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Federal Reserve Bank of Chicago
Properties of the Vacancy Statistic in
the Discrete Circle Covering Problem
Gadi Barlevy and H. N. Nagaraja
REVISED
March 2015
WP 2013-05
Properties of the Vacancy Statistic in the
Discrete Circle Covering Problem
Gadi Barlevy∗ and H. N. Nagaraja†
March 23, 2015
Abstract
Holst (1985) introduced a discrete spacings model that is related to the
Bose-Einstein distribution and obtained the distribution of the number of
vacant positions in an associated circle covering problem. We correct his
expression for its probability mass function, obtain the first two moments,
and describe their limiting properties. We then examine the properties of
the vacancy statistic when the number of covering arcs in the associated
circle covering problem is random. We also discuss applications of our
results to a study of contagion in networks.
Key Words: occupancy problems; spacings; Bose-Einstein distribution; sampling without replacement; sampling with replacement.
2010 Mathematics Subject Classification: Primary: 60C05; Secondary:
62E15.
1
Introduction
This paper examines the discrete version of the circle covering problem first
introduced by Stevens (1939) in which m arcs of length a are randomly placed
on the unit circle, and the question of interest is the fraction of the circle left
uncovered by any arc. The discrete version of this problem can be described as
follows. Consider r (≥ 2) boxes arranged in a ring numbered 0, 1, ..., r −1. Draw
m − 1 boxes by simple random sampling without replacement from the boxes
numbered 1, 2, ..., r − 1, where 2 ≤ m ≤ r. Let 1 ≤ R1 < · · · < Rm−1 ≤ r − 1 be
the drawn numbers, and set R0 = 0 and Rm = r. Define
Sk = Rk − Rk−1
∗ gbarlevy@frbchi.org; Economic Research Department, Federal Reserve Bank of Chicago,
230 South LaSalle, Chicago, IL 60604, USA.
† nagaraja.1@osu.edu; College of Public Health, Division of Biostatistics, The Ohio State
University, 1841 Neil Avenue, Columbus, OH 43210-1351, USA.
1
for k = 1, 2, ..., m, i.e. Sk are spacings. Next, for an integer b where 1 ≤ b ≤ r,
define
m
X
V =
(Sk − b)+
(1)
k=1
where (x)+ = max {x, 0}. This setup can be interpreted as follows. We can
think of {R0 , . . . , Rm−1 } as m distinct starting points whose location among
the r boxes is chosen at random. From each starting point, we designate the
next b boxes including the starting point as covered. Any remaining uncovered
boxes are designated as vacant. The random variable (rv) V represents the total
number of vacant boxes. For reference, it will be convenient to also define
N=
m
X
I (Sk > b)
(2)
k=1
where I(C) = 1 if condition C is true and 0 else. Thus, N represents the number
of distinct blocks of vacant boxes.
Our interest is in characteristics of V , specifically its distribution, some of its
moments, and its behavior when m is random. Holst (1985) derived the marginal
and joint distributions of {Sk }m
k=1 and showed they are exchangeable. He also
explored the connection between these random variables (rv’s) and the BoseEinstein distribution. Feller (1968, Sec II.5(a)) provides a nice introduction to
the Bose-Einstein urn model.
As anticipated by our comments above, in the limit as r → ∞ while b = ar
for some constant a < 1 this problem converges to the circle covering problem in
which m points are chosen uniformly from the circumference of a circle, and each
of the m points forms the end point of an arc of length a. The latter problem
has been extensively analyzed; see for example, Siegel (1978). The limit of V /r
in our discrete setup corresponds to the fraction of the circumference that is
uncovered. However, the finite version of the problem has been less studied,
even though as we discuss later this version arises in certain applications.
We derive in Section 2 an explicit expression for the probability mass function
(pmf) of V including an exploration of the range of its values; in this process
we correct an error in the expression for the pmf given in Holst (1985). We give
explicit expressions for the first two moments of V in Section 3 using several
properties of the joint distribution of Sk derived by Holst (1985). We discuss
an extension of the model in Section 4 in which the number of starting points
m is random, allowing us among other things to discuss an alternative version
of our model in which the boxes are chosen with replacement. We establish
limiting properties of V in Section 5 and link our results to those of Siegel
(1978). In Section 6, we discuss an application concerning financial contagion
that coincides with the discrete circle covering problem under certain conditions.
This application suggests generalizations of the circle covering problem that have
not been explored in previous work.
2
2
Exact Distribution of V
2.1
The Range
The value of V must be non-negative, and the lowest value it can assume is
r − mb. Further, the largest possible value of V occurs when the chosen boxes
are consecutive (implying N ≤ 1) and V takes on the value r − m − b + 1.
Thus, the support of V is the set {(r − mb)+ , . . . , (r − m − b + 1)+ }, and so V
is degenerate at 0 whenever r < m + b. Further when r − mb ≥ 0, the total
number of points in the support of V is (m − 1)(b − 1) + 1 independently of
r. Hence when b = 1, we have a single support point at r − m > 0. We now
examine the form of the pmf P (V = x) for various x values when r ≥ m + b
and b > 1.
2.2
Probability Mass Function of V
Holst (1985; Theorem 2.2) argues that P (V = x) is given by
m m−y
X
m X
y=1
y
t=0
(−1)
t
m−y
x − 1 r − (y + t) b − x − 1
r−1
. (3)
t
y−1
m−y−1
m−1
Note that expression (3) may include improper binomial coefficients nk
where either n < 0 or k ∈
/ {0, . . . , n}. Such terms are traditionally set to 0. We
now argue that this convention may yield an incorrect expression for P (V = x)
for x = 0 and for x = r − mb, and offer correct expressions for P (V = x) for
these cases.
Observe first that for x = 0, the right-hand side of(3) would under the usual
convention equal to 0 due to the presence of the x−1
y−1 term. But this is at odds
with the fact that P (V = 0) = 1 whenever r < m + b.
To obtain a proper expression for P (V = 0), we use the observation noted
by Holst that
m
X
P (V = 0) = P
I(Sj > b) = 0 = P (N = 0).
