Brownian Motion an the Stock Market We also have the dispersion of Y r)-V(Y)-|8(Y) √∑:叫=Vk=Vn/6σ Hence o=o'/v6 Is the dispersion developed at the end of unt time, Ie, In a number 1/8 of trades The central lmt theorem assures that Y(r)will approach a normal distribution for large k whatever the distrbution function of the y(i) The above considerations have an obvious analog in the diffusion of a molecule undergoing collIsions wIth its neighbors Regardless of the Intermolecular force law (ef the factors ignored in hypothecating the Weber-Fechner law), the dispersion in the probabilty distribution of posl tion of a particle nItially (at time t)located at some point wIll increase as the square root of the time interval T after t The phenomenon of the persistence of velocities, whIch the stock market also possessesdoes not alter this conclusion THE OBSERVATIONAL DATA Let us now examine the data to see in w hat particulars the above expec tations are fulfilled Figures 7 and 8 support, at least approximately, the conclusion of normality for Y(), at least for intervals =1 month and I year We would now hke to estImate how the distribution function of Y(r) changes wIth the tIme Interval Owing to the wealth of data, and for purposes of computational sImplicIty, we shall not evaluate or( dr rectly Instead, we shall evaluate the semI-interquartile range(sqr )of Y(r), or one-half of the range(n ratio units) between the 25 per cent and 75 per cent points of a sample For a normally distributed population, which Figs 7 and 8 Indicate is quIte closely the case here, sqr=prob- able error'of Y()=06745 or(r) In any event the s gr has a definite statistical interpretation, whether the distribution 1s normal or not To evaluate the sgr we take a random sample of common stocks at some random starting date t, write down the prices of each on that date and the prices of the same stocks at intervals r of a day, week, month, two months, ete, later All of these prices are then divided by their corre- sponding starting()prIces, and the ratios of each plotted for the vanous ntervals One-half of the interquartile range is obtained by taking one- half of the natural logarithm of the ratio of ratios from the 25 per cent to 75 per cent point of the sample This s q r is then plotted in Fig 9 Simular data on a tradang-tume scale appears in Fig 10 As can be seen from an Inspection of these two figures for various starting dates from