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Random Walks: the mathematics in 1 dimension
- zodiac 11y agoIsn't the expected distance (undirected) given by E[|d|], while sqrt(n) is the value of sqrt(E[d^2])?
- pash 11y agoThey each measure the same thing, more or less, but it's easier to work analytically with squares than absolute values. Similarly, we tend to work with the variance rather than with expected absolute deviations, we calculate sums of squares rather than sums of absolute values, etc. More fundamentally, root-mean-square is the norm induced by the expectation inner product in the space of random variables. Norms generalize the geometric notion of length, so intuitively RMS is an appropriate measure of the "stochastic distance" from the origin of a random walk after a set number of steps. RMS can likewise be used as an analogue for geometric length for other purposes in a stochastic context, e.g., in calculating the similarity dimension of fractal stochastic processes like Brownian motion.
- Xcelerate 11y agoOne might ask the question: what is the probability that you will return to your starting position over the course of an infinite random walk? On a 1 dimensional or 2 dimensional lattice, that probability is 1. What's crazy though is that for a 3D lattice, the probability is not 1 — it's about 0.3405.
- iaw 11y agoWow, do you have any proofs for this? I'm especially curious about the generalized n-dimensional case.
- hemmer 11y agoThere is some more information and references here: http://mathworld.wolfram.com/PolyasRandomWalkConstants.html http://mathworld.wolfram.com/PolyasRandomWalkConstants.html
- sdoering 11y agoThanks a lot. Had to save this for further reference the moment I saw it.
- Ntrails 11y agoIf you find it GP, I'd love to see where the integral constructed comes from, since that's the clever part rather than the evaluation.
- emfree 11y agoHere's a reference I found for one way to do it: http://www.math.nus.edu.sg/~matsr/ProbII/Lec6.pdf http://www.math.nus.edu.sg/~matsr/ProbII/Lec6.pdf (Theorem 2.1). You define the Green's function G(x, y) = \sum_n Pr_x(S_n=y), where x and y are 3-vectors and Pr_x(S_n=y) is the probability that an n-step random walk starting at x ends up at y. If you have an infinite random walk starting at 0, then G(0, 0) is the expected number of times that the walk returns to 0. That's what the mathworld link calls u(3). You can use Fourier inversion to compute G(0, 0) -- the link gives the gnarly details. It's pretty cool.
- Ntrails 11y agoYou're a scholar and a gentleman, merci buckets
- ccvannorman 11y agoThe probability of returning to the origin follows a very smooth logarithmic curve for dimensions 3 - 8 [copied values from the mathworld link] image: http://imgur.com/CL8MXej http://imgur.com/CL8MXej
- tristanj 11y agoI really love this proof. It's a great example of using maths to prove a counter-intuitive result. They way to prove it is rather clever, and made me appreciate what mathematicians do a lot more. Shame I've never seen it shared online. I was actually hoping the submitted article was a proof of this, but you can't have everything in life.
- deleted 11y ago[deleted]
- awalGarg 11y agoHere is a related lecture from MIT https://youtu.be/56iFMY8QW2k https://youtu.be/56iFMY8QW2k which mathematically proves how it is pretty much impossible to go "happy" from gambling in a club even though intuition says otherwise.
- Dylan16807 11y agoWhat does "happy" mean? It's not hard to set up a bet that gives you an arbitrarily high chance of gaining money, despite an expected value of less than 1.
- sdoering 11y agoOverview why the question on (and need for explanation of) random walks arise: http://www.mit.edu/~kardar/teaching/projects/chemotaxis(AndreaSchmidt)/home.htm http://www.mit.edu/~kardar/teaching/projects/chemotaxis(Andr...
- agumonkey 11y agoHow much of this led to early stages of life ?
- amelius 11y agoRandom walks have been used also to numerically solve differential equations. See e.g. [1] [1] http://www.jstor.org/stable/3612176 http://www.jstor.org/stable/3612176 "A Proof of the Random-Walk Method for Solving Laplace's Equation in 2-D"
- justifier 11y agolove this random walk..dance?.. video: neutral dynamics https://www.youtube.com/watch?v=5P6Dihkrvus https://www.youtube.com/watch?v=5P6Dihkrvus