(4)
j=1
Holst (1985) derives an expression for the right hand side of (4) in part (a) of
his Theorem 2.2. Using his result, we can deduce that for x = 0, (3) must be
replaced by
m
X
m r − jb − 1
r−1
P (V = 0) =
(−1)
.
j
m−1
m−1
j=0
j
(5)
We next turn to the case where r ≥ m + b and x > 0, and examine the
range of values for y and t for which the associated terms on the right side of
(3) are all positive, i.e. when 0 ≤ t ≤ min{m − y, (r − m − x − (b − 1)y)/b}, and
3
1 ≤ y ≤ min{m − 1, x, (r − m − x)/(b − 1)} for b > 1 and 1 ≤ y ≤ min{m − 1, x}
for b = 1.
We shall now argue that for this range, (3) holds except when x = r − mb ≥
m. To see this, we begin with the observation by Holst in proving his Theorem
2.2 that if Ik = I(Sk > b), then P (V = x) must equal
!
y
m m−y
X
X
m X
t m−y
(−1)
P
(Sk − b)+ = x, I1 = · · · = Iy+t = 1 . (6)
y t=0
t
y=1
k=1
Expression (6) can in turn be rewritten as
m−1
X
y=1
!
m−y−1
y
X
X
m
t m−y
(−1)
P
(Sk − b)+ = x, I1 = · · · = Iy+t = 1
t
y
t=0
k=1
!
y
m
X
X
m
m−y
(−1)
P
(Sk − b)+ = x, I1 = · · · = Im = 1 . (7)
+
y
y=1
k=1
Holst then computes the following probabilities based respectively on parts
(E) and (D) of his Theorem 2.1:
r−(y+t)b−1
m−1
r−1
m−1
P (I1 = · · · = Iy+t = 1) =
and
P
y
X
!
(Sk − b)+ = x I1 = · · · = Iy+t = 1
x−1
y−1
r−(y+t)b−x−1
m−y−1
r−(y+t)b−1
m−1
=
k=1
(8)
and thus concludes that
P
y
X
!
(Sk − b)+ = x, I1 = · · · = Iy+t = 1
k=1
x−1
y−1
=
r−(y+t)b−x−1
m−y−1
r−1
m−1
.
(9)
The expression for the conditional probability in (8) is valid and nonzero whenever y ≤ m − 1, and y + t < m.
Next we consider the last sum on the right in (7). Since
Py the event {I1 = · · · =
Im = 1} implies Si > b for i =P1, . . . , m, the sumP k=1 (Sk − b)+ is strictly
y
m
increasing in y for y ≤ m and k=1 (Sk − b)+ < k=1 (Sk − b)+ ≡ r − mb.
This means
!
y
m
X
X
m
m−y
(−1)
P
(Sk − b)+ = x, I1 = · · · = Im = 1
y
y=1
k=1
is equal
Pm to 0 when x > r − mb and is equalto the last term in the sum,
P
k=1 (Sk − b)+ = r − mb, I1 = · · · = Im = 1 , when x = r − mb. Since each
4
Pm
term in k=1 (Sk − b)+ is at least one, the sum should be at least m. In other
words, the only nonzero term in the last sum on the right side of (7) is
Pm
P
k=1 (Sk − b)+ = r − mb, I1 = · · · = Im = 1
= P (I1 = · · · = Im = 1) = P (S1 > b, · · · , Sm > b) = P (S1 > mb)
. r−1
= r−mb−1
m−1
m−1
(10)
provided r − mb ≥ m. Upon collecting all of our findings (in (4), (6), (8), and
(9)), we have the following modification of Theorem 2.2 of Holst (1985).
Theorem 1. The support of the rv V representing the length of the vacant
region is given by {(r − mb)+ , . . . , (r − m − b + 1)+ }. When r < m + b, V is
degenerate at 0. When r > m and b = 1, V is degenerate at (r − m). When
r ≥ m + b and r − mb ≤ 0, P (V = 0) is given by (5). In all other cases,
P (V = x) is given by
(m−1 m−y−1
X m
X
x − 1 r − (y + t) b − x − 1
t m−y
(−1)
t
y
y−1
m−y−1
y=1
t=0
r − mb − 1
r−1
+I(x = r − mb ≥ m)
.
(11)
m−1
m−1
Remark 1: The actual range of values for y and t for which the associated
terms are positive is more restricted than given by the limits in the double
sum in (11) in a way that depends on x. For example, when x = r − mb, the
lowest value V can assume, the terms are positive for all 1 ≤ y ≤ m − 1 and
0 ≤ t ≤ min{m − y − 1, m − y − (m − y)/b}. In contrast, when x = r − m − b + 1,
the highest value V can assume, (y, t) = (1, 0) is the only combination that
produces a positive term. In that case, (11) yields
r−1
P (V = r − m − b + 1) = m
,
m−1
a quantity free of b.
Table 1 below provides the pmf of V for r = 10 and m = 5. It shows how
the probability mass shifts towards values close to 0 as b increases.
3
Moments of V
Instead of using the pmf for V to compute the first two moments of V , we take
advantage of an exchangeability argument to derive them from those of Sk . We
will need the following expressions for the first two moments of nonnegative
integer valued rv’s X and Y .
E(X) =
∞
X
i=0
5
P (X > i);
(12)
Table 1: The pmf P (V = x) for r = 10, m = 5 for various values of b
b
1
2
3
4
5
6
0
0
0.008
0.405
0.802
0.960
1
x
2
0
0.476
0.159
0.040
1
0
0.159
0.397
0.159
0.040
E(X 2 ) = 2
∞
X
3
0
0.317
0.040
4
0
0.040
iP (X > i) + E(X);
5
1
(13)
i=0
E(XY ) =
∞
∞ X
X
P (X > i, Y > j).
(14)
i=0 j=0
The first two are well-known. Equations (12) and (13) are given in, for example, David and Nagaraja (2003), p. 43, and go back to Feller’s classical work.
Expression (14) is similar to known results for the continuous case; see, for example, the expression for the covariance in Barlow and Proschan (1981, p. 31),
and the idea goes back to Hoeffding (1940); see Wellner (1994).
Here we give a short proof of (14) when X and Y are nonnegative integer
valued rv’s.
