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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 , iffor all . An this concept can be related to sums and maxima (see section 6.2.6 in Embrechts 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 ifIf is not that simple to check for such convergences, it is still possible to use graphs to study the behavior of the empirical version of those quantities. Consider the following function to visualize convergence of empirical ratios,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 samplesLIMITDIST=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 haveCONVERGENCE(rexp)For variables with a lognormal distribution,CONVERGENCE(rlnorm)And finally, consider the case of a Pareto distributionrpareto=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 ifAgain, it is possible to use the function defined above,CONVERGENCE(rexp,p=2)orCONVERGENCE(rexp,p=3)or evenCONVERGENCE(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)orCONVERGENCE(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 distributionCONVERGENCE(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)versusLIMITDIST(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).