RANDOM WALK OVER A HYPERSPHERE

In a recent paper the author had shown that a special case of S. M. Joshi transform (so named after the author’s reverent father) of distributions .a (Sbf)(X) (y), 1Fl(a0 bo;+/-Xy) iFl(a’b’-2ixy)> t a characteristic tuiction of a spherical distribution. Using the methods developed in that paper; the problem of distribution of the distance CD, where C and D are points niformly distributed in a hypersphere, has been discussed in the present paper. The form of characteristic function has also been obtained by the method of projected distribution. A generalization of Hammersley’s result has also been developed. The main purpose of the paper is to show that although the use of characteristic functions, usin the method of Bochner, is available in problems of random walk yet distributional S. M. oshi transform can be used as a natural tool has been proved for the first time in the paper. kl6Y WORDS AN[) PHRASI..S. Random walk, Distributional transform, Characteristic function, lypersphere, S. M. Joshi transform, spherical, distribution. 1980 MATHEMATICS LIBJECT CLASSIFICATION. Primary 62, Secondary 60. ;. INTRODUCTION. Bochner [] observed that the Fourier transform of a radial function can be written as a Hankel transform. Though the use of characteristic functions in problems of addition of independent random vectors (using the results of Bochner) is available in the literature, it is not made clear that the Hankel transform is a natural tool. Further, it is not made clear that there is a theory for spherical distributions (i.e. when all directions are equally probable and the distribution of magnitude is independent of direction) in n dimensions which uns parallel to the theory of the addition of random variables in one dimension. Moreover, in actual problems it is frequently necessary to evaluate the Hankel t’ansform involved (in the formula for the probability function), but the ordinary methods of numerical integration are difficult to apply since the Bessel functions make tle integrand oscillate irregularly between positive and negative values. When we try to evaluate the probability function) by epansion methods, the terms contain confluent hypergeometric function,s. A new account would, therefore, appear useful (involving confluent hypergeometric function kernel transform), especially as the use of Distributional Joshi transform

;. INTRODUCTION.Bochner [] observed that the Fourier transform of a radial function can be written as a Hankel transform.Though the use of characteristic functions in problems of addi- tion of independent random vectors (using the results of Bochner) is available in the literature, it is not made clear that the Hankel transform is a natural tool.Further, it is not made clear that there is a theory for spherical distributions (i.e. when all directions are equally probable and the distribution of magnitude is independent of direction) in n dimensions which uns parallel to the theory of the addition of ran- dom variables in one dimension.Moreover, in actual problems it is frequently necessary to evaluate the Hankel t'ansform involved (in the formula for the probability function), but the ordinary methods of numerical integration are difficult to apply since the Bessel functions make tle integrand oscillate irregularly between positive and negative values.When we try to evaluate the probability function) by epansion methods, the terms contain confluent hypergeometric function,s.
A new account would, therefore, appear useful (involving confluent hypergeometric function kernel transform), especially as the use of Distributional Joshi transform [2,3] and of its special case-the Distributional Fourier transform-in certain problems of mathematical physics is now known (See [4]).
In the present note we have tried to discuss the problems of the distribution of the distance CD, when C and D are points uniformly distributed in a (hyper) sphere of radius c in dimensions.
The treatment is based on general methods developed in one o my earlier papers [3].Also a method other that that in [3] has been given here to obtain the form of the characteristic function.
Hammersley's result [5] has been shown to be a natural corol lary of our result.
Let C and D have the vector representation I and 2 in n-dimensional space.Then we require the distribution of CD I 2' where I and 2 are random vectors with identical uniform spherical distribution.But, because of the spherical symmetry 2 has the same distribution as 2 ), the problem is equivalent to finding the dibtribution of I + 2" This we shall attempt presently.In [3]   have shown that the characteristic function of a spherical distribution is an S.M. Joshi transform (so called after my reverent father, who initiated me into researchoutlook [2] S() / P(r) iFl(ao;bo;ir ) iFl((n-l); n-I" -2ir )dr where P(r) P(r <II r + dr); (2.5) But the C.F. of the distribution of I + 2 is the product of characteristic functions of i and 2, as is well known.Multiplying these characteristic functions and invert- ing we get the expression for the probability function for I + $2 as (2.6) : [(1/2(n+l); 1/2(n+3)/2; (I--)] x 2where (2) in p (2) indicates that the sum of two random vectors is being considered.
3. PROOF BY THE METHOD OF PROJECTED DISTRIBUTIONS.
We could derive (2.6) in yet another way-by the method of projected distributions discussed in [3].
If we project from a spherical distribution of random vectors with origin as centre onto a space of lower dimension (vectors being projected orthogonally) we get another spherical distribution.We have the result (see [3]), in terms of SMJ operators, wllere Pn+2k(r) and Pn(r) are probability functions of spherical distributions in n+2k and n dimensions respectively.
In the distribution in n+2k dimensions be uniform over the sphere of radius c, tl,cn (3.|) gives P (r) For k=l, (3.'2) reduces to (2.4) and thus it is evident that a uniform distribution through the volume of an n-dimenslonal sphere can be obtained by a projection from a uniform distribution over the surface of an (n+2)-dimensional sphere (each having the ame radius c).
Further for the sum of two vectors we have, for the (n+2)-dimensional sphere, p Fing (3.3) and (3.l), with k=I we get P(2)(r) as in (2.6).
in the genera] case of projection from the space of (n+2k) dimensions we have k=l this yields, again, (2.6).