E(XY )
=
=
=
∞ X
∞
X
ijP (X = i, Y = j)
i=0 j=0
∞ X
∞
X
i
P (X = i, Y > j)
i=0 j=0
∞ X
∞
X
P (X > i, Y > j)
i=0 j=0
upon using the idea of the form on the right side of (12) twice.
The moment expressions simplify further by the use of the following wellknown identity: For positive integers c ≤ a,
a
X
k−1
k=c
c−1
a
=
.
c
(15)
Harris, Hirst, and Mossinghoff (2008) derive this identity using an induction
argument (see their equation (2.10)). We give a simple probabilistic proof.
6
Proof. Multiply both sides by (1/2)a . Then the right side, ac (1/2)a represents
the probability that in a tosses of a fair coin there are exactly c heads. Now
if we have c heads, this event can be written as the union of disjoint events
Ec , . . . , Ea where Ek is the event that we have exactly c heads and the cth head
appears at the kth toss. By the negative binomial type argument we know that
this probability is
k−1
k−1
c
a−c
(1/2) (1/2)
=
(1/2)a .
c−1
c−1
2
Now sum this over k from c to a.
Theorem 2. Let Wi = (Si − b)+ , for i = 1, 2. The Wi are exchangeable and
for r ≥ m + b
r−b
r−1
E(W1 ) =
,
(16)
m
m−1
and
r−b+1
m+1
r−1
m−1
E(W12 )
= (2(r − b) + 1)E(W1 ) − 2m
.
(17)
For r ≥ m + 2b,
E(W1 W2 ) =
r − 2b + 1
m+1
r−1
m−1
,
(18)
and E(W1 W2 ) = 0 if r < m + 2b.
Proof. Exchangeability follows from Theorem 2.1 of Holst (1985). Now
E(W1 )
=
=
r−m−b
X
P (W1 > i) [from (12)]
i=0
r−m
X
P (S1 > j)
j=b
=
r−m
X
j=b
r−j−1
r−1
[from Th 2.1(B), Holst]. (19)
m−1
m−1
From (15), the numerator on the right side of (19) reduces to r−b
m .
To establish (17), we use the expression for the second moment in (13).
7
Consider
∞
X
iP (W1 > i)
r−m
X
=
i=0
(j − b)P (S1 > j)
j=b
r−m
X
=
{r − (r − j)}P (S1 > j) − bE(W1 )
j=b
= r
r−m
X
P (S1 > j) −
j=b
(r − j)P (S1 > j) − bE(W1 )
j=b
(r − b)E(W1 ) −
=
r−m
X
r−m
X
j=b
r−j−1
r−1
(r − j)
(20)
m−1
m−1
from the expression for P (S1 > i) in Theorem 2.1, Part (B) of Holst (1986).
The numerator in the second term in (20) above can be expressed as
m
r−m
X
i=b
r−i
m
r−b+1
X
=m
j=m+1
j−1
m
r−b+1
=m
,
m+1
(21)
where the last equality follows from (15). Upon using (13) with W1 = X and
applying (20) and (21) we obtain (17).
Using (14) with W1 = X and W2 = Y , and applying Theorem 2.1 Part (E),
and Part (B) of Holst (1985) in succession, we obtain
XX
E(W1 W2 ) =
P (S1 > i, S2 > j)
i≥b j≥b
=
r−m−b
X r−m−i
X
i=b
=
r−m−b
X
X r−m−i
i=b
=
P (S1 > i, S2 > j)
j=b
P (S1 > i + j)
j=b
r−m−b
X r−m−i
X
i=b
j=b
r−i−j−1
r−1
.
m−1
m−1
Now, with k = r − i − j,
r−m−i
X
j=b
r−b−i
X k − 1 r − b − i
r−i−j−1
=
=
m−1
m−1
m
k=m
from (15). Hence
r−m−b
X r−m−i
X
i=b
j=b
r−m−b
X r − b − i
r−i−j−1
=
.
m−1
m
i=b
8
With k = r − b − i + 1, the above sum can be expressed as
r−2b+1
X
k=m+1
k−1
m
=
r − 2b + 1
.
m+1
Hence the claim in (18) holds. Clearly, when r < m + 2b, we cannot have both
W1 , W2 be positive simultaneously and hence E(W1 W2 ) = 0.
2
From (1) and the exchangeability of the Wi we see that
E(V )
= mE(W1 )
= mVar(W1 ) + m(m − 1)Cov(W1 , W2 )
Var(V )
= m[E(W12 ) − {E(W1 )}2 ] + m(m − 1)[E(W1 W2 ) − {E(W1 )}2 ]
= mE(W12 ) + m(m − 1)E(W1 W2 ) − m2 {E(W1 )}2
(22)
where the expectations on the right side of (22) are given by Theorem 2. Thus
we have the following result.
Theorem 3. If r ≥ m + b, the first two moments of the rv V representing the
number of vacant boxes are given by
r−b
E(V )
Var(V )
= m
=
m
r−1 ,
m−1
(23)
m(2(r − b) + 1)
(
−m2
r−b
m
r−b+1
m+1
r−1
m−1
− 2m2
+ m(m − 1)
r−2b+1
m+1
)2
r−b
m
r−1
m−1
(24)
where the coefficient of m(m − 1) in (24) is taken to be 0 whenever r < m + 2b.
Notes:
1. After deriving the expression for E(V ), we discovered it was previously
reported in Ivchenko (1994, p. 108). However, he does not derive a formula
for the variance of V .
2. Ivchenko (1994, p. 108) also derives an expression for E(N ). Using similar
exchangeability arguments, we can derive the same expression as well as
an expression for the variance of N . In particular, from Theorem 2.1 of
Holst (1985), we see that
r−b−1
r−1
E(I1 ) = E(I12 ) = P (S1 > b) =
≡ p1 ,
m−1
m−1
and
E(I1 I2 ) = P (S1 > b, S2 > b) =
9
r − 2b − 1
m−1
r−1
≡ p2 .
m−1
Thus from (2) we obtain
r−b−1
r−1
E(N ) = mE(I1 ) = m
m−1
m−1
and
Var(N )
=
mVar(I1 ) + m(m − 1)Cov(I1 , I2 )
=
mp1 (1 − p1 ) + m(m − 1)(p2 − p21 )
=
m(m − 1)p2 + mp1 − (mp1 )2
(25)
where the pi ’s are given above.
