Karl Pearson introduced the term "Random Walk". He was interested in describing the spatial/temporal evolutions of mosquito populations invading cleared jungle regions. He found it too complex to model deterministically, so he conceptualized a simple random model.
Pearson posed his problem in Nature (27 July 1905):
A man starts from a point 0 and walks l yards in a straight line; he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times. I require the probability that after n of these stretches he is at a distance between r and r + δr from his starting point.
The question was answered the following week by Lord Rayleigh, who pointed out the connection between this problem and an earlier paper of his published in 1880* concerned with sound vibrations. Rayleigh pointed out that, for large values of n, the answer is given by
This acutally has the shape of a normal distribution, centered at the origin.
In Nature, on 10 August 1905 Karl Pearson wrote, in relation to Rayleigh's letter and reference to his earlier work:
I ought to have known it, but my reading of late years has drifted into other channels, and one does not expect to find the first stage of a biometric problem provided in a memoir on sound..
He went on to comment on the solution:
The lesson of Lord Rayleigh's solution is that in open country the most probable place of finding a drunken man who is at all capable of keeping on his feet is somewhere near his starting point.
Rayleigh first solved the one-dimensional problem where the walker can only go forward or backward. Then he solved the more difficult case when n/2 steps are in the x direction, and n/2 steps are in the y direction. Near the end of his life, he returned to his problem, but this time in three-dimensions, a problem called random flight. Just as Pearson missed Rayleigh's work, Rayleigh missed Smoluchowski's 1906 paper on the motion of colloidal particles, in which he introduces the random flight idea.
"A drunk man will find his way home, but a drunk bird may get lost forever"**.
So what did professor Kakutani (Yale) mean by this? Well, in the one and two-dimensional case the drunkard will eventually come back to the starting point (with probability one). But in ≥ 3-dimensions, there is a positive chance he will never come back. Mathematically speaking: A random walk is recurrent if d=1 or d=2, and transient if d ≥ 3. So please don't get drunk in outer space. A proof of this property and a nice introduction to Markov chains can be found here.
Source:
Random and self-avoiding Walks, Tony Guttmann
*see also Brownian Motion / Wiener Process / Random Walk
** Olivier Hardouin Duparc notes (via email): Professor Kakutani's statement actually refers to Georg Pólya's theorem 1921:
Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz Math. Ann., 84:149-160, 1921
Professor Polya used to say that in a plane, 'all ways lead to Roma'.
Pearson posed his problem in Nature (27 July 1905):
A man starts from a point 0 and walks l yards in a straight line; he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times. I require the probability that after n of these stretches he is at a distance between r and r + δr from his starting point.
The question was answered the following week by Lord Rayleigh, who pointed out the connection between this problem and an earlier paper of his published in 1880* concerned with sound vibrations. Rayleigh pointed out that, for large values of n, the answer is given by
This acutally has the shape of a normal distribution, centered at the origin.
In Nature, on 10 August 1905 Karl Pearson wrote, in relation to Rayleigh's letter and reference to his earlier work:
I ought to have known it, but my reading of late years has drifted into other channels, and one does not expect to find the first stage of a biometric problem provided in a memoir on sound..
He went on to comment on the solution:
The lesson of Lord Rayleigh's solution is that in open country the most probable place of finding a drunken man who is at all capable of keeping on his feet is somewhere near his starting point.
Rayleigh first solved the one-dimensional problem where the walker can only go forward or backward. Then he solved the more difficult case when n/2 steps are in the x direction, and n/2 steps are in the y direction. Near the end of his life, he returned to his problem, but this time in three-dimensions, a problem called random flight. Just as Pearson missed Rayleigh's work, Rayleigh missed Smoluchowski's 1906 paper on the motion of colloidal particles, in which he introduces the random flight idea.
"A drunk man will find his way home, but a drunk bird may get lost forever"**.
So what did professor Kakutani (Yale) mean by this? Well, in the one and two-dimensional case the drunkard will eventually come back to the starting point (with probability one). But in ≥ 3-dimensions, there is a positive chance he will never come back. Mathematically speaking: A random walk is recurrent if d=1 or d=2, and transient if d ≥ 3. So please don't get drunk in outer space. A proof of this property and a nice introduction to Markov chains can be found here.
Source:
Random and self-avoiding Walks, Tony Guttmann
*see also Brownian Motion / Wiener Process / Random Walk
** Olivier Hardouin Duparc notes (via email): Professor Kakutani's statement actually refers to Georg Pólya's theorem 1921:
Über eine Aufgabe der Wahrscheinlichkeitsrechnung betreffend die Irrfahrt im Straßennetz Math. Ann., 84:149-160, 1921
Professor Polya used to say that in a plane, 'all ways lead to Roma'.
Mahalanobis - am 2004-05-28 05:41 - Rubrik: mathstat
