ASYMPTOTICS OF THE LURIA-DELBRÜCK DISTRIBUTION

Helmut Prodinger

Technical University of Vienna Department of Algebra and Discrete Mathematics

We consider the Luria–Delbrück distribution, as described in [2] and previous writings, cited therein. It is given by the probability generating function

G(z) = (1 − z)^{m(1−z)/z} = ∑_{n≥0} p_{n}z^{n},

where m is a parameter.

We show that with singularity analysis of generating functions [1] (in principle) a full asymptotic expansion of p_{n} and P_{n} := ∑_{k>n} p_{k} can be achieved. In this way we generalize results given in [2].

The basic idea of this powerful method is to realize that G(z) has radius of convergence 1 and 1 as the only singularity on the radius of convergence. Then there is a local expansion about z = 1 (produced by Maple),

G(z) = 1 − m(1 − z) log \frac{1}{1 − z} − m(1 − z)^{2} log \frac{1}{1 − z}

+ \frac{m^{2}}{2} (1 − z)^{2} log^{2} \frac{1}{1 − z} + O ((1 − z)^{3} log^{3} \frac{1}{1 − z} ) ,

(1)

which can be translated term–by–term into an asymptotic expansion of the coefficients under some mild technical conditions which are satisfied in out instance; we refer again to [1] for a full treatment.

Therefore we only have to know about the expansion of the standard functions appearing in (1).

We use the following notions: [z^{n}]f(z) means the coefficient of z^{n} in (the Taylor expansion of) f(z), the symbol n\underline{k} means n(n − 1) … (n − k + 1) and H_{n} = 1 + \frac{1}{2} + \frac{1}{3} + ⋯ + \frac{1}{n} means the nth harmonic number. Then we have for n large enough

[z^{n}](1 − z)^{i} log \frac{1}{1 − z} = \frac{(−1)^{i}i!}{n\underline{i+1}},
[z^{n}] log^{2} \frac{1}{1 − z} = \frac{2}{n}H_{n−1},
[z^{n}](1 − z)^{2} log^{2} \frac{1}{1 − z} = \frac{2}{n}H_{n−1} − \frac{4}{n − 1}H_{n−2} + \frac{2}{n − 2}H_{n−3}.

Typeset by AMS-TEX

Hence

p_{n} = \frac{m}{n\underline{2}} + \frac{2m^{2}}{n\underline{3}} + \frac{m^{2}}{2} (\frac{2}{n}H_{n−1} − \frac{4}{n − 1}H_{n−2} + \frac{2}{n − 2}H_{n−3}) + O (\frac{log^{2} n}{n^{4}}) .

Maple also translates this expansion into more traditional terms,

p_{n} = \frac{m}{n^{2}} + 2m^{2} \frac{log n}{n^{3}} + \frac{2m^{2}γ − 3m^{2} − m}{n^{3}} + O (\frac{log^{2} n}{n^{4}}) .

To get the asymptotic expansions for the tails P_{n}, we just note that its generating function is

\frac{1}{1 − z} − \frac{1}{1 − z} G(z) = m log \frac{1}{1 − z} + m(1 − z) log \frac{1}{1 − z}

− \frac{m^{2}}{2} (1 − z) log^{2} \frac{1}{1 − z} + O ((1 − z)^{2} log^{3} \frac{1}{1 − z} ) ,

from which we conclude

P_{n} = \frac{m}{n} − \frac{m}{n\underline{2}} − \frac{m^{2}}{2} (\frac{2}{n}H_{n−1} − \frac{2}{n − 1}H_{n−2}) + O (\frac{log^{2} n}{n^{3}})

= \frac{m}{n} + m^{2} \frac{log n}{n^{2}} + \frac{m^{2}γ − m^{2} − m}{n^{2}} + O (\frac{log^{2} n}{n^{3}}) .

References

Mathematics 3 (1990), 216–240.

828.

Helmut Prodinger Technical University of Vienna Wiedner Hauptstrasse 8–10 A-1040 Vienna Austria

1. P. Flajolet and A. Odlyzko, Singularity Analysis of Generating Functions, SIAM J.Discrete 2. A.W. Kemp, Comments on the Luria–Delbrück Distribution, J. Appl. Prob. 31 (1994), 822-Department of Algebra and Discrete Mathematics

E-mail address: proding@rsmb.tuwien.ac.at