I
Ie
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
.-
I
PATHS AND CHAINS OF RANDOM STRAIGHT-LINE SEGMENT
by
N. L. Johnson
University of North Carolina
Institute of Statistics Mimeo Series No. 424
March 1965
Classical problems associated with paths composed
of straight line segments with independent orientations,
which have a continuous distribution, are discussed.
~ment formulas are obtained for the distance between
initial and terminal points, and a new approximation
discussed. The Rayleigh approximation is used to derive approximations to other characteristics of the
random path.
This reseach was supported by the Air Force Office of
Scientific Research Grant No. AF-AFOSR-76o-65.
DEPARTMENT OF STATISTICS
UNIVERSITY OF NORTH CAROLINA
Chapel Hill, N. C.
I
Ie
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
.-
I
Paths and Chains of Random Straight-Line Segments
By
N. L. Johnson
University of North Carolina
1.
Introduction and Historical Outline
The motion of a point traveling along a succession of straight-line
segments of equal length (see Figure 1, section 2), but randomly and independently oriented, has been studied, in various contexts, by a considerable
number of workers over a considerable period of time.
Rayleigh (1880, 1899,
1905, 1919 a,b) studied such motions, firstly in a plane and later in three
dimensions (and also in one dimension).
In the 1880 paper, he obtained appro-
priate formulas for the distribution of the distance between the initial
and terminal points of a path of N segments.
In
1919 he gave exact explicit
distributions (for the three dimensional problem) for some small values of N.
The two-dimensional problem was posed, in connection with the study of random
migration, by K. Pearson (1905) who also noted the relevance of Rayleigh's
solution (K. Pearson (1905, 1906) also Rayleigh (1905».
In response to this
statement of the problem, Rayleigh (1905) gave an outline of his analysis, and
an exact solution, in the form of an integral, was obtained by Kluyver (1905).
Numerical evaluation of this integral was not effected, however, until the
work of Greenwood and Durand (1955).
An exact explicit general solution for
the three-dimensional case ,~s given by R. A. Fisher (1953).
This expresses
the probability integral of the distribution of distance in terms of a set of
polynomials, each polynomial corresponding to a different interVal of values
of the distance.
I
At about the same time that Pearson and Kluyver 1fere working on
the t,'lo-di:mensional problem, Smoluchowski
(1906) was working on the three-
dimensional problem, in connection vuth the study of Brownian motion (with
finite time between charges of path direction).
He also considered the
rather more difficult problenl in which the angle between successive segments
is fixed (though orientation in three dimensions is assumed random, subject
to this constraint).
Smoluchowski stated that his attention was drawn to the
problem by two papers by Einstein
(1905, 1906). However, the methods used by
the latter were of a different nature, being based on continuous variation
with each coordinate of the traveling point varying accordingly to a Wiener
process (as it would now be called).
Kuhn
(1934), apparently independently, initiated stUdy of the three-
dimensional problem, regarding the "path" as a chain cOITq>osed of links of
fixed length, but independent random orientations, which he proposed asa
model for certain kinds of cl1ain molecules.
later papers by Kuhn and GrUn
Kuhn I s paper was followed by
(1942, 1946) and Kuhn (1946).
In these papers
a number of problems were solved approximately, including the problem studied
by
Sn~luchowski,
in which the angle between successive links (here regarded
as tile "valence angle") is fixed.
The accuracy of approximate solutions has been discussed, for the
two-dimensional case, by Horner
(1946), Slack (1946, 1947) and, in consider-
able detail, by Greenwood and Durand
(1957).
Lord
(1948) commented generally
on accuracy of approximation, pointing out that accuracy might be expected to
increase with number of dimensions.
Stephens
(1962 a,b) has studied the
accuracy of approximations for both two- and three-dimensional cases.
The case of fixed unequal segment lengt"hS was discussed by Rayleigh
(1919 a). A formal system of analysis (based on a method ascribed to
2
..I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
I
I
Ie
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
A. A. l'ifarkov) ,applicable to random varying (though independent) segment
lenGrtlls
si~le
"laS
developed by Chandrasekl1ar (1943).
explicit results in certain special cases, such as
tl~
case of segment
vectors '\'lith independent, id.entically normally distributed components.
