DanielWarlock

I saw you have posted multiple times. The thing is we cannot quite tell you without much more information. The major factor here is that international students' application success rate depends a lot on whether their undergrad schools are well-known to US schools. For instance, IIT, ISI are well-known to US schools. US PhD programs admit students from these schools regularly so they can compare you with students they admitted from the past as well as compare you with your peers at the same year. For instance, if you are from ISI and your grades are top at your program, you can apply to top 20 schools (with some back-up schools as well). But the best answer to your question given current information is to go to your professors and ask them how past students have performed in their phd application and how they compare to your standing now. Secondly, it is best to have some idea of what kind of probability theory you want to do. For example, go look at OOPS summer school videos on youtube and see what are the cutting edge stuff in research (This year's program is still on-going!) Form some ideas of where and whom you want to apply to based on your interests. To clarify, there are typically only 2,3 people doing probability at a university so you need to be careful with what you want to study and apply accordingly.

I personally don't think advanced calculus on transcript would make a huge difference given that you have taken mathematical statistics I, II and theoretical statistics I, II. It seems to be more productive to take probability if you haven't. The graduate version should give you all the analysis background for studying statistics including basic measure theory, Lp spaces, convergence (stochastic) etc. But of course it's always good to learn more.

Seems your main motivation of getting a doctorate is to teach at college levels. Have you considered apply to graduate school of education? Some statisticians (e.g. Prof. Luke Miratrix) are actually professors of education school who does "regular" statistical research. I imagine you will be a much better candidate at education schools if that also fits your eventual goal of getting a doctorate.

@catarctica Are you taking classes with Prof. Sheldon Lin at UofT? He is a world-class actuarial, a professor in the statistics department, and had done his PhD in mathematical analysis. I can't think of a better person to resolve your conundrum. He is very friendly to undergrad/master students and knows uoft curriculum inside out (including Zhou's class and all those analysis classes) -- he used to spend an hour discussing my application with me even though he barely knows me. You should try to talk to Prof. Lin and listen to what he says. He should know as an actuarial, where, who and what you should apply to and what you should say in your letter. I'm not sure you will get good results applying to bio-stats if you have done nothing about it. Unless it is your absolute true passion, I caution you to consult Prof. Lin first. In fact, I think your best chance is to apply to departments with people who have joint interest in actuarial/finance and statistics. By the way, in Canada, it's best that you send an inquiry emails to specific profs you would like to work with before applying. You are already a grad student at uoft, why don't you work with someone first and transition to their phd? This is the easiest route to land a phd offer at top ranked university. Why risking going to a low-ranked or even unranked school in the US?

Regarding the objections, I must reiterate that it depends on (i) math maturity (ii) how that class is taught at that particular year. The difference could literally be 6 hour/week v.s. 60 hrs per week for the same class with different instructors. Same thing goes with one's math maturity. From what I know, if Zhou Zhou or Rosenthal still teaches grad probability at UofT, it might be very doable. I heard that Zhou Zhou is drier/technical but Rosenthal on the other hand should stick to his book "A first look..." and has a reputation of being lenient and less technical. Take a read and see if that works for you. There seems to be a separate offering at math department now by D. Panchenko. Never took a class from the man but his books are among the best expository materials, written in astounding lucidity. He himself is a brilliant researcher with deep work in spin glass and inequalities. I personally would cherish an opportunity to take class with a master like that. Don't be too obsessed with grades. No ideas on real analysis.

I don't think it's impossible but rather depend on OP's "math maturity" and how these classes are taught at that particular year. OP is an actuarial not a musician. Actuarial studies is a mathematics degree at Waterloo. They do already know a great deal about theoretical things like SDEs. With certain math maturity, one can certainly take these de factor self-contained classes that tend to be taught from scratch even at graduate level. Every year, a couple sophomores or even freshmen take grad probability and grad real analysis with us. Last year we even had a junior as teach fellow for graduate real analysis class. So it is very likely OP would be able to take these classes. If not, OP will always have a chance to drop out. I don't think it's a big deal.

