I have a question in record statistics. The probability of having n record in time interval [0,t] is: $P_n (0,t) = \frac{1}{t}\frac{(\log t)^{n-1}}{(n-1)!}$
As you can see is just a Poisson distribution with parameter $\log t$ . I want to demonstrate that if we take $n\gg1$ the probability of n records in a time window $[t_1,t_2]$ is: $P_n (t_1,t_2) = \frac{t_1}{t_2}\frac{(\log \frac{t_2}{t_1})^{n}}{n!}$ . To do so I though about using the difference of poisson processes. Doing so lead to the Skellam distribution, that I cannot menage to simplify enough even using the Stirling approximation (since $n\gg1$).
Can someone help me out with this? Thanks