1
$\begingroup$

The question is simple to state:

Let $S_n = \sum_{i=1}^n X_i$, where $X_i$ are independent random variables with a common distribution $F$. For simplicity, assume that $X_i \geq 0$ with prob. 1, and that $F$ is absolutely continuous. (Think of $S_n$ as a random walk).

For a fixed $N$, what can we say about the distribution of $X_1$ (or $X_i, i = 1, 2, \cdots$ in general) given an observed value of $S_N$?

In words, if we observe the location of the random walk $S$ after $N$ steps, what does this tell us about the conditional distribution of the summands $X_1, \cdots, X_N$?

I am aware of the well known connection between partial sums of Exponential random variables and Uniform order statistics.

I'm wondering what is known in general.

Improve this question
$\endgroup$
6
  • $\begingroup$ In the case that $F$ is a normal distribution, $X_1$ subject to sum $S$ is normal with mean $S/n$ and variance a bit less than the variance of $F$. Presumably other nice distributions also have nice answers. In general, it is not even true that the variance of the components decreases when conditioning on the sum. $\endgroup$
    – Brendan McKay
    Commented Jul 29, 2022 at 5:20
  • 2
    $\begingroup$ Suppose $X_n$ is bounded above by $x$. Then the conditional probability of $X_n\leq x$ is 1, but $P(X_1+\cdots+X_{n-1}\geq S-x)$ can be anything. $\endgroup$
    – Brendan McKay
    Commented Jul 29, 2022 at 13:57
  • $\begingroup$ @VarunVejalla I couldn't save my previous comment with a ping to you, trying again. $\endgroup$
    – Brendan McKay
    Commented Jul 29, 2022 at 13:58
  • $\begingroup$ @BrendanMcKay You're right. I've deleted my earlier comment. $\endgroup$
    – Varun Vejalla
    Commented Jul 29, 2022 at 16:43
  • $\begingroup$ For some conditional limit theorems see cambridge.org/core/journals/journal-of-applied-probability/… $\endgroup$
    – esg
    Commented Jul 31, 2022 at 19:13

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .