Statistics of random processes

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1. graycatSep 2019
   To get very far in stochastic processes, usually need a good background in measure theory. Here is an overview of about two semesters in a fast, often rare, grad course plus much more:

   The integral of freshman calculus is the Riemann integral. Too soon/easily it was seen to be clumsy to inadequate in theoretical developments. Near 1900 in France, E. Borel student H. Lebesgue improved the situation; he did his Lebesgue integral. Don't worry: For every function integrated in freshman calculus, Lebesgue's integral gives the same answer. Otherwise Lebesgue's integral is much more general and, it turns out, is the solid foundation of probability and stochastic processes.

   Lebesgue's work starts with measure which is just a notion of area. So, yes, on the real line, the Lebesgue measure of the closed interval [0,1] is 1, just the same as we learned in grade school. But with Lebesgue's work, it's tough to think of a subset of the real line that is not measurable (has a measure or area in Lebesgue's work) -- the usual example of a subset of R that is not measurable needs the axiom of choice. With measure defined on R, Lebesgue can define the Lebesgue integral of functions

   f: R --> R

   For this we need a meager assumption (which essentially always holds), but lets charge on:

   For the generalizations, crucial for probability and stochastic processes, start with a non-empty set M. R is an example of a such a set M. Define a measure on M, say, m. Then for some subsets A of M, we have the measure of set A as m(A). That is, we have the area (volume, whatever) of A. But to do much, we need more assumptions: We want the set of all subsets A of M to which we assign measure m(A) to be a sigma algebra, say, F. As at

   https://en.wikipedia.org/wiki/Sigma-algebra

   we want M to be an element of F, for each A in F we want the complement, say, the set M - A (obvious but sloppy notation) to be an element of F, and for A_i in F for i = 1, 2, ..., we want the union of the A_i to be an element of F. So, we want F closed under complements and countable unions. No, we don't permit uncountable unions, a point that keeps popping up in stochastic processes. Permitting uncountable unions results in, in one word, a mess.

   The sigma part of sigma algebra is supposed to refer to the countable unions as in the capital letter sigma used in calculus, etc., for countable sums.

   So, we have the set M, the sigma algebra of subset F, and the measure m. Then the pair (M, F) is a measurable space and the triple (M, F, m) is a measure space. In probability theory and stochastic processes, the measure space used is usually denoted by space, capital omega, the sigma algebra, script F, and the measure, P.

   Right -- I just let the cat out of the bag: Probability P, as we saw in elementary treatments of probability, is a measure on the measure space with Omega, script F, and P.

   So for A a subset of the space Omega and an element of sigma algebra script F, P(A) is the probability of set A. Set A is called an event. Now we are back to what you saw in elementary treatments of probability.

   So, it went too fast -- we mentioned measure theory with (M, F, m) and then took a special case Omega, script F, P, a probability space, and got back to P(A) we saw in elementary treatments.

   This jump from a measure space to a probability space was made by A. Kolmogorov in 1933 and has been the foundation of 99 44/100% (as in Ivory soap!) of advanced work in probability and stochastic processes since then.

   So, back to the case of measure space (M, F, m). Suppose we have a function

   f: M --> R

   We want to integrate f as in freshman calculus. Well, for each subset A of M and an element of sigma algebra F, we have the measure (area) of A, that is, m(A). Well with one more assumption, and what Lebesgue did, that's enough! And it's assuming less than Riemann did and, thus, getting more generality in some quite good ways.

   Then for the case of Lebesgue integrating some

   f: R --> R

   compared with the Riemann integral, Lebesgue partitions on the Y axis and Riemann partitions on the X axis. So, Riemann partitions on the domain of the function and Lebesgue partitions on the range. In both cases we are working with lots of little rectangles that get smaller as we converge. So, Lebesgue lets the domain be much more general, e.g., enough for probability theory.

   So, back to a probability space and a function

   f: Omega --> R

   With the meager assumption (that f is measurable -- to be brief I'm skipping over the details here is only because nearly all functions are measurable -- tough to construct one that isn't, so general is Lebesgue's work), then f is a random variable. It is more common to denote random variables by X, Y, Z, etc. Now, finally, maybe after getting lost in a swamp for some years, you have a solid definition of a random variable. Then each point in Omega is one probability or statistical trial.

   Then, sit down for this amazing fact: The expectation of real valued random variable X, denoted by E[X], is just the Lebesgue integral of the function

   X: Omega --> R

   So, in this way, Kolmogorov borrowed from Lebesgue and got a solid foundation for probability and stochastic processes.

   Now you know why Lebesgue's measure theory is key to advanced work in probability and stochastic processes -- E[X] is just Lebesgue's integral generalization of the calculus Riemann integral.

   Long standard very careful, polished, treatments of measure theory are in Royden, Real Analysis and the first half of Rudin, Real and Complex Analysis. Continuing with probability is Breiman, Probability, Neveu, Mathematical Foundations of the Calculus of Probability, Loeve, Probability, etc. Uh, Loeve was long at Berkeley, and both Breiman and Neveu were Loeve students.

   Can use this work to make good sense out of Brownian motion, and there are plenty of books there.

   And can do more on stochastic processes, e.g.,

   Ioannis Karatzas and Steven E. Shreve, Brownian Motion and Stochastic Calculus.