GroG;)ean
(1960) allo'\'lcd for random variation in segment length, and for a general
orientation distribution, thougll he retained. the
asst~tions
of independence
bet-;ieen sevnents, and betlreen length and orientation of the same segment.
Apart from the distribution of distance between the initial and
ter;:rl.nal points, the distribution of relative orientation of seGments of the
paUl :nas been the object of study.
Kuhn and GrUn (1942) considered the
distribution (in three dimensions) of the angle between a randomly chosen
linl: and tIle "terminal axis" joininG the initial and terminal points of the
chain.
It is interesting to note that the approximate distribution they
obtained had been introduced (is the study of magnetism) by Langevin (1905)
about the time that Pearson, IQ.uyver and Smoluchowski were working on the
distribution of distance between initial and terminal points.
Statistical
properties of the Langevin distribution were studied by R. A. Fisher (1953).
Recently, a number of statistical procedures based on this distribution have
been developed by G. S. Watson and co-workers (Watson and Williams (195 6 ),
Watson (1960), Stephens (1962 b».
The corresponding two-dimensional distribution, described by Mises
(1918), has been discussed and named the "circular normal" distribution by
Gumbel et al. (1953).
It also haS recently been used as the basis for
statistical procedures developed by Watson and Williams (1956), Watson (1961,
1962) and Stephens (1962 a). Breitenberger (1963) has discussed ways of
describing the genesis of both the two- and three-dimensional distributions •
•e
I
This Ims found. to lead to
3
I
In tnis paper, relatively simple metnods will be used to study
random paths and chains.
These methods can be adapted, in a straightforward
fashion, to cases where the segments (or links) have lengths which are
randomly distributed (and even correlated).
We will, at first, be particu-
larly concerned with the distribution of distance between initial and terminal
points, but some other, related problems will be discussed later.
We will
not, here, be directly concerned with the development of tests of significance,
and other statistical procedures connected with these problems.
2.
Distribution of Terminal Distance when Segment Lengths are Fixed
P~1"+1
J"
,d
/
~
,
I
/
p.-1
L1
(7
'P
, i-2.
,
,/
,/
In Figure 1, Po represents the
initial position of the point (or beginning
of the chain) and P. lP, represents the
J-
J
,J,th straight line segment.
'/
"p~I/
"
that the ,J,th segment is of length J.,
P.
o
11
FrGl1RE
It is supposed
J
~mich
'1
may be fixed or may be represented by a
1
random variable.
Figure 1 is applicable, whatever the number (v > 1) of dimensions,
though the distribution of the angle ej(=p P - Pj ) depends on v.
o j l
distance POP
j
between the initial point and the end of the Jth segment will
be represented by r j •
From the triangle Po P j - l Pj
r.2
J
2
= r j _l - 2rj _l .e.J cos Bj
or, equivalently
(1)
The
r2
j
2
= r.J- 1
+ r.J- 1 J j u. +i:
J
J
4
+
l:J
..I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
I
I
..I
I
where the .t j , sand u j • s are nmtually independent random variables, and r j
is independent of .ti and u i for i > j, but not for i S j.
Each u has the
j
same distribution, taking values in the interval -2 < u. < 2.
-
density function of u when v
j
J-
The probability
= 2 (two dimensions) is
I
I
I
I
I
I_
I
I
I
I
I
I
I
.e
I
When v = 3, it is
In each case the fornnlla applies for the interval -2
S u j S 2 only. Whatever
the value of v, u j is distributed symmetrically about zero, so E(U;)
= 0 if
s be odd.
Adding together (N-l) equations like (1) (with
j
= 2,3, ••• ,N), and
noting that r l equals .t , we obtain
l
z2_
N -
N
N_2
1: r. 1 .t. u. + 1: r:j
j=2 JJ J
j=l
•
We now proceed to evaluate the mments of
.tj's having fixed values.
