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I’m looking for a kind of book which gives the elements of Real Analysis for Mathematical Statistics and Econometric Theory, etc.

I’m planning to work with the 3 following textbooks on advanced statistical theory:

  • Probability Essentials by Jacod and Protter
  • Theoretical Statistics: Topics for a Core Course by Keener Robert
  • Mathematical Statistics by Jun Shao

But they are quite heavy on Real Analysis although they all do give an introduction in meausure-theoric probability, so I just lack Real Analysis to begin. I have a background in Economics (Licence 3 in European LMD system), so even if I have done some Calculus, Linear Algebra, Elementary probability and statistics, But being an economics major, I have by no means a background in « proof-based » maths, but I do want one, especially in statistical theory.

I have done some research, and find those books:

  • An Introduction to Mathematical Analysis for Economic Theory and Econometrics by Corbae, Stinchcombe and Zeman.
  • Real Analysis with Economic applications by A. Efe
  • Advanced Calculus for Economics and Finance: Theory and Methods by Giulio Bottazzi

But I think that would be better to ask some authorities to be sure and don’t waste time for something too heavy for me or not complete for Jun Shao’s type of book.

Do some of you have some good references for self-study real analysis, adpated for economist, engineers, etc?

Thank you and sorry for my terrible english.

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  • $\begingroup$ I think it would help if you explain why you have decided to read eg Shao's book as a non mathematician. Maybe another book is more suitable for your goals? eg googling "statistics for economists" gives press.princeton.edu/books/hardcover/9780691235943/… - which is aimed at phd students in econometrics (and uses essentially no measure theory) $\endgroup$
    – seanv507
    Commented May 25 at 15:03
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    $\begingroup$ Seán Dineen has a nice little book called Probability Theory in Finance that builds up the necessary probability theory to derive the Black-Scholes formula starting with no prior knowledge of analysis. Thus, he builds up the basic concepts of real analysis, measure theory, and probability theory necessary for this task -- quite a feat, especially in a little book. What you learn in this book can take you a long way, though it's not a thorough intro to real analysis, measure theory, or probability. $\endgroup$
    – zxmkn
    Commented May 25 at 17:02
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    $\begingroup$ One thing that is probably missing from the sources that you are considering is the extrapolation from statistical inference to inference about the real world. There are many pitfalls and dangers in that gap. You might find a useful primer within this chapter: link.springer.com/chapter/10.1007/164_2019_286 $\endgroup$ Commented May 25 at 22:01
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    $\begingroup$ I'm not sure what you mean by "I don’t think that I’m a « proof-based » maths guy." If you have no interest in mathematical rigor (i.e. proofs), then there's no need to worry about real analysis and measure theory. But perhaps you just mean that it's not your goal to do original research in mathematics/mathematical statistics. $\endgroup$
    – zxmkn
    Commented May 26 at 13:17
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    $\begingroup$ @zxmkn I think you did not get me. I don’t have a training and a background in proof-based maths (I’m an economics major) but I want to get that training, especially for statistics. I want to do Statistics with solid foundations. I have now correct the first post. $\endgroup$ Commented May 26 at 13:40

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To study books like that of Shao's, Keener's or some other books of similar breed, you need to have a robust concept of not only real analysis but measure theory, general topology and functional analysis too.

My personal favorites are (as evident in many of my answers):

  • Real Analysis: Theory of Measure and Integration by J. Yeh, $2011$.

This book builds up the measure theory maintaining a digestible rigor and goes through all the standard topics with exceptional details, that can sometimes lead to proof two-three pages long. While it also has a chapter on locally compact spaces and their integration, unfortunately it doesn't delve too far. Convolution on $L^p$ spaces has been dealt with beautifully. Then there is a chapter on Hausdorff measure and other topics. The problems are quite standard and have been solved by the author (in an another book). I can assure you this is an ideal piece for self-study.

  • Real Analysis and Probability by Robert Ash, $1972$.

While most of the standard topics in measure theory have been included, there is also comprehensive treatment of topological vector spaces and weak $^*$ convergence, Daniell integration (something that Yeh didn't cover and in fact, the last editions excluded this part). The author then concentrates on probability. Best is the section of conditional expectation and regular conditional probability, the parallels of which I haven't seen anywhere. Again the problems, while not too many, have been solved by the author. (There is additional topic on stochastic integrals in later editon too, but I haven't read that).

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  • $\begingroup$ but should OP be studying Shao's book at all- if they are not a mathematician/proof based $\endgroup$
    – seanv507
    Commented May 25 at 14:37
  • $\begingroup$ @seanv507, that could be a valid argument. Shao's book is bit terse but approachable once you have the necessary prerequisites. I am not going ro discourage anyone to read Shao, but they must bear in mind the tools needed to go through it. $\endgroup$ Commented May 25 at 14:41
  • $\begingroup$ I have read one French statistician told that the chapter 2 of Jun Shao’s book, is the best chapter ever written in statistical theory, if we are looking for a deep understanding of the field. $\endgroup$ Commented May 25 at 15:00
  • $\begingroup$ First of all, thank you. But do you guys, don’t have another reference on real analysis, which gives just what is necessary to go into Measure-theoric probability at the level of Jacod-Protter and Keener's book? As far as I know, the rigorous probability texts just take from measure theory what is necessary — maybe there exists a book like that for the relationship between real analysis and measure-theoric probability? It looks like the book of Yeh is very complete — but maybe too? I’ll however, look at his book. $\endgroup$ Commented May 25 at 15:02
  • $\begingroup$ @Hiba__Nouhoum__Djeneba, Ash would provide enough material for you that would be needed in measure theoretic probability theory. You can use the combination of two books above without any hassle as these are few of the authors who don't leave details for exercise or condense things claiming "it's obvious" (things that Shao and Keener often resort to :-)). Measure theory itself is a very broad and general topic and Yeh covers only a part of that, tbh, but in a sufficiently comprehensive way for you to transition to other advanced sources subsequently, if needed. $\endgroup$ Commented May 26 at 6:37
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Jun Shao's book and Keener's book are both proof based, and would require background in analysis. My favorite text for basic analysis is "Introduction to Analysis in One Variable" by M. Taylor. It is available at https://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/anal1v.pdf. This book contains the minimum of background that I would recommend before reading a measure based probability theory book.

As a side note, my favorite text for probability theory is "Probability: Theory and Examples" by R. Durrett, available at https://services.math.duke.edu/~rtd/PTE/PTE5_011119.pdf. This book is at a similar level of Jacod and Protter's book and Jun Shao's book, and is what I based my analysis book recommendation on.

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