I take the difference between 2 time series (each with 200,000 observations) drawn from the same ARMA(2,1) process and find that (at least the first 1000 observations of) this difference looks like White Noise (visually; for what it is worth; see top plot) but the ACF plot and Breusch-Godfrey test suggest it is not White Noise. Shouldn't I expect the difference to be White Noise? If so, where have I made the mistake(s)?
library(forecast)
library(lmtest)
par(mfrow=c(2,1))
set.seed(1000)
simA1<-arima.sim(list(ar=c(0.88,-0.4),ma=c(-0.22)),sd=0.1,n=200000)
set.seed(2000)
simA2<-arima.sim(list(ar=c(0.88,-0.4),ma=c(-0.22)),sd=0.1,n=200000)
dif<-simA1-simA2
plot(dif[1:1000],type="l")
Acf(dif,lag.max=100,ylim=c(-0.02,0.02))
lmtest::bgtest(dif~1,order=100)