Hi, I meet a problem on large deviation, see the document attached. 

I need to bound the probability of certain events, and I have provided a (kind of tedious and stupid) proof for it.  

The conclusion seems to be easy. I suspect that it may be just a lemma/theorem in some textbooks or papers. 

I am looking for experts on probability theory /large deviation theory. Could you give me some comments/suggestions or references?

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