E(r~ I [.t} )
Since E(U~)
=
N
1:
r;, conditional on the
= 0 if s be odd,
l:
j=l J
where [.t} denotes the set of fixed values .tl,.t2' ••• '~.
central mment
0f
2 .
rN
1S
N
~ (r~) I (.t)) = E( [1: rj_l.t j u j ]21[.t}) •
s
j=2
5
The sth conditional
I
Putting s
..I
= 2,
I
(4)
(using (j»
2
"There E(u ) =
E(U~) (for any j), and
N
N-l
N
E . In later fornmlas,
i=l j=i+l
higher order summations will be introduced, using similar conventions.
r.I:
means
I
E
Thus
I
I
I
11'1. l' U 1• , [l} ) [ .e} ) •
I
1<.1
The calculations for s=3 and s=4 follO"l similar lines.
?
J.L. (~
.~
I{.t})
= 3E [E
i
N
2
E
<
= 3E (u )
2
No'" E(r. 1 1
J-
1'1. l'
j
r.
1-1
.t. u. (1'.
1
= r.1-1
J J
2
I{.e} ]
N
2__
2
E E l . l--:E ( l' • lU • E ( r. 1
i < j
1 J
1- 1
J-
.2
u., Ce})
J-
1 1
ll. u . )
+ r. l.t·u. +
1-
1 1
j-l 2
E lh'
h=i
_I
I
and so
(since E(u.) = 0)
1
Sirular calculations lead to
(6)
2 2
+ 6 [E (u )]
E
N
E
E
hfi,hfj,i<j
6
L~ 2 2
4 N 4 4
thr: l. + E (u ) E U . .e.
1
J
i<j
1
J
I
I
I
I
I
I
-.
I
I
..I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
.e
I
The second term on the right-hand side of (6) is actually
N
N
N
4
2 2
2
(6[E(u )J (E E E + E E E) + E(u )[2+E(u )]E E E) I~!~!~
h<i<j
i<h<j
i<j<h
J
but, for our purposes, it can be written in the form shown, since E(u2 )
and E(u4 ) = 48[v(V+2) ]-1 and so 6[E(u2 )J2 • E(u4 )[2+E(u2 )J.
If
(using
(8)
I
each straight-line segment is of the same fixed length
in place of (llto denote 11
• ••• •
IN • / ).
212
4
var(rNI/) • 2 E(u ) N(N-l)!
2
~3(rN'/) •
(10)
=12
2 J2 N(N-l) (N-2)/6
2l[ E(u)
~4(r~,/) • (t[E(u2 )]2[1+E(u2 )JN(N-l)(N-2)(N-3)
+ 3[E(u2 )]2N(N_l)(N_2) + ~ E(u4 ) N(N-1»)18 •
In two dimensions
(11)
var(rill) • N(N-l)14
(12)
~3(ri
(13)
~4 (ril f) • 3N(N-l)(3~-1lN+ll)f8 •
I) = 2N(N-1) (N_2)/6
In three dimensions
(14)
2
4
var(r21/
N ) . '3 N(N-1)f
(15)
~3(r~ll)
•
~
(16)
~4(r~ll)
•
~ N(N-1)(35~-115N+108)/8
N(N-1) (N_2)/6
7
I,
= 4v- l
then
I
Approximations to Distribution of r
~
2
N
For the case of constant segment length the asymptotic distributions
of
(V/(NJ2»r~,
as N increases, is that of x2 with v degrees of freedom.
(Rayleigh (1880), Pearson (1905), Kluyver (1906), Smoluchowski (1906), Kuhn
(1934) etc.) This result can be obtained (see Stephens (1962 a, pp. 473-4)
(t ~ 1,2, ..• ,v) for P , with x
N
tO
for all t, so that Po is at the origin of coordinates, and noting that
by introducing Cartesian coordinates x
tN
=0
(i)
x
is the sum of independent, identically distributed random
tN
t
variables v3 ) (the projections of links P j - P on the x axis) which each
l j
t
have a finite range of variation, expected value zero and variance f2 jv ,
and
(ii)
x
lN
' x2N ' ••• , x VN are uncorrelated (though not independent).
It may be noted that a similar argument applies also when the fj's
are independent random variables with identical distributions with finite
expected value and standard deviation.
ment of
The only modification is the replace-
f2 by E(f2).
Horner (1946), Slack (1946,1947) and Greenwood and Durand (1955)
found that the asymptotic distribution did not give very good results for N
less than 10 when v
=2
(i.e., in two dimensions).
