AUTOCORRELATION FUNCTIONS OF WHITE NOISE AND RANDOM WALK PROCESSES
set.seed(233254)
n = 1000
r0 = 0
r = rep(0,n)
phi = 1
sig = 1

if (abs(phi)<1){
  stdev.u = sqrt(1/(1-phi^2))
}else{
  stdev.u = 0
}
a = rnorm(n,0,sig)

ts.plot(a)

r[1] = phi*r0+a[1]
for (t in 2:n)
  r[t] = phi*r[t-1]+a[t]

par(mfrow=c(2,2))
ts.plot(a)
acf(a)
ts.plot(r,ylim=range(r,2*stdev.u,-2*stdev.u))
if (abs(phi)<1){
abline(h=0+2*stdev.u,lty=3)
abline(h=0-2*stdev.u,lty=3)
}
acf(r)








# stationary AR(1) and random walks
set.seed(233254)
n = 100
r0 = 0
r = rep(0,n)
sig = 1
phi=1
a = rnorm(n,0,sig)
r[1] = phi*r0+a[1]
for (t in 2:n)
  r[t] = phi*r[t-1]+a[t]
par(mfrow=c(1,1))
ts.plot(r,ylim=c(-50,50))

for (i in 1:99){
a = rnorm(n,0,sig)
r[1] = phi*r0+a[1]
for (t in 2:n)
  r[t] = phi*r[t-1]+a[t]
lines(r,col=i)
}