First, you need to tell us what school you went to and what percentile you ranked there in your cohort. Cornell, CMU, Duke, UPenn are all ranked at 101~150 in math but these schools are much more highly regarded compared to Hunan University, Chongqin univeristy from China which are ranked similarly on ARWU math. Whether you school is known and most importantly whether students from your program are admitted regularly to good PhD programs in US makes a huge difference. For example, indian statistical institute (ISI), Zhejiang Univeristy sent many students top phd programs but are ranked at 100-159 and ~300 respectively on the ARWU ranking. This means that if you are a top student from ISI or Zhejiang university, your application result would be drastically better than say if you are an average student from Chongqin university. I would recommend you to apply for top 50 or top 30 universities if you fall into the former category (top student from known international schools) but if you are from somewhere unknown and placed mediocre in your cohort, I would recommend you be cautious and emphasize on 50+ range. The main problem is that you appear to have little background in statistics. In theory, this is fine, but you will be at a disadvantage when compared to other candidates, many of whom have worked with famous statisticians and acquired in-depth knowledge of a subfield. Basically, it is difficult for you to write your SOP without actual experience working with some statistics-related projects. Say, why do you want to apply for statistics and what kind of statistics you want to do and what you know or have done about it? You could probably do some projects on statistics-related topics before you apply--even short, simple ones that take a few weeks will drastically improve your profile. Otherwise, if you like to work with probabilists, make sure to say that in your SOP and do a little bit of research of the profs you may potentially work with. That said, you appear to have strong math background--some schools do dig that. I don't know what medals you have won but if it is something like Putnam, then it can help your profile as well. I also recommend you to take GRE math subject test. You are a math major so this should not require too much preparation for you. Getting a 90%+ rank can drastically improve your chances.

I doubt that one course in real analysis will change things drastically. I had overlapping courses with actuarial students at UofT including the "Elements of analysis": MATH 336 H1. This is the real analysis class for actuarial students at uoft and I'm worried you may take it. Don't! Take MATH 357H1 instead. I also had complex variable (334) btw. Got 100 in both of these classes --no help to my application at all. The truth is that most people in those classes are definitely not math-savvy and have no clue so the instructor has to go extremely slow and review calculus stuff all of the time. The most advanced thing we learned was just calculus materials like sequence convergence, series, mean value theorem and we don't learn those very well. In fact, a lot of classes at uoft is made easy and "useless" for you future academic careers as a PhD in stats, except for those for pure math specialist: e.g. MATH 357H1, Math 347H1. MATH 336 H1 don't even teach standard real analysis material like Azera-Ascoli, Weierstrass but 357 does. I thought admission at other schools don't know the difference but I was wrong. I was immediately questioned for taking "computation based" math classes. Someone even said I would be better off taking hard, proof-based classes with a less perfect score. Absolute truth. I got 95% on my linear algebra class (designed for engineers)--I didn't even understand eigenvalues beyond the definition. So you see how those marks you got on your transcript are questionable . Also a definite way for us non-math majors is to take GRE math subject tests. I can tell you that it will definitely help boost your profile if you score anywhere above 90%. A hard task but getting high mark is not the only objective. I took it twice with one year span in between. Did poorly both times (74% and 79%) so didn't end up submitting it. But I don't regret studying for it one bit as it really prepares you for grad school if you are not solid in calc and linear algebra. Similar to you, I worked in risk management and most my work consisted of excel and writing simple programs. Taking GRE math really taught me calculus and linear algebra before grad school. I self-studied from classic books like linear algebra done right, baby Rudin, Dummit and Forte, Munkres etc. Of course, a "crash education" in math like this is not comparable to a 4-year, solid math education but it's absolutely helpful for my grad school and allowed me to read some theoretical papers.

Completely agree with this assessment. I went to U of T for my undergrad and also had background from finance. I did my masters at Harvard with full A's in standard phd sequence (math stats, probability). My GPA is much better (near 4.0), with strong letters. Still I was rejected at schools at the rank of ~50, e.g. University of Florida as well as mid-ranged schools such as UWM. A major flaw is my math background which is still stronger than yours. The point is mid to low ranked schools care A lOT about math abilities such as real analysis but I don't have it. You are definitely NOT safe at ~50 rank level. And I would say UT Austin, Penn state, UWM level school is the "pipe dream"/"top choice" level.