However, as Lord (1948)
pointed out, the approximation would be expected to be better as v increases.
Rayleigh
(1919
a,b), Pearson
(1906)
and Kuhn
(1934)
investigated possible
2
improvements of the X approximation, and various methods of approximation (in
the case v=2) were compared by Durand and Greenwood (1957).
For relatively small values of N, a natural method of approximation
2
is to use a Pearson Type I curve with the same first two moments as r , and
N
2
the same range of variation (from 0 to (Nf)). This gives exactly correct
results when N
I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
= 2.
8
I
I
..I
I
I
I
I
I
I
This method is equivalent to approximating the probability density
function of YN
= (rN/(N{))2
by
1
-1
ex • 2
\I-N j
with
13 =
( ) (1
N-l
The third and fourth central moments of r
2
2
N
\I-N
-1
) •
corresponding to the approximating
distribution (1'7) are
I_
The ratios of approximating to exact values of the third and fourth central
I
I
I
I
I
I
I
.-
I
moments are
(r;1 {)
~3(ril {)
ii
3
(for N > 2).
~4 (ri l {)
~4(r;I{)
Some values of these ratios are shown in Table 1.
In assessing these ratios
as indices to the accuracy of approximation it should be borne in mind (see
Pearson (1963)) that variation in high order moments may have little effect on
computed probabilities.
ardized" ratios
(It may be more informative to consider the "stand-
::Jfi3/~3 and ~ft4/~4 .)
9
I
..I
Table 1
Ratios of Approximate to Exact Central
Mo~nts__~!
Third ltt>ment
N
0.750
0.800
0.833
0.875
0.909
0.952
3
4
5
7
10
20
N
Fourth Moment
v=3
0.818
0.857
0.882
0.913
0.937
0.968
~
2
r
v=2
0.900
0.818
0.806
0.823
0.856
0.910
I
I
v=3
0.933
0.896
0.888
0.895
0.915
0.950
Application of this approximation requires the evaluation of the
incon~lete
beta function ratio I (CX,f3) with x::= (r/(Nl)2. Lower percentage points can be
x
obtained (using linear harmonic interpolation) from the recently published
tables of Vogler (1964).
This is, unfortunately, not so for the upper percentage
points, so no direct comparison has been made with the upper 5 %and 1% points
given by Greenwood and Durand (1955).
Special calculations gave the follovring
comparisons with exact values of the probability integral of r, for N ::= 7 and
v
=2
given by these authors
Table 2
Probability Integral of r (V=2)
7
Pr[r
R/ (Nt)
1.0
2'.0
3'.0
!~.o
6.0
0.125
0.418
0.714
0.900
0.9979
7
:s R]
~ approx.
Type ,I approx.
0.133
0.435
0.724
0.898
0.9942
0.145
0.425
0.704
0.895
0.9992
For small values of R the present approximation is not even as good as the
approximation, but the approximation improves for larger values of R.
i
I
I
I
I
_I
I
I
I
I
I
I
I
-.
10
I
I
~
I
I
I
I
I
I
I
aI
I
I
I
I
I
I
x2
The
approximation may also be used in obtaining approximations to
other properties of random paths and chains (see, e.g., Stephens (1962 a,b)).
In section 5 of this paper there will be found some applications of this kind.
2
The accuracy of these approximations depends closely on that of the X approximation on which they are based; they improve as v and/or N increase.
4.
Variable segment Length
If the
I's
are not constants, but random variables, the moments about
zero of r; can be obtained by finding the expected values of the corresponding
conditional moments,
(The corresponding relationship for central moments is, of course, not generally
true. )
If the
12 ,s
are mutually independent and identically distributed,
with finite moments (about zero) ~~, then the moments of r; can be expressed
in terms of these "s
u"s •
Thus
(18)
2
~l'(rN) =
~2'(rN2)
N
E( E
j=l
2
Ij )
= N~'
1
N
= E(E(u2 )E E 1:/2j
~
i<j
= E([2+E(u
2
N
+ ( E 12j )2)
j=l
N
)]E
E/~I~
i<j
.e
I
11
N 4
+ E 1j )
j=l
I
J
In a similar fashion we obtain
(20)
and
(21)
Expressing equations (19)-(21) in terms of central moments (for both
r
2
and ,.,2 ), we find
N
(22)
(24)
+
2'l[ E(u4 ) + l2E(u2
) +]
6 N(N-l) 1122 + 6E(u2 )N(N-l)1l Ili +
3
So, in two dimensions
(26)
Nlll~ •
I
I
I
I
I
I
I
et
I
I
I
I
I
I
I
-.
