Yesterday, we have discussed briefly sums and maximas of i.i.d. random variables using the concept of *subexponential* distributions. Today, we will introduce the concept of regular variation: a positive function is said to be regularly varying (at infinity), denoted , for some , if

*et al*. (1997)). Consider i.i.d. positive random variables : let and . Then it can be shown easily that

- if and only if

- for some if and only if the exists a non-degenerate variable such that

- with if and only if

CONVERGENCE=function(g,p=1,n=500000){ set.seed(1) X=g(n);X1=g(n);X2=g(n);X3= g(n);X4=g(n) Tp =cummax(X^p)/cumsum(X^p) Tp1=cummax(X1^p)/cumsum(X1^p) Tp2=cummax(X2^p)/cumsum(X2^p) Tp3=cummax(X3^p)/cumsum(X3^p) Tp4=cummax(X4^p)/cumsum(X4^p) plot(Tp4,type="l",ylim=c(0,1),log="x", xlim=c(100,n),ylab="",col="light blue",xlab="") lines(Tp1,col="light green") lines(Tp2,col="yellow") lines(Tp3,col="pink") lines(Tp,lwd=2) abline(h=0:1,col="red",lty=2) }

or the following to study the “asymptotic” distribution of the ratio on simulated samples

LIMITDIST=function(g,p=1,n=500000,ns=1000){ set.seed(1) T=rep(NA,ns) for(i in 1:ns){ X=g(n) T[i]=max(X^p)/sum(X^p) } hist(T,breaks=seq(0,1,by=.05),probability=TRUE, col="light green",ylab="",xlab="",main="") }

In the case of exponentially distributed variables, we have

CONVERGENCE(rexp)

For variables with a lognormal distribution,

CONVERGENCE(rlnorm)

And finally, consider the case of a Pareto distribution

rpareto=function(n){runif(n)^(-1/1.5)-1} CONVERGENCE(rpareto)

Here, it looks like those three distributions have finite variance (and actually, they do). To go one step further, for , define and . Then analogous results can be derived,

- if and only if

- for some if and only if the exists a non-degenerate variable such that

- with if and only if

CONVERGENCE(rexp,p=2)

or

CONVERGENCE(rexp,p=3)

or even

CONVERGENCE(rexp,p=10)

If the power is not too high, it looks like the ratio goes to zero. But when it becomes larger, it looks like more simulations might be necessary to say something relevant.

CONVERGENCE(rlnorm,p=2)

or

CONVERGENCE(rlnorm,p=3)

Here also, it looks like we have a light tailed distribution (and actually, it is the case). And finally, if we consider the case of a Pareto distribution

CONVERGENCE(rpareto,p=2)

Then it looks like it is an heavy tailed distribution. In order to get a better understanding, plot the distribution of the ratio obtained from 1,000 simulated samples (of size 500,000),

LIMITDIST(rpareto,p=1)

versus

LIMITDIST(rpareto,p=2)

So obviously, something is going on between 1 and 2 (recall that the power parameter of the Pareto distribution is 1.5).