# Mathematical knowledge needed for learning upper level statistics [duplicate]

im a math student who wants to study statistics in depth for a master degree(maybe doctor degree). And I was wondering what kind of math knowledge might be needed for upper level statistics. I wanna learn statistics in a mathematically rigorous way and understand all the maths behind statistics knowledge.

I have searched relevant questions on a couple platforms and the answers are linear algebra, calculus, and analysis. But what analysis exactly do I need to study? I can think of real analysis, functional analysis, measure theory and maybe a bit of topology?(not very sure if these are actually needed) Are these enough? Is there anything else I need to study in depth or is there anything I don't need to spend too much time on?

Also any math books recommended for learning statistics in a theoretical and mathematical way? I have a couple books in hand such as mathematical methods of statistics by Cramer, but its simply not for me yet. I found it a bit challenging to read(since im still doing my undergrad degree in math) and that's why im asking for some math books to read first and then get back to this book for further study.

• This sounds like the question answered at stats.stackexchange.com/questions/118112. Closely related are stats.stackexchange.com/questions/402333 and stats.stackexchange.com/questions/463248.
– whuber
Sep 21, 2022 at 21:58
• Something worth remembering is that, while many people will enter statistics master’s and doctoral programs with bachelor’s degrees, probably not many will already have a PhD in mathematics. Statistics programs seem to train their PhD students just fine without making them earn math PhDs first.
– Dave
Sep 21, 2022 at 22:00
• Unless you really pivot towards something like topological data analysis, I don't think a lot of topology is required. Some results require compactness, but I wouldn't recommend trying to master everything related to topology (e.g. Poincare-Hopf theorem) just because topology is occasionally used in statistics. Unless you specialize to something using topology, of course. Sep 21, 2022 at 22:36
• @Galen Thank you so much for your advice! Sep 21, 2022 at 23:08
• @Dave Thank you Dave Sep 21, 2022 at 23:09