Suppose $X\sim \text{Bin}(2n,p)$ and $X_1,X_2\sim\text{Bin}(n,p)$ are independent, with $X_1+X_2=X$. I'm interested in the rate of convergence for the absolute difference $$ \left\vert P(X>c|X_1\leq a,X_2\leq a)-P(X>c)\right\vert $$ where $a$ and $c$ are constants in a neighborhood of $np$ and $2np$, respectively. Specifically, $a\in[np,np+o(n^{2/3})]$, and $c\in[2np,2a)$. These ranges will help control the rate of convergence. Assume that the conditioning set has probability 1 asymptotically.

The reason I'm interested in bounding the quantity above is because I need to approximate the conditional probability with the normal tail probability (e.g. by Berry-Esseen bound).

Question Are there any references for this result? I've tried writing out the probabilities explicitly, but can't seem to get anywhere with the calculations.

Edit By some elementary probability, I can show that on one hand $$ P(X>c|X_1\leq a,X_2\leq a)=\frac{P(X>c,X_1\leq a,X_2\leq a)}{P(X_1\leq a,X_2\leq a)}\leq \frac{P(X>c)}{P(X_1\leq a,X_2\leq a)}\leq P(X>c)(1+o(1)) $$ where $o(1)$ is the rate as $n\to\infty$. On the other hand, \begin{align*} P(X>c\mid X_1\leq a,X_2\leq a)&=\frac{P(X>c,X_1\leq a,X_2\leq a)}{P(X_1\leq a,X_2\leq a)}\\ &= \frac{P(X>c)}{P(X_1\leq a,X_2\leq a)}- \frac{P(X>c,\{X_1> a\}\cup\{X_2> a\})}{P(X_1\leq a,X_2\leq a)}\\ &\geq P(X>c)- \frac{P(X>c,\{X_1> a\}\cup\{X_2> a\})}{P(X_1\leq a,X_2\leq a)}\\ &\geq P(X>c)(1-o(1)) \end{align*} where I claim that $P(X>c,\{X_1> a\}\cup\{X_2> a\})/P(X_1\leq a,X_2\leq a)=o(P(X>c))$.

This suggests the stronger result that $$ \lim_{n\to\infty}\frac{P(X>c\mid X_1\leq a,X_2\leq a)}{P(X>c)}=1 $$

Is the work above correct? Have I missed something or been careless about the convergence rates $1\pm o(1)$?

  • $\begingroup$ Your calculation cannot be correct because you don't use that $c<2a$. And it seems natural that $\mathbb{P}(X>c|X_1\leq a,X_2\leq a)\ll \mathbb{P}(X>c) $ if $c-2a$ is too small. Therefore I think you should give a more precise hypothesis on $a$ and $c$. $\endgroup$ – RaphaelB4 Jan 21 at 12:29
  • $\begingroup$ @RaphaelB4 I don't think that $c<2a$ is needed to make the calculations go through. Further, I think that this will work for any $c<2a$. The event $\{X_1\leq a,X_2\leq a\}$ happens with high probability as $n\to\infty$, so that I think even if $c-2a$ is small, the conditional and marginal should be asymptotically the same. $\endgroup$ – stats134711 Jan 21 at 16:15

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