683
[nternat. J. Math. & Math. Sci.
Vol. 8 No. 4 (1985) 685-688
RANDOM WALK OVER A HYPERSPHERE
J. M. C. JOSHI
Shipra Sadan
G. I. C. Road, Pittogarab 262501, U. P. India
(Received November 10, 1983 and in revised form July 23, [985)
In a recent paper the author had shown that a special case of S. M. Joshi
ABSTRACT.
transform (so named after the author’s reverent father) of distributions
.a
(Sbf)(X)
t
(y),
1Fl(a 0 bo;+/-Xy) iFl(a’b’-2ixy)>
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 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.
it is not made clear that there is a
Further,
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
684
J.M.C. JOSHI
[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
of radius
in one
o
C
D
and
are points uniformly distributed in a
(hyper) sphere
The treatment is based on general methods developed
dimensions.
in
c
my earlier papers [3].
Also a method other that that in [3] has been given here to obtain the form of the
Hammersley’s result [5] has been shown to be a natural corol
characteristic function.
lary of our result.
2.
PROBLEM OF D[S]’RIBUTION ON THE HYPERSPHERE.
Let
space.
C
D
and
I
have the vector representation
I
CD
Then we require the distribution of
2
and
2’
random vectors with identical uniform spherical distribution.
2
spherical symmetry
has the same distribution as
I
to finding the dibtribution of
+
in n-dimensional
I
where
2 ),
2
and
the problem is equivalent
This we shall attempt presently.
2"
are
But, because of the
In [3]
have shown that the characteristic function of a spherical distribution is an S.M. Joshi
transform (so called
S()
/
P(r)
2
after
P(r)
my reverent father, who initiated me into researchoutlook [2]
iFl(ao;bo;ir
(2.1)
) iFl((n-l); n-I" -2ir )dr
witl
-n+2
{F(1/2n)} -2
O
S()(r)
x
n-I
F
(ao;bo;
-ir ) x
iFl((n-l); n-l" 2ir)
(2.2)
d,
where
P(r)
P(r
r
(2.3)
+ dr);
and not of
being characteristic function of n-dimensional distribution of
S(
l-dimensional
of
<II
and
a
b
II-
r
O
at
uniform in a sphere of radius
P(r)
nr
IXl,
Here
while
c
where
a
a
+
X
.
is variable and
+ I, b
a
c
r
0
r
c.
+
are the values
ao,b O
Now for a distribution
we have
n-l-n
c
(2.4)
O,
Hence (2.1) gives
S()
iFl(ao;bo;iC)
F ((n+l); n+l; -2ic)
But the C.F. of the distribution of
of
i
and 2,
as is well known.
I
+ 2
is the product of characteristic functions
Multiplying these characteristic functions and invert-
ing we get the expression for the probability function for
n-1 -n
2
r
nr
c
(n+l) x
(I
(n+l)
p(2)(r)
B(1/2(n+l) 1/2)
x
:
2-
(2.5)
}-I
---)
4c
[(1/2(n+l); 1/2(n+3)/2; (I-
2
I +
$2
as
(2.6)
-)]
where (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].
RANDOM WALK OVER A HYPERSPHERE
685
If we project from a spherical distribution of random vectors with origin as centre
space of lower dimension (vectors being projected orthogonally) we get another
onto a
We have the result (see [3]), in terms of SMJ operators,
spherical distribution.
s"
P (r)
n
P+2 k (r)]
n, k
B(n k)
Pn+2k(r)
wllere
n+2k
and
n
r
n-I
r
Pn(r)
and
r2) k-1 4-n-2k+2d4
are probability functions of spherical distributions in
dimensions respectively.
In the distribution in
tl,cn
(3.1)
Pn+2k (r)(2
dimensions be uniform over the sphere of radius
n+2k
gives
(3.|)
F(21])(!n+k)(k)
P (r)
n
c-n-2k+2(c2-r)k-lrn-l’
_>
c
r
(3.2)
=0,
For
c,
c
r.
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
ame radius
(n+2)-dimensional sphere (each having the
Further for the sum of two vectors we have, for the (n+2)-dimensional
c).
sphere,
p(2)
n+2
21-n
(r)
-’n n
r (4c
B(1/2(n+l),)
r
2 n-1/2
<_
r
2c
(3.)
()’
Fing (3.3) and (3. l), with
k=I
we get
P(2)(r)
as in
in the genera] case of projection from the space of
p(2)(r)
2a
(,
2c.
r
f
(4c
k+(n-3)
2
(4
r2
k-1
(2.6).
(n+2k)
dimensions we have
-2k+2d
(3.4)
r
k=l
]or
4.
this yields, again, (2.6).
GENERALIZATION OF HAMMERSLEY’S RESULT.
Before giving a generalization of
Hammersley’s result [5],
result (2.) different from the one presented in [3].
we append a proof of the
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.
THEOREM 4.1.
I + 42
Let
bution function
D(r)
D(r)
21-n
(n)
with probability functions
of
n
141
r
(n+l)
PI(I)
and
P2(42)
then the distri-
is given by
f S()
n-1
o
F (a ;b ;-ir) x
0 0
(4.1)
x
where
()
1Fl(1/2(n+l)
1()$2()"
PROOF.