(26)
12
I
I
~
I
I
I
I
I
I
I
I_
and in three dimensions
(28)
~4(r~) ~ ~N(N-1)(35~-115N + 108)~i4
+
~N(N-1)~~
.e
I
8N(N-1)~3~1
+
~ N(N-l)(25N-36)~2~i2
N~4
.
It can be seen that, for sufficiently large N, the most important
term in ~s(r~) is that containing ~ls.
corresponding formula for fixed
I
I
I
I
I
I
I
+
+
~'~.
This term can be obtained from the
11 = t 2 = ... = IN = I,
by replacing
fS
by
It is to be expected that ri will tend to its asymptotic distribution
N~l
(that of (-V-) x (X
2
with v degrees of freedom)) more rapidly when ~2' ~3' ~4
are small compared with
~l.
The moments of ri can be evaluated in a similar way when the
are not independent.
If the
12 ,s
12 ,s
all have the same moments, and the corre-
lations among them are
then
while
13
I
J
In these two cases the limiting values of the moment-ratios as the same (as N
increases) as when there is no correlation.
same correlation, that is
p(li,
I~)
= p for
If,
2
u
howeve~ all ~'s
have the
all i ~ j.
and the limiting value of the variance of V;/N is increased in the ratio.
The other moments are affected also.
2
It is possible to investigate cases where individual Ij's have
different distributions, using methods of the kind described above.
5.
Uses of the Chi-Square Approximation to Distribution of Length
It can be seen from the results in the last section that the moment-
2
ratios of the distribution of r~ tend to those of X with v degrees of freedom
as N increases.
Although the work of Greenwood and Durand (1955) indicated
that this approximation is not completely satisfactory for values of N even
as large as 20, it can be used with confidence for larger values of N (say
40-50 or more) and can provide useful insight into the nature of the distribution of the end-points of segments of the path or chain.
As an example, suppose we wish to consider the distribution of the
distance d
of P from the "terminal axis" POP joining the extremities of
N
n
n
the (path or) chain (n < N). Assuming that there is no dependence between
lengths of segments, and that each f~ has the same distribution we have
J
14
I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
I
I
\
I
I
I
I
I
I
I
d
2
n
2
r n-n
rN
'
sin ¢
=--------r -2r r '
cos¢+r '
N-n
n
n N-n
where ¢ (the angle POPnP N), r n (poP ) and rN_n(PnP ) are mutue.lly independent
n
N
-2 cos ¢ has the same distribution as u, and r~, rN~n are
random variables:
distributed as r~ with N replaced by n, N-n respectively.
We will now drop
the prime in r ' •
N-n
The distribution of d
following demonstration, for
~
n
can be obtained by direct analyses.
=3
The
dimensions, is particularly simple.
I_
provided D < min(r ,r
I
I
I
I
I
I
I
.-
I
n
N-n
).
Otherwise the probability is zero.
Since (for ~s 3) cos ¢ is uniformly distributed between -1 and +1
(for D < min(r , r
n
N-n
)
and
jr~-n_D2 p(r2
r
(since r 2 and r 2
are mutually independent).
N-n
n
N-n
2
)dr
)
N- n -Nn
Now introducing the approximate
2
probability density function (corresponding to the asymptotic x distribution)
1
2'2
(r N)
2
3r N
exp[-2N~' ]
1
15
I
J
we obtain from (35)
2
2
2
3ND
Pr[d n > D J . exp(- 2n(N-n)J..L'
)
=
1
00
Je (
'2 -Kz dz = 2e
1 -K~L-2
z-e) e
-K
(Note that
From
:3
1
vr-=1t'
•
)
(37) we see that d2n is approximately distributed as
In(l-~)J..Li x
~nJ..Li x
2
(x
(x
2
with two degrees of freedom), as compared with
with two degrees of freedom)
for the distribution of the square of the distance of P from an arbitrary line
n
through Po'
2
The approximate expected value of dn , calculated from
2n(~;n)
• J..Li'
If N is even and n
=~
N, so that P
path or chain, the approximate expected value is
(37), is
is the mid-point of the
n
NJ..Li' or
rf2 i f each link
g
i
is of length [, agreeing with Kuhn and GrUn (1942) and Kuhn (1946).
