Analyzing A/B test results which are not normally distributed, using independent t-test
The distribution of your data doesn't need to be normal, it's the Sampling Distribution that has to be nearly normal. If your sample size is big enough, then the sampling distribution of means from Landau Distribution should to be nearly normal, due to the Central Limit Theorem. 

So it means you should be able to safely use t-test with your data.

Example

Let's consider this example: suppose we have a population with Lognormal distribution with mu=0 and sd=0.5 (it looks a bit similar to Landau)



So we sample 30 observations 5000 times from this distribution each time calculating the mean of the sample 

And this is what we get 



Looks quite normal, doesn't it? If we increase the sample size, it's even more apparent



R code

x = seq(0, 4, 0.05)
y = dlnorm(x, mean=0, sd=0.5)
plot(x, y, type='l', bty='n')


n = 30
m = 1000

set.seed(0)
samp = rep(NA, m)

for (i in 1:m) {
  samp[i] = mean(rlnorm(n, mean=0, sd=0.5))
}

hist(samp, col='orange', probability=T, breaks=25, main='sample size = 30')
x = seq(0.5, 1.5, 0.01)
lines(x, dnorm(x, mean=mean(samp), sd=sd(samp)))


n = 300
samp = rep(NA, m)

for (i in 1:m) {
  samp[i] = mean(rlnorm(n, mean=0, sd=0.5))
}

hist(samp, col='orange', probability=T, breaks=25, main='sample size = 300')
x = seq(1, 1.25, 0.005)
lines(x, dnorm(x, mean=mean(samp), sd=sd(samp)))