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Im wondering if anyone could recommend a book (or a few books) about statistics/probability for someone at the advanced undergraduate level who has taken some real analysis (at the level of baby rudin) and some mathematical statistics and probability (only with calculus and intro to proofs as prerequisite). I would like to really start from the beginning and approach statistics from a rigorous (rudin-esque) sort of approach, a lot of the statistics I've encountered so far has been a lot of hand waving and lacking in rigorous proofs. Im hoping for a few books to really build a rigorous foundation and intuition, not just introduce me to the topics as quickly as possible. Thanks for any help and suggestions. Please let me know if what I'm asking for is ridiculous.

Note: I haven't taken measure theory, but I would be happy to learn it.

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Billingsley: Probability and Measure is a good choice. It covers probability measures on metric spaces very well including all you need to know about Fourier transforms. With the prerequisites you mention in your question you can nicely read it.

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I fear you will get a variety of answers with no way to evaluate the credentials of those who offer them. Why not browse course materials posted online at nearby or well-known PhD programs in statistics to see what texts they are using? That way, you can judge the degree of rigor of the course, and thus find a text that suits you.

Billingsley was my thesis adviser and I personally like his book very much, but I wonder if it is the best choice to begin the kind of self-study you have in mind.

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