# convpow-methods

##### Distribution of the sum of univariate i.i.d r.v's

Method `convpow`

determines the distribution of the sum of N univariate
i.i.d r.v's by means of DFT

- Keywords
- distribution

##### Usage

```
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)

##### 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)`

.

##### Value

- Object of class
`"AbscontDistribution"`

,`"DiscreteDistribution"`

,`"LatticeDistribution"`

resp.`"AcDcLcDistribution"`

##### concept

- convolution for distributions
- arithmetics for distributions
- info file
- FFT
- DFT

##### 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

##### See Also

`operators`

, `distrARITH()`

##### Examples

`convpow(Exp()+Pois(),4)`

*Documentation reproduced from package distr, version 2.0.2, License: LGPL-3*