One further application may be noted here, though we shall not pursue
the matter in great detail.
It is possible to write down an approximate formula
2
2
for the joint probability density function of r~ , r
••• , r N ' where
1
N2 ,
s
N < N < •.• < N and N , N -N , ••• , N -N 1 are each large enough for the
2 l
s ss
l
l
2
2
x approximation to be useful.
We first consider the conditional distribution of r~ , given
xIX '
1
... ,
X
VN •
Using the random variables
1
u~t)
defined in s:ction 4, we
have
The conditional distribution of r~ given xlN ' ••• , x VN is approximately that
211
of
16
I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
I
I
..I
I
I
I
I
I
I
I_
V-l(N2-Nl)~i x (Non-central x2 with v degrees of freedom and noncentrality parameter vri [(N2-Nl)~iJ-l).
1
Hence this is also the conditional distribution of r; given r; , and using
2
1
2
the x approximation to the distribution of ri '
1
222
The approximate joint distribution of r N ' r N ' and r N is obtained by multi1
2
3
plying (38) by the approximate conditional probability density function
p(r~ Ir; ) and so on. When v= 3 (three dimensions) the relationship
3
I
I
I
I
I
I
I
.e
I
2
leads to considerable simplification, giving
where
a
s
:: (N -N 1)
s s-
-1
•
The right-hand side of (39) can be written as a weighted sum of terms
of the form
17
I
(40)
exp[quadratic function of r N ' ""
1
J
r N J,
s
Approximate evaluation of the probability of a given class of configurations
of the set of values (rN ' ""
r N ) can be effected by term-by-term intes
2
2
gration of expressions of type (40) with respect to r ' "" r N •
N
1
s
Among other interesting approximate results which can be obtained
I
using the methods of this section may be mentioned the distribution of the
N,
distance between the terminal points P , P od two independent chains with
N
initial points a distance R apart, The square of this distance is approximately
distributed as
V-l(Nf2+N'f,2) x (non-central
x2
with v degrees of freedom and
non-centrality parameter VR2(Nf2+N'f,2)-1)
_I
using an obvious notation.
6.
Distribution of Angle of Inclination to Terminal Axis
So far, we have been mainly concerned with the distance between the
initial and terminal
point~
I
I
I
I
I
I
I
Po' PN respectively. We now consider the distri-
bution of the angle between a randomly chosen segment P. IP" and the terminal
J- J
axis POP N, This distribution will eVidently depend on the length of POPN(r ),
N
and it is this conditional distribution that we will consider. We will confine
our attention to the case of segments of constant length,
f,
in three dimensions.
Since any ordering of the N segments, given a set of N orientations,
leads to the same position for the terminal point; and since each such ordering
is, under our assumptions, equally likely, it follows that the distribution of
the angle between Pj-lP j and the terminal axis is the same for all values of j.
It is convenient to consider the angle between the final segment, PN-IP , and
N
the terminal axis POP • Denoting this angle by ~, we have from the triangle
N
18
I
I
I
I
I
I
I
-.
I
I
'-I
I
I
I
I
I
I
(41)
cos
Hence the conditional distribution of cos
from that of
it can be seen that (in three dimensions)
Hence
I_
I
I
I
I
I
I
I
Hence, from (41)
I
2
given r , is directly derivable
N
r~_l' given r~. From the relation (see (1))
and
.e
~,
(42)
2
Now introducing the x approximation for the distribution of r;_l we find
(-loS cos ~ oS 1)
approximately.
