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Suppose we pick $m$ vectors i.i.d from the surface of a $d$-dimensional unit sphere (they all have length 1). What would be the expected length of their sum?

Equivalently, we can ask about the expected length of their average.

I managed to solve it for 2 vectors in 2 dimensions, but I need a generic solution...

If there's any "known solution" for this problem, even a link will be good enough.

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  • $\begingroup$ Hint: use linearity of expectation. $\endgroup$
    – Igor Rivin
    Commented Jan 4, 2011 at 0:56
  • $\begingroup$ Do you need an exact answer, or just an approximation? If $d$ is large then you could try approximating the vectors by rescaled Gaussian vectors; the sum of the vectors will itself be a Gaussian vector, and the length will be the square root of a rescaled chi-squared r.v. $\endgroup$
    – Yemon Choi
    Commented Jan 4, 2011 at 0:58
  • $\begingroup$ I need an exact answer $\endgroup$
    – Oren
    Commented Jan 4, 2011 at 0:59
  • $\begingroup$ @Igor I'm not seeing how to use your hint. Have you actually thought through a complete solution? I see that this gives me an easy way to compute E(length^2), but this doesn't tell me what E(length) is. $\endgroup$
    – David E Speyer
    Commented Jan 4, 2011 at 1:03
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    $\begingroup$ This isn't very closely related, but I can't resist mentioning the amazing fact that the probability that this length is $<1$ in 2 dimensions is $1/(m+1)$. This might mean there's some hope of a not-too-ugly formula in 2 dimensions. $\endgroup$
    – Peter Shor
    Commented Jan 4, 2011 at 17:52

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OK, to atone for my rash comment, I recalled that I needed something related in:

@article {MR2356429, AUTHOR = {Rivin, Igor}, TITLE = {Surface area and other measures of ellipsoids}, JOURNAL = {Adv. in Appl. Math.}, FJOURNAL = {Advances in Applied Mathematics}, VOLUME = {39}, YEAR = {2007}, NUMBER = {4}, PAGES = {409--427}, ISSN = {0196-8858}, MRCLASS = {52A38 (28A75 33C65 60F99)}, MRNUMBER = {2356429 (2008k:52012)}, DOI = {10.1016/j.aam.2006.08.009}, URL = {http://dx.doi.org/10.1016/j.aam.2006.08.009}, }

(see page 414). And found it in the book by Mathai and Provost:

@book {MR1192786, AUTHOR = {Mathai, A. M. and Provost, Serge B.}, TITLE = {Quadratic forms in random variables}, SERIES = {Statistics: Textbooks and Monographs}, VOLUME = {126}, NOTE = {Theory and applications}, PUBLISHER = {Marcel Dekker Inc.}, ADDRESS = {New York}, YEAR = {1992}, PAGES = {xxii+367}, ISBN = {0-8247-8691-2}, MRCLASS = {62H10 (62E15)}, MRNUMBER = {1192786 (94g:62110)}, MRREVIEWER = {Donald R. Jensen}, }

It should be in the vicinity of page 62. The argument uses Laplace transforms, and is not trivial.

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  • $\begingroup$ Any chance you want to provide a brief explanation for why your comment was rash? It of course seemed reasonable to someone as naive as me. $\endgroup$
    – Deane Yang
    Commented Jan 4, 2011 at 3:16
  • $\begingroup$ As @David Speyer pointed out, I was computing the expectation of the SQUARE of the length (which equals $m$, btw), but this does not tell you what the expectation of the length is... $\endgroup$
    – Igor Rivin
    Commented Jan 4, 2011 at 3:22
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At http://www.carma.newcastle.edu.au/~jb616/walks2.pdf Jonathan Borwein and co-authors discuss $$\int_{[0,1]^n}\left|\sum_{k=1}^ne^{2\pi ix_k}\right|^s\thinspace dx$$ which you may find has some relevance to your question. That paper also has references to others where such integrals are discussed, and there's another paper on Jonathan's website on a closely related topic.

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