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I have a Masters in math and I am currently doing a Masters in statistics. I am disappointed at the level of rigor in the proofs (and often outright omission of the proofs just a statement saying "in advanced analysis classes it can be shown that...") in the text we are using (Hogg and Craig, "Intro to Math Stats"). Looking at other books, (like Casella and Berger), I find their treatment somewhat better but still lacking.

Does anyone know of a good (non-measure theoretic) book on probability theory that gives rigorous proofs of analysis based facts?

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    $\begingroup$ Neither Hogg & Craig nor Casella & Berger are texts on probability theory. Are you asking for texts on mathematical statistics from a rigorous (measure-theoretic) viewpoint? If you don't want any skimping on the proofs, ultimately you'll have to face up to some measure-theoretic ideas. $\endgroup$ – cardinal Sep 23 '14 at 15:09
  • $\begingroup$ I am looking for a textbook on mathematical statistics that provides rigorous proofs of results from advanced calculus/real analysis that are not measure theoretic in natures. There a number of elementary results involving variable transformations, mgfs (Laplace transformations) etc that do not involve measure theory. $\endgroup$ – Matt Brenneman Sep 23 '14 at 15:29
  • $\begingroup$ @Cardinal ... unless you want to dip into Edward Nelson's Radically Elementary Probability Theory. By using nonstandard analysis, he only needs to deal with finite probability spaces. (All his disclaimers aside, though, that book requires even more mathematical sophistication than a rigorous measure theory text would, IMHO.) $\endgroup$ – whuber Feb 3 '17 at 14:58
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When I first studied probability and statistics many decades ago, a probability course based on William Feller's "An Introduction to Probability Theory and Its Applications, Volume 1" was prerequisite to the statistics course that used Hogg and Craig. With Volume 1's restriction to discrete sample spaces, Feller achieved considerable rigor without needing measure theory. Even in Volume 2, his use of measure theory was generally approachable. Both Volumes are still available, over 40 years since the author's death.

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  • $\begingroup$ Very good idea, but IIRC, I thought Feller limited his treatment to discrete RVs $\endgroup$ – Matt Brenneman Sep 23 '14 at 15:30
  • $\begingroup$ The limitation to discrete RVs is in volume 1; Volume 2 covers continuous RVs, with a manageable, self-contained, practical use of measure theory. $\endgroup$ – EdM Sep 25 '14 at 17:00
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Although perhaps its focus is elsewhere, maybe you should check out

Corbae, D., Stinchcombe, M. B., & Zeman, J. (2009). An introduction to mathematical analysis for economic theory and econometrics. Princeton University Press.

The authors write

The three major innovations in this book relative to mathematics textbooks are: (i) we have gathered material from very different areas in mathematics, from lattices and convex analysis to measure theory and functional analysis, because they are useful for economists doing regression analysis, working on both static and dynamic choice problems, analyzing both strategic and competitive equilibria; (ii) we try to use concepts familiar to economists - approximation and existence of a solution - to understand analysis and measure theory; and (iii) pedagogically, we provide extensive simple examples drawn from economic theory and econometrics to provide intuition necessary for grasping difficult ideas. It is important to emphasize that while we aim to make this material as accessible as possible, we have not excluded demanding mathematical concepts used by economists and that, aside from examples assuming an undergraduate background in economics, the book is self-contained (i.e. almost any theorem used in proving a given result is itself proved).

Chapter titles:
1. Logic
2 Set Theory
3 The Space of Real Numbers
4 The Metric Spaces $\mathbb R^{\ell}, \ell=1,2,...$
5 Convex Analysis in $\mathbb R^{\ell}$
6 Metric Spaces
7 Measure Spaces and Probability
8 The $L^p(\Omega, \mathcal F, P)$ and $\ell^p$ spaces, $p\in [1,\infty]$
9 Probabilities on Metric Spaces
10 Convex Analysis in Vector Spaces
11 Expanded Spaces

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How about grimmett & stirzaker: probability & random processes

New sections on Markov chain Monte Carlo, coupling and its applications, geometrical probability, spatial Poisson processes, Stochastic calculus and the Itô integral, Itô's formula and applications (including the Black-Scholes formula), networks of queues, and renewal-reward theorems and applications.

A separate volume including worked solutions to the problems and exercises will be available.

Minimal prerequisites (basic algebra and calculus)

The third edition of this successful text gives a rigorous introduction to probability theory and the discussion of the most important random processes in some depth. It includes various topics which are suitable for undergraduate courses, but are not routinely taught. It is suitable to the beginner, and provides a taste and encouragement for more advanced work. There are four main aims: 1) to provide a thorough but straightforward account of basic probability, giving the reader a natural feel for the subject unburdened by oppressive technicalities, 2) to discuss important random processes in depth with many examples. 3) to cover a range of important but less routine topics, 4) to impart to the beginner the flavour of more advanced work. The books begins with basic ideas common to many undergraduate courses in mathematics, statistics and the sciences; in concludes with topics usually found at graduate level. Highlights include new sections on sampling and Markov chain Monte Carlo, geometric probability, coupling and Poisson approximation, large deviations, spatial Poisson processes, renewal-reward, queueing networks, stochastic calculus, Itô's formula and option pricing in the Black-Scholes model for financial markets.

  1. Events and their probabilities
  2. Random variables and their distribution
  3. Discrete random variables
  4. Continuous random variables
  5. Generating functions and their applications
  6. Markov chains
  7. Convergence of random variables
  8. Random processes
  9. Stationary processes
  10. Renewals
  11. Queues
  12. Martingales
  13. Diffusion processes
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