convpow-methods: Distribution of the sum of univariate i.i.d r.v's
Description
Method convpow
determines the distribution of the sum of N univariate
i.i.d r.v's by means of DFTUsage
convpow(D1,...)
## S3 method for class 'AcDcLcDistribution':
convpow(D1,N)
## S3 method for class 'AbscontDistribution':
convpow(D1,N)
## S3 method for class 'LatticeDistribution':
convpow(D1,N)
## S3 method for class 'DiscreteDistribution':
convpow(D1,N)
Arguments
D1
an object of (a sub)class (of) "AbscontDistribution"
or
"LatticeDistribution"
or of "UnivarLebDecDistribution"
...
not yet used; meanwhile takes up N
N
an integer or 0 (for 0 returns Dirac(0), for 1 D1)
Value
- Object of class
"AbscontDistribution"
, "DiscreteDistribution"
,
"LatticeDistribution"
resp. "AcDcLcDistribution"
concept
- convolution for distributions
- arithmetics for distributions
- info file
- FFT
- DFT
Details
in the methods implemented a second argument N
is obligatory;
the general methods use a general purpose convolution algorithm for
distributions by means of D/FFT. In case of an argument of class
"UnivarLebDecDistribution"
, the result will in generally be
again of class "UnivarLebDecDistribution"
. However, if acWeight(D1)
is positive, discreteWeight(convpow(D1,N))
will decay exponentially
in N
, hence from some (small) $N_0$ on, the result will be of
class "AbscontDistribution"
. This is used algorithmically, too, as
then only the a.c. part needs to be convolved.
In case of an argument D1
of class "DiscreteDistribution"
,
for N
equal to 0,1 we return the obvious solutions, and for N==2
the return value is D1+D1
. For N>2
, we split up N
into
N=N1+N2
, N1=floor(N/2)
and recursively return
convpow(D1,N1)+convpow(D1,N2)
.References
Kohl, M., Ruckdeschel, P., Stabla, T. (2005):
General purpose convolution algorithm for distributions
in S4-Classes by means of FFT.
Technical report, Feb. 2005. Also available in
http://www.uni-bayreuth.de/departments/math/org/mathe7/RUCKDESCHEL/pubs/comp.pdf