inequalities in probability: convergence w/ rates Pr(A−a>ε1)≤ekn and Pr(B−b>ε2)≤esn

[Markov]

inequalities in probability: convergence w/ rates Pr(A−a>ε1)≤ekn and Pr(B−b>ε2)≤esn

I have two independent random variables A and B that converges to a and b with rate:[FONT=MathJax_Math-italic]P[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Math-italic]A[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]a[FONT=MathJax_Main]>[/FONT][FONT=MathJax_Math-italic]ε[FONT=MathJax_Main]1[/FONT][/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]k[/FONT][/FONT][FONT=MathJax_Math-italic]n[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT] and [FONT=MathJax_Math-italic]P[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Math-italic]B[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]b[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Math-italic]ε[FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]s[/FONT][/FONT][FONT=MathJax_Math-italic]n[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT] where k and s are some constants.[/FONT]
Now I need to find [FONT=MathJax_Math-italic]P[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Main]([FONT=MathJax_Math-italic]A[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]B[/FONT][FONT=MathJax_Main])[FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]b[/FONT][FONT=MathJax_Main])[FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Math-italic]ε[FONT=MathJax_Main]3[/FONT][/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]≤?[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]

 

[Deleted member 4993]

I have two independent random variables A and B that converges to a and b with rate:[FONT=MathJax_Math-italic]P[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Math-italic]A[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]a[FONT=MathJax_Main]>[/FONT][FONT=MathJax_Math-italic]ε[FONT=MathJax_Main]1[/FONT][/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]k[/FONT][/FONT][FONT=MathJax_Math-italic]n[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT] and [FONT=MathJax_Math-italic]P[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Math-italic]B[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]b[/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Math-italic]ε[FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]≤[/FONT][FONT=MathJax_Math-italic]e[FONT=MathJax_Math-italic]s[/FONT][/FONT][FONT=MathJax_Math-italic]n[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT] where k and s are some constants.[/FONT]
Now I need to find [FONT=MathJax_Math-italic]P[FONT=MathJax_Math-italic]r[FONT=MathJax_Main]([FONT=MathJax_Main]([FONT=MathJax_Math-italic]A[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]B[/FONT][FONT=MathJax_Main])[FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Main]([/FONT][FONT=MathJax_Math-italic]a[/FONT][FONT=MathJax_Main]−[/FONT][FONT=MathJax_Math-italic]b[/FONT][FONT=MathJax_Main])[FONT=MathJax_Main]2[/FONT][/FONT][FONT=MathJax_Main]>[/FONT][FONT=MathJax_Math-italic]ε[FONT=MathJax_Main]3[/FONT][/FONT][FONT=MathJax_Main])[/FONT][FONT=MathJax_Main]≤?[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]

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[Markov]

I tried first to prove it with series and not probabilities, i.e:
I know a series A converges to a
and a series B converges to b
and I proved that

[FONT=MathJax_Main]([FONT=MathJax_Math-italic]A[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]B[FONT=MathJax_Main])[FONT=MathJax_Main]2[/FONT][/FONT][/FONT][/FONT][/FONT] converges to [FONT=MathJax_Main]([FONT=MathJax_Math-italic]a[FONT=MathJax_Main]−[FONT=MathJax_Math-italic]b[FONT=MathJax_Main])[FONT=MathJax_Main]2[/FONT][/FONT][/FONT][/FONT][/FONT][/FONT]
[/FONT]
I attached the proof.
I don't know if it's a good way to start with, and if it is, I don't know to make the transition between the calculus and the probabilities.