3. For m ≤ r − b,
E (V )
=
r−b
r−1
r−b
r
m
=r
m
m−1
m
m
(26)
=
(r − b)! (r − m)!
.
(r − b − m)! (r − 1)!
(27)
As seen from the expression in (27), E (V ) is symmetric in b and m, even
though the pmf for V is not symmetric in these parameters. The symmetry
also does not hold for the second moment.
4. As mentioned in Theorem 1, if r < m + b, P (V = 0) = 1. Thus if b ≥
r − m + 1 or m + b > r, all the Si are b or less. Thus, E(V ) = Var(V ) = 0
whenever r < m + b. When b = 1 and r > m, V is degenerate at r − m
and in that case E(V ) = r − m and Var(V ) = 0.
4
Distribution and Moments with Random m
Up to now we assumed the number of starting points m is fixed and restricted
to values in {2, . . . , r}. We now consider an extension in which the number of
starting points is a rv M with support {0, ..., r} and pmf P (M = m) = pM (m).
We let VM denote the number of vacant boxes in {0, . . . , r − 1} in this case to
highlight that the number of starting points in this case is allowed to be random.
Formally, we construct VM as follows. We first draw a value for M . If
M = 0, we designate all boxes as vacant and set VM = r. If M = m > 0, we
draw m starting points without replacement from the boxes labelled 0, . . . , r −1.
Let {R0 , . . . , Rm−1 } denote the identities of these starting points. For each
k ∈ {0, . . . , m − 1} we designate the boxes
{Rk , (Rk + 1)
mod r, . . . , (Rk + b − 1)
mod r}
as covered. Any box that is not covered is labelled vacant. Define Ji for each
i = 0, . . . , r − 1, as equal to 1 if box i is vacant and 0 if covered. Then the
10
number of vacant boxes is
VM =
r−1
X
Ji .
(28)
i=0
The discrete circle covering problem we started with is thus a special case of
this formulation in which M is degenerate with full mass at a single value m.
We now describe two different approaches for deriving the mean and variance
of VM in the general case where M can have a nondegenerate distribution. We
also obtain an expression for the pmf of VM .
4.1
A Conditioning Approach
When 2 ≤ m ≤ r, the rv VM given M = m has the same distribution as V in the
circle covering problem with m starting points. When M = 0 and M = 1, the
rv VM has a degenerate distribution will full mass at r and r − b, respectively.
We can therefore use a simple conditioning argument to obtain the following:
Theorem 4. The pmf, mean, and variance of VM are respectively given by
P (VM = x)
E(VM )
=
=
r
X
m=0
r
X
P (Vm = x)pM (m),
(29)
E(Vm )pM (m)
m=0
= r
r−b
X
m=0
r−b
m
r
m
pM (m),
(30)
and
V ar(VM )
=
V ar(E(VM |M )) + E(V ar(VM |M )),
(31)
where, for 2 ≤ m ≤ r, P (Vm = x) is given in Theorem 1 and expressions for
E(Vm ) and V ar(Vm ) are given in Theorem 3. Further, V0 is degenerate at r
and V1 is degenerate at (r − b)+ .
In (30) we have used the second form of E(Vm ) in (26) and the fact that
r
E(V1 ) = (r − b)+ = r r−b
1 / 1 .
We now describe three specific mechanisms that are covered by the above
result. The first two models have attracted attention in the application to financial contagion discussed later in Section 6, although the first one turns out
to be of more general interest as we discuss below.
Binomial
M . Let M assume a Binomial(r, p) distribution, i.e. pM (m) =
m
r
p
(1
−
p)r−m , 0 ≤ m ≤ r. This is equivalent to assuming that each box
m
i ∈ {0, . . . , r − 1} is chosen as a starting point with probability p independently
11
of whether any other box is chosen as a starting point. The expression for the
mean in (30) simplifies substantially;
r−b
X
r−b m
E(VM ) = r
p (1 − p)r−m = r(1 − p)b .
(32)
m
m=0
While an expression for V ar(VM ) can be obtained using (31), the algebra is
quite involved; we compute it using another approach in Section 4.2.
Mixture of Binomials. As above let M be Binomial(r,p) and assume p is also
a rv with support (0, 1), i.e. M is a mixture of binomials. In this case, we can
compute the conditional mean of VM given p as rE(1 − p)b . If p has a Beta
distribution, this expectation can be easily calculated and expressed in terms of
a beta function.
Sampling With Replacement.
Suppose we choose starting points by taking
n (≥ 1) draws from {0, . . . , r − 1} with replacement. The resulting number of
starting points M will be random with support {1, . . . , n}, and its pmf is given
by (see, e.g., Feller, 1968, p. 102)
X
m
r 1
j m
(−1)
(m − j)n
pM (m) =
j
m rn j=0
r(r − 1) · · · (r − m + 1)
S(n, m)
(33)
rn
where S(n, m) is the Stirling’s number of second kind (see, e.g., Johnson and
Kotz, 1977, pages 110, 8 for the Stirling number connection).
=
When the values of all the parameters are known, the expressions for pmf,
mean, and variance (29)–(31) given in Theorem 4 can be numerically evaluated
in a straightforward manner. However, we now introduce another approach that
greatly simplifies the evaluation of these moments.
4.2
An Alternative Approach for Moments
An alternative approach to computing moments for VM makes use of (28) which
defines VM as the sum of the indicator variables Ji where Ji = 1 if box i is vacant.
As long as the sampling mechanism is completely symmetric with respect to the
r available boxes, these Ji are identically distributed (but not exchangeable) and
E(VM )
V ar(VM )
= rP (J0 = 1),
= rP (J0 = 1)P (J0 = 0) + r
r−1
X
Cov(J0 , Jk );
(34)
k=1
and Cov(J0 , Jk ) = P (J0 = 1, Jk = 1) − [P (J0 = 1)]2 . We will now illustrate the
utility of this approach with the examples considered above.