Kuhn and Grun (1942) studied this problem, by dividing the range of
variation of
~
into a large number of small intervals and max:lmizing the
probability, among sets of WI s constrained to give a specified value of r N•
The limiting relative frequencies of different values of
V,
obtained in this
way were found to lead to a distribution of the same type as (lJ.3) but with
19
I
3rN[(N-l)fJ-l replaced by the inverse Langevin function of rN/(NI); i.e., the
value of
~
satisfying.
(44)
F',r small values of rN/(Nf), (43) implies t3 .;. 3r /(Nf).
N
expected value and standard deviation of r
N
are both of order
rN/(Nf) will usually be small when N is large.
VN,
Since the
the ratio
Our result is thus in qUite
good agreement with that of Kuhn and Griin.
7.
Continually Varying Orientation as a Limiting Case
It would appear natural to try to approach the situation in which
the point moves with a continuously varying orientation by considering the
limit of the case considered in section 2 (With corrnnon fixed segment length f)
as N tends to infinity and
t
to zero,
Nt
imagined to move with a fixed velocity,
point is in motion.)
remaining constant.
Nt
(If the point be
=0
for all s > 1.
As the limit is approached the
probability of the terminal point being in any neighborhood, however small,
of the initial point tends to one.
A non-degenerate limit can be obtained by assuming (see Chandrasekhar
(1943, p. 19), also Einstein (1905)) that
Nt 2 remains
constant--equal to ~ say.
Then the limiting values of the moments of r~ are
lim
E(r~lt) = ~j lim ~2(r~lt) = ~ E(u2 )¢2;
.
21·
11m ~3(rN
lim
t)
= 21
[
2 J2 3
E(u) ¢
~4(rilt) = ~[E(u2)J2[1+E(u2)J¢4
20
I
I
I
I
I
I
I
_I
is proportional to the time the
However, from (7( and (8) we have
and in fact lim ~s(r~lt)
J
I
I
I
I
I
I
I
-.
I
I
~
I
I
I
I
I
I
I
I_
I
I
I
I
I
I
I
.e
I
and the limiting distribution of r~ is that of (~/v) x (x
2
with
v degrees of
freedom).
If the Ij's are independent identically distributed random variables
and ~
has a proper limiting distribution ri also has a proper limiting
tj
2
distribution, but it need not be of X form.
REFERENCES
[1]
Breitenberger, E. (1963)Ii-Analogues of the normal distribution on the
circle and the sphere,' Biometrika, 50, 81-88.
[2]
Chandrasekhar, S. (1943), "Stochastic problems in physics and astronomy,1I
Rev. Mod. Physics, 15, 1.89.
[3J
Durand, D. and Greenwood, J. A. (1957), IIRandom unit vectors II: II Usefulness of Gram-Charlier series in approximating distributions," Ann. Math.
Statist., 28, 978-986.
[4J
Einstein, A. (1905), "Uber die von der molekularkinetischen Theorie der
Warme geforderte ~wesung von in ruhenden Flussigkeiten suspendierten
Teilchen," Ann. d. Physik, 17, 549-560.
[5] Einstein, A. (1906), "Zur theorie der Brownschen Bewegung, II Ann. d. Physik,
19, 371-381.
[6]
Fisher, R. A. (1953), "Dispersion on a sphere,1I Proc. Roy. Soc. London,
Series A, 217, 295-305.
[7]
Greenwood, J. A. and Durand, D. (1953), "The distribution of length and
components of the sum of n random unit vectors," Ann. Math. Statist.,
26, 233-246.
[8J
Grosjean, C. S. (1953), IlSo1ution of the non-isotropic random flight
problem in the k-dimensional space," Physica, 19, 29-45.
[9J
Gumbel, E. J., Greenwood, J. A. and Durand, D. (1953), "The circular
normal distribution: Theory and tables, II J. Amer. Statist. Assoc.,
48, 131-152.
[10]
Horner, F. (1946), IIA problem in the summation of simple harmonic functions of the same amplitude and frequency, but of random phase,1I Phil.
Mag., 7th Series, 37, 145-162.
[11]
Kluyver, J. C. (1905), "A local probability problem,1I Proc. Kon. Nederl.
Akad. v. Wetensch., 8, 341-350.
[12J
Kuhn,
w.
(1934), "Uber die Gestalt fadenformige Molekule in Losungen,1I
Kolloid-Zeit., 68, 2-15.