GENERALIZATION OF HAMMERSLEY'S RESULT.
Before giving a generalization of Hammersley's result [5], we append a proof of the result (2.) different from the one presented in [3].The method of proof is relevant to the discussions in earlier sections in that it deals with the distribution of the sum of two independent random vectors.
The only defect of this method is that it does not bring out the importance of characteristic functions as such.We require the probability that [41 141 + 42[ r, or 2 -2r r2cos 0) < r where p (r + r 2 ll P, IiI r I, t21 r 2 and 0 is the angle between $I and $2" Then it i known that tile desired probability is given by 7 pl(l) p2( 2) OA( I, $2)d$1d2, where OA() is any (mathematical) characteristic function of the set A viz. ([4] p.l) @A(X) I, x A 0, A, (4.3) where A is the set in which the desired condition is satisfied.Now if we let the characteristic function @A be the integral (which is equal to zero when P r and equal to one when P r) and evaluate the inner integral in (4.2) with the present value of @A ($I' $2 ), we obtain the value of the inner integral for a fixed value of I as n n-1 i( where $2() is the (statistical) characteristic function of the probability function P2(r2) corresponding to the distribution of $2 (n-dimensional), while P(r) in (2.1) refers to the probability function of the distribution of II r which is one dimensional.
The two are related by the well-known result P(r) p(r) rn-l 2gnr n-I n p(r)/r(n) (an is the surface of the unit sphere in n-dimensions).(4.6)In this evaluation we have used the fact P2($2)d$2 P2(r2)r-ldr2dn P2r-lsinn-2OdOdfl where we have used 0 as a co-latitude in defining the position of $2" (See [6] p. 76).
This incidentally proves S(X) SI(X) S2(X) (4.11) i.e. the characteristic function S(X) of the sum of two independent random vectors and 2 is equal to the product of the characteristic functions SI(X of I and S2(X) of 2 inverting and using D'(r) P(r) we get (2.1).

HAMMERS|.EY'S RESULT.
By showing i! + (2 is asymptotically distributed in a normal distribution with 2 mean 2c and variance c /2n as n tends to infinity, Hammersley [5] has proved that for large va|ues of n the distance between two points in a sphere is nearly always equal to 21/2c-the diagonal of the rectangle determined by the orthogonal radii.Now as shown in [3] the distribution (3.2) has the characteristic function iFl(aO'bo'iC) IFl(n+k-" n-I + 2k; -2ic). (5.1) But by [7]  Thus the distribution, as is clear from (5.1), is asymptoticaIlynormalwith 2--n+2------" (5.4) Also, taking k=l (3.2)we get (2.4)and thus (2.4) is asymptotically normalwithvariance 2 nc (n+2) The distribution of 1 + 2 is, therefore, asymptotically, normal with variance double that of the above.

Again taking k
O, we see that the distribution uniform over the surface of the sphere with radius d is asymtotic to normal distribution with variance d 2. Comparing this variance with the variance just obtained for I + 2' we notice that the distri- butation of I + 42 is asymptotic to the distribution over the surface of the sphere of radius (2n)c/(n+2) 1/2, which again is asymptotically equivalent to 2c(I + ).Thus have Hamersley's result.
IXl, where X is variable and ao,b O are the values of a and b at O while a a + + I, b a + .Now for a distribution uniform in a sphere of radius c we have n;bo;iC) F ((n+l); n+l; -2ic) with probability functions PI(I) and P2(42) then the distri- bution function D(r) of 141 is given by 212(n+l) n+l; 2irK)dg, distribution function (or so called probability integral, D(r) + r 1F ((n-1); n-I; -2irlK) 1Fl((n+l); n+l; -2ir)dK drld