12
Binomial M (Contd.) Note that box 0 is vacant if and only if none of the boxes
labeled 0, r − 1, . . . , r − b + 1 are starting points, i.e. {R0 , . . . , RM −1 } ∩ {0, r −
1, . . . , r − b + 1} = ∅. Recall that when M is binomial, each box i is chosen as
a starting point with probability p independently of whether any other box is a
starting point. Hence, P (J0 = 1) = q b , where q = 1 − p, and E(VM ) = rq b as
seen earlier in (32). When 2b ≤ r, we have
b+k
if 1 ≤ k ≤ b − 1
q ,
b+r−k
P (J0 = 1, Jk = 1) = q
, if r − b + 1 ≤ k ≤ r − 1
2b
q ,
if b ≤ k ≤ r − b;
b+k
− q 2b ,
if 1 ≤ k ≤ b − 1
q
b+r−k
Cov(J0 , Jk ) = q
− q 2b , if r − b + 1 ≤ k ≤ r − 1
0,
if b ≤ k ≤ r − b.
and consequently, we obtain
r−1
X
Cov(J0 , Jk )
=
k=1
b−1
X
q b+k +
k=1
r−1
X
q b+r−k − 2(b − 1)q 2b
k=r−b+1
b
=
2q b
q−q
− 2(b − 1)q 2b .
p
(35)
It follows that when 2b ≤ r + 1,
V ar(VM ) = rq
b
2q
b 2
1+
− q ( + 2b − 1) .
p
p
(36)
For larger b values, the associated geometric series can be summed similarly.
The rv VM is never degenerate whatever be b. For b ≥ r, VM has a two point
distribution with E(VM ) = rq r , and V ar(VM ) = r2 q r (1 − q r ).
Sampling With Replacement (Contd.) We will now use (34) and obtain the first
two moments of VM when M is determined by drawing n times with replacement
from all boxes. Note that P (J0 = 1) = {(r − b)/r}n ; thus
n
b
E(VM ) = r 1 −
, 1 ≤ b ≤ r.
(37)
r
Since
n
r−(b+k)
, if 1 ≤ k ≤ b − 1
r
k−b n
P (J0 = 1, Jk = 1) =
,
if r − b + 1 ≤ k ≤ r − 1
r
r−2b n ,
if b ≤ k ≤ r − b,
r
13
(38)
we obtain
n
2n
r−(b+k)
− r−b
, if 1 ≤ k ≤ b − 1
r
r
2n
k−b n
Cov(J0 , Jk ) =
− r−b
,
if r − b + 1 ≤ k ≤ r − 1
r
r
2n
r−2b n
− r−b
,
if b ≤ k ≤ r − b.
r
r
Thus,
r−1
X
Cov(J0 , Jk )
k=1
=2
n
b−1
X
r−b−k
k=1
r
+ (r − 2b + 1)
r − 2b
r
n
− (r − 1)
r−b
r
2n
,
and for 2b ≤ r + 1, (34) yields the following expression for V ar(VM )/r:
n
n
n
2n
b−1
X
b+k
2b
b
b
+2
1−
+ (r − 2b + 1) 1 −
−r 1 −
. (39)
1−
r
r
r
r
k=1
For b ≥ r, VM is degenerate at 0; for other b values, the third term above vanishes and k is summed from 1 to (r − b) in the second term.
Without-Replacement Sample. In this setup, we have
r−b
r
P (J0 = 1) =
m
m
leading to the second form for E(V ) given in (26). Using a representation
for P (J0 = 1, Jk = 1) that parallels (38), an expression for the variance can
be written using (34). We will not pursue it as we already have a compact
expression available in (24).
5
Limiting Properties of V
5.1
Limiting Distributions
We now return to the case where the number of starting points m is fixed. Holst
(1985; Theorem 3.2) argues that as r, b → ∞ with b/r → a for some 0 < a < 1,
d
V /r → Va where Va has the same distribution as the length of non-covered
segments when m arcs of length a are dropped at random on a circle with
unit circumference. Siegel (1978; Theorem 3) has shown that the distribution
of Va can be expressed as the mixture of a degenerate and a continuous rv.
Specifically, he shows that P (Va (m) = (1 − ma)+ ) = pa (m) where
m−1
X
m
m−1
pa (m) =
(−1)i
(1 − ia)+
, ma > 1
(40)
i
i=0
=
(1 − ma)m−1 , ma ≤ 1,
14
(41)
and with probability 1 − pa (m), Va (m) behaves like a continuous rv Wa (m)
having the pdf f (w; a, m) given by
f (w; a, m)
m
1 − pa (m)
=
m m−1
X
X
(−1)i+j
i=1 j=1
m−1 m−1
i−1
m−j−1
wj−1 (1 − ia − w)+
,
i−1
j
j−1
(1 − ma)+ < w < 1 − a.
(42)
with the convention that (1 − ia − w)0+ is interpreted as 1 if 1 − ia − w ≥ 0, and
as 0, otherwise. We now show the following.
Lemma 1. If r, b → ∞ such that b/r → a, 0 < a < 1, then
P {V = (r − mb)+ } → pa (m) ≡ P {Va = (1 − ma)+ },
given by (40) when ma > 1, and by (41) when ma < 1. When ma = 1, both
(40) and (41) reduce to 0.
Proof. When ma > 1, r − mb is eventually negative, our interest then is in the
limiting form of P (V = 0) given in (5). Consider the jth term there, excluding
the factor (−1)m m
j :
r−jb−1
m−1
r−1
m−1
=
(r − jb − 1) · · · (r − jb − m + 1)
,
(r − 1) · · · (r − m + 1)
if r − jb = r(1 − j(b/r)) > m − 1; and it is 0 if r(1 − j(b/r)) ≤ m − 1. So if
m−1
b/r → a with 1 − ma < 0, the above ratio converges to (1 − ja)+
. Thus the
limit is given by (40).
Whenever ma < 1, since r − mb = r(1 − m(b/r)), r − mb eventually exceeds
any fixed m. In that case the term (10) converges to (1−ma)m−1 . The remaining
finite number of terms in the numerator on the right in (11) are finite and each
is of o(rm−1 ) whereas the denominator is O(rm−1 ). Thus, the only nonzero
term in the limit is that of (10) and it coincides with (41).
