,if any.
Random Walk Theory  Efficient Market Hypothesis

Hollenbeck on Random Walk Hypothesis  YouTube
He concluded that the random walk model best fits the data, but found leptokurtosis in the distribution of returns.

Random Walk Theory Definition & Example  …
He (correctly) focussed on the concept of a martingale, rather than a random walk (as in Fama (1965)).

The Random Walk Hypothesis and Correlation in the …
Random walk  Wikipedia
hence indicating a random walk market.
Holbrook Working was an agricultural economist with Stanford University’s Food Research Institute. In a 1934 study, he compared time series of historical commodity price changes to time series of random numbers. He wanted to determine what nonrandom price patterns might be exploited by traders to realize speculative profits. Using statistical techniques, he was unable to distinguish the series of price changes from the series of random numbers. He concluded that there were no predictive patterns in the price changes—that the prices were entirely random. He showed graphs of the time series to professional commodity traders. They too were unable to distinguish the series of price changes from the series of random numbers.
Maurice Kendall was one of the great statisticians of the 20th century. In 1953, he published a groundbreaking empirical study of weekly changes in nineteen indices of British industrial share prices and in spot prices for New York cotton and Chicago wheat. His goal was to advance the field of technical analysis by introducing statistical rigor. He was startled to find that the random component of prices swamped any autocorrelations. Frustrated, he concluded
The series looks like a wandering one, almost as if once a week the Demon of Chance drew a random number from a symmetrical population of fixed dispersion and added it to the current price to determine the next week’s price.
Random walk theory  SlideShare
If you simulate a random walk process (for example, by building aspreadsheet model that uses the RAND() function in the formula for generatingthe step values), you will typically find that different iterations of the samemodel will yield dramatically different pictures, many of which will havesignificantlooking trends, as shown in the mentioned above.
Random Walk Hypothesis: Evidence from Market Efficiency …
The random walk hypothesis is not so much a hypothesis as it is a model that has been found to be surprisingly useful for describing the behavior of prices in various markets, including major equity, fixed income and commodities markets. It states that price series do not exhibit predictive patterns over time but can best be described with a random walk. Accordingly, the random walk hypothesis is a rejection of technical analysis.
An early version of the random walk hypothesis was proposed by Louis Bachelier (1900) in his famous doctoral thesis Théorie de la Spéculation. Bachelier studied the market for forwards and options on French government bonds, finding a number of important results. As part of that work, he discovered the mathematics of Brownian motion—five years before Albert Einstein independently did so. With regard to the markets, Bachelier noted
The influences that determine the movements of the exchange are innumerable; past, current and even anticipated events that often have no obvious connection with its changes … it is thus impossible to hope for mathematical predictability.
He went on to conclude that, if the market’s movements cannot be predicted,
The mathematical expectation of the speculator is zero.
Bachelier’s thesis was decades ahead of its time. It was ignored for over a half century, but other researchers, acting independently, started to draw similar conclusions.