We require the probability that
[41
141 + 42[
r, or
n+l; 2irK)dg,
686
J.M.C. JOSHI
p
(r
ll
2
+ r 2 -2r
2
IiI
P,
r2cos
r
< r
0)
t21
I,
r
where
0
and
2
is the angle between
$I
and
$2"
Then it i known that tile desired probability is given by
7
where
any (mathematical) characteristic function of the set
OA() is
@A(X)
I,
A
([4] p.l)
(4.3)
is the set in which the desired condition is satisfied.
r
n
F (1/2n) F (n+1) /
o
iFl(1/2(n+l)
X
n+l;
(which is equal to zero when
X
En-le(n-P)iE
(4.4)
-2irE)iFl((n-l)
P
n-l; 2iPE)dE,
and equal to one when
r
I
the inner integral for a fixed value of
r
X
n
((n+1)) I
(n)
o
iFl(1/2(n-1)
n-l"
fo IFI
$2()
be the integral
@A
r)
and evaluate the
we obtain the value of
as
+
n-1 i(r
$2()
P
@A ($I’ $2 ),
inner integral in (4.2) with the present value of
$2()
viz.
A,
Now if we let the characteristic function
where
A
A
x
0,
where
(4.2)
pl(l) p2(2) OA( I, $2)d$1d2,
e
-2ir1) 1Fl(1/2(n+l);
(ao’bo;(i$2"r2)P2(r2)dr 2.
X
r2)
n+l;
-2ir2)d,
Here, as isevident from R.H.S. of (4.5),
is the (statistical) characteristic function of the probability function
corresponding to the distribution of
$2
(n-dimensional), while
to the probability function of the distribution of
II
P2(r2)
P(r) in (2.1) refers
r which is one dimensional.
The two are related by the well-known result
P(r)
rn-l n 2gnr n-I p(r)/r(n) (an
p(r)
is the surface of the unit sphere
(4.6)
in n-dimensions).
In this evaluation we have used the fact
P2(r2)r-ldr2dn P2r-lsinn-2OdOdfln-l’
P2($2)d$2
0
where we have used
as a co-latitude in defining the position of
(4 7)
$2"
(See [6] p. 76).
Evaluating the final integral in (4.2), using
pt(r)r-Idn,
pl(i)d$1
(4.8)
r
we obtain the distribution function
as
(or so called probability integral, D(r)
n
D(r)
r__ {F(n) r(1/2(n+l))
2
e
i(r
+
integration over
n-1
r
n
P(r)dr)
n-1
;
"Pl(rl)2(K)rI-1K
n
(4.9)
1F
2
o
((n-1); n-I;
-2irlK) 1Fl((n+l);
gives rise to the factor
an
n+l; -2ir)dK
drld n
and that with respect to r
yieids
RANDOM WALK OVER A HYPERSPHERE
{F(n)}
{ Pl(r
2 n-i
(2
-n
Thus, finally, we have
by
)r
?-I
(n-I
l)(r)
n-I
F
687
((n-l);n-l;-2irl)dr
SI( ).
as in
(4.10
(4.1) which is (2.2.12) of [5]" with S() replaced
S1()$2().
This incidentally proves
SI(X) S2(X)
S(X)
(4.11)
i.e. the characteristic function S(X) of the sum of two independent random vectors
and
of
2
is equal to the product of the characteristic functions
2
inverting and using D’(r)
5.
i!
By showing
2c
+
(2
va|ues
21/2c-the
equal to
I
and
S2(X)
P(r) we get (2.1).
is asymptotically distributed in a normal distribution with
and variance
that for large
of
RESULT.
HAMMERS|.EY’S
mean
SI(X
c
2
/2n
of n
as
n
tends to infinity, Hammersley [5] has proved
the distance between two points in a sphere is nearly always
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] (4.11) is asymptotially equivalent to
2
2 (n+2k)
(5.2)
e
But again, we know [3] that a normal distribution has the probability
P(r)
Ar
n-I
nr 2
2> 2
e
(5.3)
and has the characteristic function
S ()
e
1/222/n
(5.4)
Thus the distribution, as is clear from (5.1), is asymptoticaIlynormalwith
nc
2
2-- n+2------"
Also, taking k=l (3.2)we get (2.4)and thus (2.4) is asymptotically normalwithvariance
2
nc
The distribution of
+
is, therefore, asymptotically, normal with variance
(n+2)
1
2
double that of the above.
Again taking
O, we see that the distribution uniform over the surface of the
k
sphere with radius
d
is asymtotic to normal distribution with variance d
this variance with the variance just obtained for
butation of
of radius
have
I 42
+
I + 2’
2.
Comparing
we notice that the distri-
is asymptotic to the distribution over the surface of the sphere
(2n)c/(n+2) 1/2,
which again is asymptotically equivalent to
Hamersley’s result.
2c(I
+
).
Thus
688
J.M.C. JOSHI
ACKNOWLEDGMENT.
am grateful to the refree for suggesting improvement in the paper
indicating the usefulness of the results proved and to the Editor for suggesting
better!
presentation.
REFERENCES
I.
BOCHNER, S. and (,HANIJRASEKHARAN, K. Fourier Transforms, Annals of Math. Studies,
19, Princeton, 1949.
2.
JOSHt, J.M.C. and UI’RETI, R. On the Representation of Functions as Generalized
LaPlace Integrals, Indian J. Pure Appl. Math., 12(7), pp. 856-864, 1981.
3.
JOSHI, J.M.C. and JOSHI, S.M. Functional as a Characteristic Function of Mathematical Statistics, Journal of Institute of Mathematics and its
(Presently I Journal of Applied Mathematics) Accepted.
Applications
4.
VLADIMIROV, V.S. Generalized Functions in Mathematical Physics Mir Publishers,
Moscow, 1979.
5.
HAMMERSLEY, J.M. The Distribution of Distance in a Hypersphere, Annals of Math.
Stat., 21, pp. 447-452, 1950.
6.
GEL’FAND,
7.
SLA]’ER, L.J. Confluent Hypergeometric Functions, Cambridge University Press,
Oxford, 1960.
[.M. and SHILOV, G.E. Generalized Functions, Vol. I, Academic Press,
New York and hondon, 1964.