If ma = 1, (41) is obviously 0 and now we show that (40) also converges
to 0 as a → (1/m)+ . For this we consider the continuous uniform spacing
problem where one chooses at random m − 1 points U1 , . . . , Um−1 from the
interval (0, 1). With spacings defined as Yi = Ui:m−1 − Ui−1:m−1 , i = 1, . . . , m,
where U0:m−1 = 0 and Um:m−1 = 1, it is known that the distribution function
of the continuous rv Y(m) representing the maximal spacing can be expressed as
(see, e.g., David and Nagaraja, 2003, p. 135)
m
X
m
P (Y(m) ≤ a) = 1 − P (Y(m) > a) =
(−1)i
(1 − ia)m−1
,
(43)
+
i
i=0
for all a in (0, 1). Since by construction the maximal spacing Y(m) exceeds 1/m
with probability 1, the right side sum in (43) is 0 whenever a ≤ 1/m or ma ≤ 1
15
and thus approaches 0 as a → (1/m)+ . The difference between this sum and the
sum in (40), (−1)m (1 − am)m−1
, is 0 whenever a ≥ (1/m). Thus we conclude
+
that as a → (1/m)+ the expression in (40) converges to 0.
2
Notes:
5. When b = ar, with ma < 1, we have seen that the expression in (10)
converges to (1 − ma)m−1 , while the other terms contributing to P (V =
r − mb) converge to 0, indicating the dominant nature of this term missing
in Holst’s Theorem 2.2 (1985). That is, the term missing from Holst’s
expression is precisely what converges to the degenerate component of the
rv in the continuous case.
6. Holst’s Theorem 3.2 gives expressions for P (Va = 0) and the pdf of the
continuous part. Lemma 1 reveals that his expressions are imprecise and
fail to properly account for the range of V .
7. Siegel’s (1978) version of (40) [his expression (3.23)] has the summation
that includes an additional term with i = m. In view of the assumption
that ma > 1, the corresponding term is 0, and hence they coincide. Further, in view of the above lemma, we can conclude that when ma = 1
both (40) and (41) hold.
8. Consider the case where M is random, specifically where M is generated
by taking n draws with replacement from {1, . . . , r}. As r → ∞ while n
is held fixed, the first factor in the expression for the pmf of M in (33)
converges to 0 whenever m < n and to 1 when m = n. Since S(n, n) = 1,
P
it follows that M → n, the sample size, and the limiting distribution of
VM is the same as the limiting distribution of V when m is replaced by
n. More generally, given any process M that converges in probability to
a degenerate distribution as r → ∞, the distribution for VM will converge
to the same limiting distribution as V with m corresponding to the value
that M collapses to.
For m = 5 and selected r values, Figure 1 and Figure 2 respectively provide the
normalized conditional pmf of V given the event {V > (r − mb)+ }, for a = 0.1
and 0.25. The case of r = ∞ corresponds to the conditional pdf f (w; a, m)
of the continuous case, given in (42). Both these figures suggest that by the
time r reaches 500, we are close to the limiting result, indicating that when the
sampling fraction is under 1%, f (w; a, m) provides a close approximation to the
conditional pmf of V .
16
12
rP(V = wr | V > (r‐mb)+)
r = 50
r = 100
r = 500
10
r=
8
6
4
2
w
0.0
0.2
0.4
0.6
0.8
1.0
Figure 1: rP {V = rw|V > (r − bm)+ } for m = 5, a = 0.1, b = ar, for selected
r; f (w; a, m), given in (42), corresponds to r = ∞.
17
4
rP(V = wr | V > (r‐mb)+)
r = 48
r = 100
3
r = 500
r=
2
1
w
0.0
0.2
0.4
0.6
0.8
1.0
Figure 2: rP {V = rw|V > (r − bm)+ } for m = 5, a = 0.25, b = ar, for selected
r; f (w; a, m), given in (42), corresponds to r = ∞.
18
5.2
Limiting Moments
Limits for Large r and b
Since V /r is uniformly bounded, convergence in distribution implies that
E(V /r)k → E(Vak ) when r → ∞ and b/r → a and ma 6= 1. Siegel has shown in
his Theorem 2 that,
E(Vak ) =
−1 X
k
k+m−1
k
m−1
m+k−1
(1 − ia)+
, k ≥ 1.
m
i
i
−
1
i=1
(44)
Hence we can obtain approximations to any moment of V when m is small and
r and b are large using the moments of Va .
Table 2: Properties of V and Va for m = 5, r = 20, 50, and selected b
values; a = b/r.
b
P (V = (r − mb)+ )
pa (m)
1
2
3
4
5
1
0.008
0.405
0.802
0.960
0.3164
0.0625
0.0040
0
0.0040
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
0.6407
0.3882
0.2189
0.1121
0.0502
0.0183
0.0047
0.0006
0.0000
0.0006
0.0047
0.0179
0.0452
0.0885
0.1467
0.2158
0.2912
0.3689
0.4455
0.6561
0.4096
0.2401
0.1296
0.0625
0.0256
0.0081
0.0016
0.0001
0
0.0001
0.0016
0.0081
0.0246
0.0545
0.0989
0.1561
0.2226
0.2944
0.3680
E(V )
r = 20
15
11.05
7.98
5.63
3.87
r = 50
45
40.408
36.199
32.348
28.832
25.628
22.716
20.075
17.685
15.528
13.587
11.845
10.287
8.897
7.661
6.566
5.601
4.752
4.010
3.363
19
rE(Va )
SD(V )
rSD(Va )
15.48
11.14
8.87
6.55
4.75
0
0.80
1.42
1.84
2.04
0.49
1.14
1.71
2.11
2.31
45.196
40.769
36.695
32.954
29.525
26.387
23.521
20.911
18.537
16.384
14.436
12.678
11.095
9.675
8.404
7.27
6.262
5.369
4.581
3.888
0
0.586
1.213
1.875
2.532
3.156
3.726
4.229
4.658
5.007
5.273
5.457
5.560
5.586
5.541
5.430
5.261
5.041
4.780
4.486
0.341
0.899
1.536
2.196
2.842
3.452
4.009
4.503
4.925
5.270
5.538
5.728
5.842
5.881
5.852
5.759
5.609
5.408
5.164
4.886
Table 2 provides some key facts about the features of the distributions of V
and the limiting rv Va for m = 5, r = 20, 50 and b values up to 20 corresponding
to a good range of a values. It shows that as a increases pa (m) decreases for
a ≤ 1/m, it is 0 when ma = 1 and then pa (m) increases. Note that whenever
a reaches 1/m from below, the lower limit of the support of Va moves towards
0 and whenever ma > 1, the lower limit remains at 0. This limiting pattern
is closely followed by V when r = 50, but not that closely when r = 20. The
moments converge fairly quickly to the limiting values. The mean is better approximated by the limit for small b (= ar) whereas for the standard deviation,
large b values tend to be slightly more efficient.
