Statistics & Probability
Letters 11 (1991) 161-165
of Bernoulli random variables
for random sums
Department of Mathematics, Royal Institute of Technology, S-10044 Stockholm, Sweden
and an appropriate
marked and thinned
Bounds for the total variation
are given. The results can be used to derive limit theorems
Some examples are given.
between the distribution of the sum of a random number
with rates of convergence
of Bernoulli summands
Keywords: Poisson convergence, random sums, total variation, point processes, thinned point processes, Hellinger integral.
matics, and in many other applications
sequence of dependent
that each event has small probability
Then, it is of interest
events which occur
However, many of the known
not give any rates of convergence.
cannot be useful in applications.
we need explicit error bounds
Let cl, Ez,. . . , 5, be Bernoulli
with P( [, = 1) = p, = 1 - P( 6, = 0), and consider
the sum S, = E, + .& + . . . t-5,. In the last three
decades there has been a continued
approximation of the distribution
by a suitably chosen Poisson
effort has been made to evaluate
approximations by measuring
For any nonnegative
We have a
to prove that the number
for the approxima-
of the sum S,
the error of such
the fit via the total
tance d by
X and Y define the total variation dis-
d(X, Y) = supIP(X~4) -P(YEA)~.
It can be shown that
d(X, Y)=+ 5
In the sequel we denote by U, a Poisson distribu-
tion with mean A. In the case of independent
Barbour and Hall (1984) showed that
d(S,,, U,) < y f: P:,
gives the following
X =p, + . . . +p,.
for any i = 1,. . , n, inequality
bound for the error
the binomial distribution
In the simplest case
by the Pois-
G (1 - eCnP)p.
0167-7152/91/$03.50 0 1991 - Elsevier Science Publishers B.V. (North-Holland)