Find $\lim_{n \rightarrow \infty} E(X_1\mid X_1+\cdots+X_n)$

Question

Let $X_1,..,X_n$ be i.i.d integrable random variables. Find $\lim_{n \rightarrow \infty} E(X_1\mid X_1+\cdots+X_n)$.

I was thinking of using the Basu's theorem in some way. If I can claim that $\frac{X_1}{X_1+\cdots+X_n}$ is ancillary , then it is independent of $\sum_{i=1}^n X_i$. Then since the $X_i$'s are i.i.d, and we know that $\sum_{i=1}^n \frac{X_i}{\sum X_i}=1 \implies E\left(\frac{X_1}{\sum X_i}\right)=\frac{1}{n}$ which goes to $0$ as $n \rightarrow \infty$. But I don't think ancillarity makes sense here. How to approach this otherwise?

Answer 1

Because the sequence $X_1,\dots,X_n$ is i.i.d, that sequence has the same joint distribution as any permutation of the sequence. Then for $k=1,\dots,n$ the values $$ a_k := \mathbb E[X_k \mid X_1+\dots+X_n] $$ are all equal, and their sum is $$ n a_1 = \sum_{k=1}^n\mathbb E[X_k \mid X_1+\dots+X_n] = \mathbb E[X_1+\dots+X_n \mid X_1+\dots+X_n] = X_1+\dots+X_n. $$ That means $$ a_1 = \frac{1}{n}\left(X_1+\dots+X_n\right) . $$ For limit as $n \to \infty$, apply the law of large numbers.