Limits for Large r and m
When b is held fixed and r, m → ∞ such that m/r → p = 1 − q, 0 < p < 1,
Holst (1985, Theorem 4.2) has shown that V is asymptotically normal and has
given the first two moments of the limit distribution. We now derive asymptotic
approximations for the expressions for E(V ) and V ar(V ) in (23) and (24) to
study their limiting properties. When b is held fixed, we have
C(r, m; b) ≡
(r − m)(r − m − 1) · · · (r − m − b + 1)
(r − b)!(r − m)!
=
≈ qb .
r!(r − m − b)!
r(r − 1) · · · (r − b + 1)
Hence
r−b
m
r−1
m−1
r−b+1
m+1
r−1
m−1
r−2b+1
m+1
r−1
m−1
=
=
=
r
r
C(r, m; b) ≈ q b ;
m
m
r(r − b + 1)
r(r − b + 1) b
C(r, m; b) ≈
q ;
m(m + 1)
m(m + 1)
r(r − 2b + 1)
r(r − 2b + 1) 2b
C(r, m; 2b) ≈
q .
m(m + 1)
m(m + 1)
Upon plugging in these approximations in the expressions given in Theorem 3,
we obtain
E(V ) ≈ rq b ,
(45)
and V ar(V )/r is
m
m−1
≈ q b 2(r − b) + 1 − 2
(r − b + 1) + q 2b
(r − 2b + 1) − r
m+1
m+1
qb
q 2b
{2(r − b) + 1 − m} −
{2r + (2b − 1)(m − 1)}
m+1
m+1
1+q
2
≈ qb
− q 2b
+ 2b − 1 .
p
p
=
(46)
We note that the approximations of the mean and variance of V above where
m grows deterministically with r match with the exact moments of VM in the
20
case where M is Binomial(r, p) with p equal to the limiting value of m/r. In
particular, (45) matches with (32) and (46) matches the expression in (36). The
expressions we obtain for the variances differ from the variance of the limiting
normal distribution reported in Theorem 4.2 of Holst (1985). While the convergence in distribution and convergence of moments are not directly related, it
appears his expression is incorrect.
Remark 2: To appreciate why the asymptotic approximations for the mean
and variance of V coincide with the exact moments of VM when M has a binomial distribution, observe that when m/r → p, if we take any pair of boxes
i and j, the probability that each is a drawn as a starting point converges to p
while the probability that both are drawn converges to p2 , i.e. the two events
are asymptotically independent. But recall that this independence is what distinguishes the case where M is binomial when r is finite. Consistent with this,
Ji converges in probability to a Bernoulli rv with success probability (1 − p)b ,
which is the exact distribution of Ji when M is binomial. In other words, when
the number of starting points m is deterministic but grows proportionately with
r, the number of vacant boxes in a block of boxes of fixed size behaves asymptotically in the same way as the number of vacant boxes in that block for the
same r where M is distributed Binomial(r, p).
Sampling with Replacement: Limits for Large r and n
Finally, consider the case where M is random and generated by drawing
n times with replacement from all boxes. As we discussed above, this case
converges to the discrete circle covering problem with n boxes. Consistent with
this, suppose r, n → ∞ such that n/r → θ, 0 < θ < 1. Using (37) and (39) we
obtain
E(VM ) ≈
Var(VM )
r
≈
re−bθ = rq b
e−bθ + 2
b−1
X
(47)
e−(b+k)θ + (r − 2b + 1)e−2bθ − re−2bθ
k=1
=
q
b1
+q
− q 2b
p
2
+ 2b − 1
p
(48)
upon simplification with − log(q) = θ, and p = 1 − q. This matches (32) and
(36). Under this scheme, the probability that a particular box is never among
the n boxes drawn is (1 − r−1 )n , and thus the expected number of distinct boxes
chosen is
1 n
E(M ) = r 1 − (1 − )
≈ r(1 − e−θ ) = r(1 − q) = rp.
r
In other words E(M ) ≈ rp.
21
6
Application to Financial Contagion
We conclude by showing how the discrete circle covering problem we analyzed
is related to the literature on financial contagion. One of the workhorse model
of financial contagion is due to Eisenberg and Noe (2001). Their model posits
that banks are connected via a directed network based on the obligations banks
owe one another. If some banks incur losses, they will be unable to meet their
required payments to other banks, inflicting losses on other banks. Thus, shocks
that affect certain nodes in a network can propagate to other nodes. Eisenberg
and Noe derive the vector of clearing payments given the network structure and
the identity of the nodes that incur direct losses. One can use this vector to
deduce which banks will be adversely affected when certain banks are hit.
Subsequent work has extended the Eisenberg and Noe model by assuming
that the number of banks that experience losses as well as their identity is random. For tractability, this literature has focused on simple network structures.
For example, Caballero and Simsek (2013), Acemoglu, Ozdaglar, and TahbazSalehi (2014) and Alvarez and Barlevy (2015) all consider contagion in circular
networks in which the network of obligations across banks is isomorphic to a
circular graph. While these networks bear little resemblance to the pattern of
obligations across banks in practice, these settings still provide useful intuition
about what determines contagion and how banks might behave when they are
uncertain about the extent of contagion. We now show how contagion in circular networks is related to the discrete circle covering problem. We further
argue that the connection between the two suggests generalizations of the circle
covering problem that to our knowledge have not been noted previously.
Suppose there are r banks, that each bank owes an amount λ to one other
bank, and that each bank is in turn owed λ by another bank. Banks can be
viewed as connected to one another via a directed network in which a bank
points to another bank if the former owes something to the latter. Formally,
index banks in the network by i ∈ {0, . . . r − 1}. A circular network is one
where bank i owes λ to bank i + 1 mod r. We drop the reference to mod r in
what follows. Following Alvarez and Barlevy (2015), we impose the following
assumptions: (1) Each bank owns µ worth of assets that it can sell to repay
its outstanding obligations if it needs to, (2) Among the r banks, a random
number M with pmf P (M = m) = pM (m) will be “bad”, meaning they incur
a loss of size φ that must be subtracted
from their initial asset holdings µ, (3)
r
Given M = m, each of the m
groups of size m is equally like to be those
r
µ. The last assumption implies bad banks
which are bad, and (4) µ < φ < m
incur losses that exceed what they can afford to pay by liquidating their assets,
but total losses across all m bad banks are still less than the combined value
of all assets held among all r banks in the network. Banks are required to pay
their full obligation λ if possible, and must sell their asset holdings if they fall
short. Although the distribution of the number of bad banks M is unrestricted,
Alvarez and Barlevy (2015) draw particular attention to the cases where M has
a degenerate distribution (i.e. pM (m) = 1 for some m), a binomial distribution,
22
and a mixture of binomials as instructive special cases.1
Let xi denote the amount bank i pays bank i + 1 and assume that the bad
banks are labeled Rk , k = 0, . . . , m−1, and these are the ones who have incurred
a direct external loss of φ and the others have no external losses. Given that
banks must pay their obligations if they can, the payments {xi }r−1
i=0 satisfy the
following system of equations
xi
=
min{(xi−1 + µ − φ)+ , λ},
=
min{xi−1 + µ, λ},
i ∈ {R0 , ..., Rm−1 }
i 6∈ {R0 , ..., Rm−1 },
r−1
r
µ, there exists a unique solution {xi }i=0
to this
with x−1 ≡ xr−1 . For φ < m
system. Bank i is said to be insolvent if xi < λ, i.e. if it cannot meet its
obligation, and solvent otherwise. Each of the m bad banks are insolvent, since
even if they received the full amount λ from the bank that is obligated to them,
the fact that µ < φ implies xi−1 + µ − φ < λ and so they would be unable to
pay in full even after liquidating their assets. Beyond these m bad banks, banks
that do not directly suffer losses may still end up insolvent because they are
exposed to bad banks either directly – meaning the bank that owes them λ is
bad – or indirectly – meaning the bank that owes them λ is good but is exposed
to a bad bank. A central question in this framework is to determine the number
of banks that are insolvent, i.e. to gauge the extent of contagion when only M
banks suffer direct losses.
The results turn out to depend on the parameter λ. Suppose λ ≤ φ − µ, and
λ = bµ, for some integer b. Then (b + 1)µ ≤ φ, and xRk = 0, and xRk +j = jµ,
j = 1, . . . , min{b − 1, Sk+1 − 1}, for k = 0, . . . , m − 1. Hence, the number of
insolvent banks starting from each bad bank is a fixed number b, unless one of
those b banks is itself bad. It should be clear that the number of bad banks
corresponds to the number of starting points M , while the number of solvent
banks corresponds to the number of vacant boxes V with b equal to µλ . Thus
our results provide the small as well as large sample properties of the number
of solvent banks in a circular network with a random number of bad banks M
when λ ≤ φ − µ.
The situation where λ > φ − µ provides a new generalization of the discrete
circle covering problem. In this case, bk , the number of insolvent banks induced
by the kth bad bank, becomes a rv that depends on the location of the other bad
banks. However, unlike in Siegel and Holst (1982) who discuss the continuous
case of the circle covering problem assuming the length of the arc b starting at
any point is an i.i.d. rv, here the number of insolvent banks starting at each
Rk will depend on the distribution of the spacings between the bad banks. To
elaborate, xRk = (xRk −1 −(φ−µ))+ is no longer identically 0, and this will affect
the number of banks immediately following bank Rk that are insolvent. That is,
the number of banks that must be covered starting from the kth bad bank is a rv
1 Specifically, the unconditional probability that a bank is bad is higher than, equal to, and
lower than the probability that a bank is bad conditional on news that another bank is good
when M is degenerate, binomial, and a mixture of binomials, respectively. These distinctions
turn out to matter when there is some possibility of news about some banks might be revealed.
23
that depends on the entire collection of spacings {Sk }M
k=1 . Alvarez and Barlevy
(2015) show that when M has a degenerate distribution so that pM (m) = 1 for
some m, if λ > m(φ − µ) then the number of solvent banks V (vacant boxes) is
degenerate and equals r − mb where b = λ/µ. In the intermediate case where
φ − µ < λ < m(φ − µ) the distribution of V is non-degenerate. We leave the
investigation of the closed-form expression for this distribution where bk is a
function of {Sk }m
k=1 for future work. More generally, results for Bose-Einstein
statistics may prove useful for analyzing contagion in networks that are not
circular but still symmetric.
References
[1] Acemoglu D, Ozdaglar A, and Tahbaz-Salehi A (2013) Systemic risk and
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University of Chicago Working Paper.
[3] Barlow RE, Proschan F (1981) Statistical Theory of Reliability and Life
Testing. To Begin With, Silver Springs, MD.
[4] Caballero RJ, and Simsek A (2013) Fire sales in a model of complexity. The
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[5] David HA, Nagaraja HN (2003) Order Statistics 3rd edn. Wiley, Hoboken
NJ.
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Vol 1, 3rd edn. Wiley, New York.
[8] Harris J, Hirst JL, Mossinghoff M (2008) Combinatorics and Graph Theory.
Springer.
[9] Holst L (1985) On discrete spacings and the Bose–Einstein distribution. In
Contributions to Probability and Statistics in honour of Gunnar Blom. Ed.
by Jan Lanke and Georg Lindgren. Lund. 169–177.
[10] Ivchenko GI (1994) On the random covering of a circle: a discrete model
(in Russian). Diskret. Mat. 6, No. 3, 94–109.
[11] Johnson NL and Kotz S (1977). Urn Models and Their Application. Wiley,
New York.
[12] Siegel AF (1978) Random arcs on the circle. J. Appl. Prob. 15, 774–789.
[13] Siegel AF, Holst L (1982) Covering the circle with random arcs of random
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24
[14] Stevens WL (1939) Solution to a geometric problem in probability. Annals
of Eugenics, 9, 315–320.
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Neerlandica, 48, 201–207.
25
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