Comparing Tail Probabilities of Sum of Dominated Random Variables

Question

I have been thinking about the following problem for quite a while, but did not get much progress of how to approach it:

Assume that

1. $X$, $Y$ are independent random variables with N(0, 1) distribution.

2. A and B are dependent bounded random variables: i.e., $|A|\leq a$ and $|B| \leq b$ for some fixed real number $a$ and $b$.

3. Moreover, $A$ and $B$ also have a dependence on $X$ and $Y$.

It is obvious that $ P(|aX|\geq M) \geq P(|AX|\geq M) $ since $|aX|\geq |AX|$ almost surely.

I am interested in comparing the tail probabilities of the sum as follows.

Are there any "if and only if" type of conditons that $$ P(|aX+bY|\geq M) \geq P(|AX+BY|\geq M) \ when \ M \rightarrow \infty$$

If so, we will have the tail dominance in the limit as M goes to infinity.

Many thanks!

Sean