Nonstandard analysis in probability theory I am quite new at nonstandard analysis, and recently I became aware of its use in probability theory mainly through the following two books: 


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*Nelson (1987). Radically Elementary Probability Theory

*Geyer (2007). Radically Elementary Probability and Statistics
Although Nelson's book is several decades old, as far as I can see, its approach has not yet caught on. Also, I couldn't find a lot of papers published in the leading probability journals on that topic. I am quite intrigued by that phenomenon. My questions are the following


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*Why hasn't nonstandard analysis been widely adopted by probabilists?

*Were there some success stories in some particular sub-fields of probability theory or statistics?

*Does there exist some known fundamental objections in probability theory to the approach in there?

 A: The answers given earlier are excellent.  I would merely like to supplement them by the observation that the success of NSA and IST in probability and related fields is attested to by the fact that new books continue to be in demand and are being published in this area, in some of the most prestigious series, such as the 2013 book by F. Herzberg entitled "Stochastic Calculus with Infinitesimals", see http://link.springer.com/book/10.1007/978-3-642-33149-7/page/1
A: Non-standard analysis has been quite successful in settling existence questions in probability theory. Hyperfinite Loeb spaces allow for several constructions that cannot be done on standard probability spaces. In particular, NSA was quite useful for the construction of certain adapted processes. There is a paper by Hoover and Keisler, Adapted Probability Distributions, from 1984, in which the authors show that many of the properties that make hyperfinite Loeb spaces so useful where due to a property they called saturation: A probability space $(\Omega,\Sigma,\mu)$ is saturated if whenever $\nu$ is a Borel probability measure on $[0,1]^2$ and $f:\omega\to[0,1]$ a random variable with distribution equal to the marginal of $\nu$ on the first coordinate, then there exists a random variable $g:\Omega\to[0,1]$ such that the distribution of $(f,g)$ is $\nu$. An example of a saturated probability space that is not a hyperfinite Loeb space is the coin-flipping measure on $\{0,1\}^\kappa$ when $\kappa$ is uncountable. A relatively readable exposition of this approach can be found in the small book Model Theory of Stochastic Processes by Fajardo and Keisler. There are also several related papers and surveys on Keisler's homepage.
In a sense, we nowadays understand fairly well how certain powerful techniques of non-standard analysis work below the surface, so we can use a lot of the constructions freed of NSA. There isn't really anything where it is necessary to use NSA. Still, NSA is a rather powerful and useful tool. A good overview over what it can do for probability theory, mainly the theory of sochastic processes, is in the article by Osswald and Sun in Nonstandard Analysis for the Working Mathematician by Loeb.
A: My colleague Ed Perkins used quite a bit of nonstandard analysis in probability theory in the early 80's.  See for example http://www.springerlink.com/content/e636h42166202387/
I don't know if he's used nonstandard analysis more recently.  In his lecture notes from the 1999 St. Flour summer school 
http://www.math.ubc.ca/~perkins/dawsonwatanabesuperprocesses.pdf
he remarked "I noticed that some of the theorems were originally derived using nonstandard analysis and I have standardized the arguments ... to make them more accessible.  This saddens me a bit as I feel that the nonstandard view, clumsy as it is at times, is pedagogically superior and allows one to come up with novel insights."
Basically I suspect that might be a summary of the general situation.  Everything 
that can be done in nonstandard analysis can be done in standard analysis, 
and thereby becomes accessible to those who don't know nonstandard analysis.  Each method
has its advantages and disadvantages, but the big advantage of standard analysis is that it is familiar to more people.
A: Probabilists are no different than the conventional analyst, at least in most places I have been, Eastern Europe (my home), and US. NSA has been seen a lot more as an alternative rather than competitor. Model theorists are more interested in studying different models than actually electing one best and propose a universal transition. The work of Nelson has been revisited. Besides the work of Geyer in your post there is a terrific recent book by Herzberg and articles by Weisshaupt (Journal of Logic and Analysis 2009 and 2011) and Andrade (Positivity, doi 10.1007/s11117-015-0333-9) and also an article by Geyer and Andrade (Journal of Logic and Analysis).
I dont agree with other post saying that Nelson's NSA is more of interest to non mathematicians. Nelson, Herzberg and Weisshaupt are mathematicians, Geyer has degree in Physics but is now statistician as is Andrade (also see work on NS Brownian motion in Physica A, 2015 Volume 429). I say interest is also high in physics and mathematical finance as illustrated by Herzberg's book as in pure maths and stats
A: True probabilists have a rather unique way of thinking. It is, if you will allow word-creation, hyper-analytic. This thought pattern seems (anecdotally!) to not be too compatible with algebraic or logical patterns. I'm not talking about basics, of course, but on a high level. I've never met a probabilist who enjoyed or personally valued the theory of modules, for example. I've never met a probabilist who would feel that the model-completeness of algebraically closed fields was super-cool. 
If you're not inclined toward such things, then the foundational advantage conferred by NSA is moot. And the intuitive advantages are already exploited without hesitation. In my experience, all probabilists think with NSA ideas by default and without self-conciousness, and without concern about how to "rigorize" the arguments.
To make my point, everyone knows that Brownian Motion is the limit of simple random walks. They don't feel the need to make this rigorous, it is just self evident. That it can be made to be almost trivial using NSA is as interesting as seeing an epsilon-delta proof of continuity. Fine for beginners, but not something for me to spend time on now.
(Edit) Disclaimer: I make no claim to have met a random sample of logicians, algebraists, probabilists, or anyone else. I was at Urbana-Champaign for a number of years, and had classes/seminars with Loeb, Henson, and Burkholder, and am married to a industrial stochastic analyst. I love NSA and find it gorgeous, and I feel the same way about probability (but not stochastic calculus, sorry). I've seen first-hand over almost 2 decades how students and professors react to NSA, but again it was not a random sample. 
We all know that essentially every mathematician has a "flavor" or two that they prefer over the others. Some of us are analysts, some algebraists; I love combinatorics, and many others don't give it much respect. All I wanted to point out was that the "flavor" of formal NSA is distinctly different from that of today's probability, while "infinitesimal" thinking is already incorporated. This combination, in my humble opinion, is why NSA has not taken hold in probability. There are of course exceptions, with Ed Perkins being the most notable but not the only.
