= Poisson count-arrival joint central limit theorem
{c}
{title2=$\rho=-\min\{\sqrt\lambda,1/\sqrt\lambda\}$}
For a rate-one <Poisson process> with <Poisson arrival times> $T_n$, suppose $t_n\to\infty$ and $t_n/n\to\lambda\in[0,\infty]$. Then
$$
\left(\frac{N_{t_n}-t_n}{\sqrt{t_n}},\frac{T_n-n}{\sqrt n}\right)\xrightarrow{\mathrm d}\operatorname{BN}(\rho),
\qquad
\rho=\begin{cases}-\sqrt\lambda,&0\leq\lambda\leq1,\\-1/\sqrt\lambda,&1\leq\lambda\leq\infty,\end{cases}
$$
where $1/\sqrt\infty=0$ and $\operatorname{BN}(\rho)$ denotes the <bivariate normal distribution> with standard <normal> <marginal distributions>, including the singular cases. In particular, equal scales give $\rho=-1$ and separated scales give $\rho=0$.
To prove this, fix real $x,y$ and put $s_n=n+y\sqrt n>0$ for large $n$. The event $(T_n-n)/\sqrt n\leq y$ is exactly $N_{s_n}\geq n$. Counts at the two deterministic times $t_n,s_n$ are sums of a shared count on $[0,\min(t_n,s_n)]$ and an independent count on the remaining interval. The <central limit theorem> for a centered <Poisson random variable> of diverging parameter follows by writing a count at a real time $u$ as its count at $\lfloor u\rfloor$, a sum of independent unit increments, plus an independent remainder with <expected value> at most one. <Markov's inequality> makes that remainder negligible after division by $\sqrt u$.
Apply this <central limit theorem> to the two independent count blocks whenever their lengths diverge. A block with bounded length, or length negligible compared with the scale of its coordinate, contributes a term tending to zero in <probability>. The standardized pair of counts therefore has a joint <normal> limit whose <correlation coefficient> is
$$
\lim_{n\to\infty}\frac{\min(t_n,s_n)}{\sqrt{t_ns_n}}
=\begin{cases}\sqrt\lambda,&0\leq\lambda\leq1,\\1/\sqrt\lambda,&1\leq\lambda\leq\infty.\end{cases}
$$
This includes $\lambda=1$: the difference block then has negligible <variance> after normalization. In the arrival event the standardized count at $s_n$ must be at least $(n-s_n)/\sqrt{s_n}\to-y$. Reversing this inequality changes the sign of the second limiting <normal random variable>, giving the asserted negative <correlation coefficient>. Continuity of the <marginal distributions> permits the moving thresholds and the weak inequalities, including at the singular limits.
Every $\rho\in[-1,0)$ is realized by $t_n=\lambda n$ with $\lambda=\rho^{-2}\geq1$, and zero by $t_n=n^4$. Conversely, even when $t_n/n$ has no limit, it has a <subsequence> converging in the extended <interval> $[0,\infty]$. If the original pair has a joint <bivariate normal distribution> limit, that <subsequence> has the same limit, forcing its <correlation coefficient> into $[-1,0]$.
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