§ Markov and chebyshev from a measure theoretic lens
I've been idly watching Probability and Stochastics for finance: NPTEL, and I came across this nice way to
think about the markov and chebyshev inequality. I wonder whether Chernoff
bounds also fall to this viewpoint.
§ Markov's inequality
In markov's inequality, we want to bound . Since we're in measure land,
we have no way to directly access . The best we can do is to integreate
the constant function , since the probability is "hidden inside" the measure.
This makes us compute:
Hm, how to proceed? We can only attempt to replace the with the to get
some non-trivial bound on . But we know that . so we should perhaps
first introduce the :
Now we are naturally led to see that this is always less than :
This completes marov's inequality:
So we are "smearing" the indicator over the domain and attempting
to get a bound.