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Fix a number m and let us take a set, say A, of unit vectors {v_1,...,v_k} in R^n. Assume that k is large, say exponentially large in n (k=e^{cn}). Let X be the euclidean length of a random sum of m vectors in A (all sums are equally likely).

It is quite intuitive that the typical length of such sum should be sqrt(m) and it is not that hard to verify it.

Question: How does the variance of X behave? More precisely, is it true that VarX=o(sqrt(m)). If so, could one get a better rate? Maybe it is m^1/4?

I would be very grateful for any information or link.

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  • $\begingroup$ This seems strange; your unit vectors, are these (1,0,...) and (0,1,0,0,...) and so on, or just vectors of length 1? If the latter, the expected mean should be the zero vector then, no? $\endgroup$
    – Per Alexandersson
    Commented Feb 17, 2013 at 14:23
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    $\begingroup$ Is not the question asking for statistics for the total travel of a random walk, each of whose steps is of length $1$? $\endgroup$
    – Joseph O'Rourke
    Commented Feb 17, 2013 at 14:58
  • $\begingroup$ It sounds like once you figure out what question you want to ask, the answer will be the Central Limit Theorem (for vector-valued summands, or for each coordinate if you don't mind losing a factor). If you mean something deeper than that, please clarify. $\endgroup$
    – Douglas Zare
    Commented Feb 17, 2013 at 19:58
  • $\begingroup$ Per Alexandersson: there are n vectors with 1 as a coordinate and we have exponentially many, so I clearly mean any vector of legth 1. Joseph O'Rourke: it is - it is the about the average length of a random sum x_1+...+x_m, that is, it's euclidean length. Douglas Zare: I think I have really meant what I have written. Anyhow, how should I use the CLT in this case? I really need just a bound on the variance of the length of such a sum, nothing more. $\endgroup$
    – TOM
    Commented Feb 18, 2013 at 3:06
  • $\begingroup$ TOM, your reply to Per adds to my confusion. You refer to a finite set of vectors (all 0 except for a single 1) then you say "any vector of length 1" (of which there are infinitely many). You put your vectors in $R^n$, not $Z^n$, remember. What is the complete definition of $A$? $\endgroup$
    – Brendan McKay
    Commented Feb 19, 2013 at 1:38

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I am assuming that the OP means that the initial $k$ vectors are themselves a sample of the uniform distribution on the sphere, so the question is both about the limiting distribution of the length (which is normal) and the speed of convergence to this distribution. This is a much studied (and highly nontrivial) question: for a nice analysis of the two dimensional case (and extensive references), see:

Random Unit Vectors II: Usefulness of Gram-Charlier and Related Series in Approximating Distributions

David Durand and J. Arthur Greenwood Source: Ann. Math. Statist. Volume 28, Number 4 (1957), 978-986.

(available for free on Project Euclid).

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  • $\begingroup$ Thank you, that is most useful and interesting! But still, the initial set is arbitrary - not random - I want to have a given large set of unit vectors (much larger than all other parameters) and then I want to take a random m-tuple of vectors from that set. Let us for simplicity agree that the sum of all vectors in A is 0, so that the mean of the random sample is 0. Then second moment of the length of a random m-tuple is at most m (straightforward calculation) and so the mean is at most sqrt(m) (by Cauchy-Schwarz). Thus we know that most of the sums lie within a ball of radius const*sqrt(m). $\endgroup$
    – TOM
    Commented Feb 19, 2013 at 5:24

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