# illustrateLLN

From distrEx v1.9
by Matthias Kohl

##### Functions for Illustrating the LLN

Functions for generating a sequence of plots of randomly generated replicates of $\bar X_n= \frac{1}{n} \sum_{i=1}^n X_i$ for sums of iid r.v. distributed according to a prescribed discrete or absolutely continuous distribution. A line for the expectation and CLT based (pointwise) 95%-confidence bands are also plotted and the empirical coverage of this band by the replicated plotted so far is indicated.

- Keywords
- methods, distribution, dynamic

##### Usage

```
illustrateLLN(Distr = Norm(),n = c(1,3,5,10,25,50,100,500,1000,10000),
m = 50, step = 1, sleep = 0, withConf = TRUE,
withCover = (length(n)
```

##### Arguments

- Distr
- object of class
`"UnivariateDistribution"`

: distribution of the summands - n
- vector of integers: sample sizes to be considered
- m
- integer: (total) number of replicates to be plotted subsequently
- step
- integer: number of replicates to be drawn at once
- sleep
- numeric: pause in seconds between subsequent plots
- withEline
- logical: shall a line for the limiting expectation
(in case of class
`Cauchy`

instead: median) be drawn? - withConf
- logical: shall (CLT-based) confidence bands be plotted?
- withCover
- logical: shall empirical coverage of (CLT-based) confidence bands be printed?
- withLegend
- logical: shall a legend be included?
- CLTorCheb
- character: type of confidence interval ---"CLT" or "Chebyshev"; partial matching is used; if this fails "CLT" is used.
- coverage
- numerical: nominal coverage of the confidence bands ---to be in (0,1)
- col.Eline
- character or integer code; color for confidence bands
- lwd.Eline
- integer code (see
`par`

); line width of the confidence bands - lty.Eline
- integer code (see
`par`

); line type of the confidence bands - col.Conf
- character or integer code; color for confidence bands
- lwd.Conf
- integer code (see
`par`

); line width of the confidence bands - lty.Conf
- integer code (see
`par`

); line type of the confidence bands - cex.Cover
- magnification w.r.t. the current setting of
`cex`

to be used for empirical coverages; as in`par`

- cex.legend
- magnification w.r.t. the current setting of
`cex`

to be used for the legend as in`par`

- ...
- further arguments to be passed to
`matplot`

,`matlines`

,`abline`

##### Details

`illustrateLLN`

generates a sequence of plots.
Any parameters of `plot.default`

may be passed on to this particular
`plot`

method.
There are default `main`

titles as well as `xlab`

and `ylab`

annotations.
In all title arguments, the following patterns are substituted:

`"%C"`

`x`

}
`"%P"`

`x`

in form of a comma-separated list of
`"%Q"`

`x`

in form of a comma-separated list of
`"%N"`

`x`

in form of a comma-separated list
`"%A"`

`x`

}
`"%D"`

`"%X"`

##### Value

- void

##### code

`lty`

##### concept

- illustration
- teaching
- LLN

##### Examples

```
illustrateLLN(Distr = Unif())
illustrateLLN(Distr = Pois(lambda = 2))
illustrateLLN(Distr = Pois(lambda = 2)+Unif())
illustrateLLN(Td(3), m = 50, col.Eline = "green", lwd = 2, cex = 0.6, main =
"My LLN %C%Q", sub = "generated %D")
illustrateLLN(Td(3), m = 50, CLTorCheb = "Chebyshev")
illustrateLLN(Td(3), m = 50, CLTorCheb = "Chebyshev", coverage = 0.75)
```

*Documentation reproduced from package distrEx, version 1.9, License: GPL (version 2 or later)*

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