I would recommend just taking same courses with PhD students at Stanford (probability, theory stats, and applied stats) and focus on doing research. If you could do PhD sequence, there is no concern about whether or not if you have real analysis. And this saves you tons of time when you become a phd because you have already done these classes. I think you failed last time because you have no relevant research experience in statistics--that *in principle* is fine but you are competing against people who have done cutting-edge stuff in stats for 2+ years in their undergrad and have accumulated quite a bit of expertise. At Harvard at least, most people come in with prior concentration on one of the subfields, say experimental design/causal inference, Bayesian computation, network data, differential privacy/minimax bound etc. It helps a lot when you have substantial letters from experts who work in your chosen subfield. This doesn't mean they published in top journals--just that they demonstrate interest and research experience in these things. If I were you, I would find say Montanari (who is also a physicist turned statistician) and start doing research on AMP/deep learning/spin glass right away. I could say with very high confidence that you can get admission from the departments that have interests in this area, e.g. Yale, Chicago, Stanford, Berkeley, Columbia, MIT (math). The reason? Not a lot of undergrads have worked in this area or even know anything about it. A physics-major from Stanford stats with a letter from Montanari/Chatterjee would look very,very good to top departments. This is just one example. Usual high dimensional stats is probably a safer choice and a letter from Candes probably will make competitive anywhere. But I would say it's a lesser strategy because there will be a lot of competition from peers with similar interests. I stress that you don't need to publish in top journals or have solved some open problems. Just exposure + experience is fine. Statistics, and probably academia in general, cares a lot about pedigree even at undergrad level. *Substantial* letter from famous scholars is the most effective way to stand out. No need to obsess with real analysis or courses in general. I think it has very little impact when your goal is top 10 departments. Usually lower-ranked departments care more about it so that they can make sure the admits can handle first-year class.

Given your interest, I think Harvard is best fit. Imai has affiliation at Kennedy school, Neil Shephard is affiliated at economics. Murphy is also a big name here doing causal inference and reinforcement learning affiliated to CS department. There is no problem that you seek additional advisors at MIT or other Harvard departments. You can easily find someone at MIT to supplement for (3) (4). Everyone is saying Stanford stats but they are mainly about highly mathematical/theoretical high-dimensional stats and probability theory. So I guess you will need to go to their CS department to find advisors? Stanford probably is not that of a good fit for you.

Both schools are focused around the theme of high-dimensional stats. But risking oversimplification, a quick summary of their difference is: CMU is more "CS"; UChicago is more "mathematical". If you consider yourself more of a mathematician/probabilist, go to Chicago. If you consider yourself a computer scientist who looks at more applied stuff, then go to CMU. I will now further explain what I mean. CMU focuses more heavily on more applied, interdisciplinary stuff like neurosciences, astrostatistics, social sciences and yes sports analytics. Of course, most of these are done under the tag of "high-dimensional statistics". But I would even go so far to say CMU stats has more of a "CS flavour" if you know what I mean. On a related note, CMU is also much stronger on causal inference, which is also more "CS". I also feel the organization is very similar to what I see at EECS at MIT: they have all these themed working groups like "astrostats group", "causal inference group". So the community based activities like colloquium/talks, reading groups will be more specifically tailored to your subfield. Chicago is more theoretical and will probably be more so in the coming years based on their new hires. Maybe "theoretical" is not a good descriptor. What I mean is that their new hires now mostly have tags such as "physics", "statistical mechanics", "random matrices", "Fourier/harmonic analysis", "combinatorics", "random graphs". Chicago is definitely more mathematical and has a taste of more probabilistic things. They even hired student of Borodin who does hard-core math.

Again, if you are talking about statistics, then I would say they are both on the same level. But the reputation of UChicago is better if you talk to someone who is not a statistician

Very interesting story. Maybe he is not dead-set on getting a PhD? Yale stats is actually very good. With guys like Harrison Zhou, Zhou Fan and Van Vu, I can even see one choosing Yale over Stanford when admitted to both. Not to mention the tremendous cost of money and time and the uncertainty of actually getting into Stanford (or even Yale itself) 2 years later. Sounds like a terrible decision.

I personally would definitely go to UChicago. It is the most reputable and give you a leg up finding jobs in not only data analysts/statistician roles but also in finance, consulting, SDE etc whereas Duke, UNC, UW Maddison are strong in statistics but not as strong in other things (like finance). That said, if you would like to do a PhD, ETH Zurich may also be a good option because European masters, as far as I know, is research-based and you could very likely transfer into their very excellent PhD program and finish much faster.