21
I
[13J
Kuhn, W. (1946), "Dependence of the average transversal on the longitudinal dimension of statistical coils formed by chain molecules," J. Polymer
Sci., 1, 380-388.
[14J
Kuhn, W. and Grim, F. (1942), "Beziehungen zwischen elastischen Konstanten
und Dehnungsdoppelbrechung hochelastischer Stoffe," Kolloid-Zeit., 101,
248-271.
[15J
Kuhn, W. and Grun, F. (1946), "Statistical behavior of the single chain
molecule and its relation to the statistical behavior of assemblies
consisting of many chain molecules," J. Polymer Sci., 1, 183-199.
[16J
Kuiper, N. H. (1960), "Tests concerning random points on a circle," Proc.
Kon. Nederl. Akad. v. Wetensch, Series A, 63, 383-397.
[17J
Langevin, P. (1905), "Magnetisme et theorie des
et de Phys., 5, 70-127.
[18J
Lord, R. D. (1948), "A problem on random vectors," Phil. Mag., 7th Series,
39, 66-71.
[19J
von Mises, R. (1918), "Uber die 'Ganzzahligkeit' der Atomgewichte und
verwandte Fragen," Phys. Zeit., 19, 490-500.
[20J
Pearson, E. S. (1963), "Some problems arising in apprOXimating to probability distributions, using moments," Biometrika, 50, 95-111.
[21J
Pearson, K. (1905), "The problem of the random walk," Nature, 72, 294 and
318.
[22J
Pearson, K. (1906), "A mathematical theory of random migration," Math. Contr.
Theory Evol. XV, Draper Co. Res. Mem., Biometric Series }.
[23J
Rayleigh, Lord (Strutt, J. W.) (1880), "On the resultant of a large number
of vibrations of the same pitch and of arbitrary phase," Phil. Mag. 5th
Series, 10, 73-78.
[24J
Rayleigh, Lord (1899), "On James Bernoulli's theorem in probabilities,"
Phil. Mag., 5th Series, 47, 246-251.
[25J
Rayleigh, Lord (1905), "The problem of the random walk," Nature 72, 318.
[26J
Rayleigh, Lord (1919 a), "On the problem of random vibrations and random
flights in one, two and three dimensions," Phil. Mag., 6th Series, 37, 32131+7.
[27J
Rayleigh, Lord (1919 b), "On the resultant of a number of unit vibrations
whose phases are at random over a range, not limited to an integral
number of periods," Phil. Mag., 6th Series, 37, 498-515.
[28J
Slack, Margaret (1946), "'The probability distribution of sinusoidal
oscillations combined in random phase," J. !nst. Elect. Eng., 90(3),
76-86.
,
22
electrons, " Ann. de Chim.
J
I
I
I
I
I
I
I
_I
I
I
I
I
I
I
I
-.
I
I
..I
I
I
I
I
I
I
I_
[29J
Slack, Margaret (1947), Letter in Phil. Mag., 7th Series, 38, 297-298.
[30 J
von Smo1uchowski, M. (1906), "Zur kinetischen Theorie der Brownschen
Mo1eku1arbewegung und der Suspensionen," Ann d. Physik, 21, 756-780.
[31J
Stephens, M. A. (1962 a), "Exact and approximate tests for directions, I,"
Biometrika, 49, 463-477.
[32J
Stephens, M. A. (1962 b), "Exact and approximate tests for directions,
Biometrika, 49, 547-552.
[33J
Watson, G. S. (1960), "More significance tests on the sphere,u Biometrika,
47, 87-91.
[34J
~'latson, G. S. (1961), "Goodness-of-fit tests on a circle, I," Biometrika,
1~8,
[35J
109-114.
Watson, G. S. (1962), "Goodness-of-fit tests on a circle, II," Biometrika,
1.1-9, 57-63.
[36J
Watson, G. S. and Williams, E. J. (1956), "On the construction of significance tests on the circle and the sphere," Biometrika, 43, 344-352.
[37J
Vogler, L. E., (1964), Percentage Points of the Beta Distribution, National
Bureau of Standards, Boulder, Colorado.
I
I
I
I
I
I
I
.e
I
n"
23