   R. S. Lipster and A. N. Shiryayev, Statistics of Random Processes.

   Ronald K. Getoor, Markov Processes and Potential Theory.

   This material is now decades old. With this material it is possible to give high end treatments of some topics in Wall Street type investing, e.g., exotic options and in particular the Black-Scholes formula.

   This material was called by Breiman "graduate probability" as he left the field and did, e.g., CART -- classification and regression trees, maybe the most important seed for the current work in machine learning. Breiman was interested in making sense out of some complicated medical data.

   The potential of such graduate probability for careers in academics and/or applications seems now to be open to question. But, with a larger view, probability and stochastic processes just will NOT go away and will remain some of the most powerful tools for making sense out of reality -- we can be sure that over the horizon we will be glad to have such a solid foundation and WILL encounter important, new applications, ours or those of others.

   There is an old point: Long it was common for the pure mathematicians, yes, to pay close attention to measure theory but essentially to ignore the role in probability theory, saying, that probability theory was just a trivial special case of measure theory for positive measures (can also have measures that have negative or complex values) where the measure of the whole space was 1.

   But: Probability theory does a LOT with the assumption of independence tough to find in texts in just measure theory. Also the Radon-Nikodym theorem (an amazing generalization of the fundamental theorem of calculus) IS treated in measure theory but the applications crucial in probability and stochastic processes to conditioning and sufficient statistics usually are not.

   So, with this overview, you might find Royden, Rudin, Breiman, Neveu (especially elegant) not so hard reading and a lot of fun.

   You will learn about the cases of convergence of sequences of random variables and the classic results of the central limit theorem, the laws, weak and strong, of large numbers, ergodic theory (pour cream into coffee and stir until the cream returns to the distribution it had in the coffee just after pour it in -- Poincare recurrence), and martingales. Margingales are amazing and provide the easiest proof of the strong law of large numbers and some of the strongest inequalities in math.

   So, amazing stuff: Go into a lab, measure something, walk out, and have the value of a random variable -- okay, that's good enough for an elementary course. But typically in math, we can't calculate just what we want in just one step so have to iterate and approximate and hopefully converge in some appropriate sense to just what we want. Well, in those classic results, we have notions of convergence. The most amazing case is of strong convergence where we have a sequence of random variables that converge to some random variable. From the elementary treatments, having a sequence of randomness converge to more randomness seems absurd. But with Kolmogorov's treatment, a random variable is just a function, and from math such as Rudin, Principles of Mathematical Analysis we know quite a lot about convergence, and in measure theory we learn more. Then the case of a sequence of random variables converging to another random variable is just a special case of one case of convergence of sequences of functions. So, we can approximate a random variable X by having a sequence of random variables converge to random variable X. How 'bout that! Thank you Borel, Lebesgue, Kolmogorov, von Neumann, etc.

   Have fun! Get rich???

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2. HilbertSpaceNov 2009
   Early in my career, I asked that question. Eventually I got some answers.

   Broadly usually the 'big bucks' are in things that are new at least in some important sense. To find such things, one recommended approach is to stand "on the shoulders of giants". Then you may still do some original work, but you have one heck of a base. My view is that Moore's law, the Internet, etc. are not nearly fully exploited, that enormous value remains, and the best way to get some of this value is to find a big unsolved problem where a good solution will be very valuable, stand on the shoulders of some giants, do some original work to get some powerful 'secret sauce' for solving the problem (which appears not to be easy to solve otherwise), start a business, please customers, get revenue, grow the business and sell it.

   Here, however, just what problem, giants, original work, secret sauce are NOT easy to see. Broadly we are trying to see, no, MAKE, the future, and usually that is not easy. While we can see what applications have been made of statistics in the past, what statistics will be applied where in the future is not easy to see.

   Yes, broadly statistics is highly promising. Or, in 'information technology' we want valuable, new information, are awash in cycles, bytes, bandwidth, infrastructure software, and data, and want to use these to create the information. Here, on appropriate problems, statistics can commonly, easily, totally 'blow away' anything else. The marriage between (A) 'information technology' in business and (B) statistics has yet to be consummated, even to reach to 'going steady' stage, assuming the traditional order of events.

   For 100 years or so, people have come to statistics from various areas of work. Usually they had some data and some questions. Some of the areas have been educational and psychological testing, experiments and testing in agriculture and medicine, industrial quality control, model building in economics, experimental work in physics and chemistry, investing, attempts to create mathematically based sciences in the social sciences, especially economics, psychology, sociology, and political science. In sociology, old examples would be James Coleman, Pete Rossi (professors of my wife when she got her Ph.D.), Leo Goodman.

   Likely the best fast, practical path into statistics is via books, courses, etc. intended for students in the social sciences. These students commonly do not have good backgrounds in mathematics. For the mathematical prerequisites, generally can get by, as a start, with just high school first year algebra. With this path can learn about probability distributions, the central limit theorem, the law of large numbers, statistical estimation and confidence intervals, hypothesis testing, cross tabulation, analysis of variance, regression analysis, principle components analysis, and more.

   For more, statistics is a rock solid part of mathematics, as solid as any part of pure mathematics, e.g., topology, geometry, analysis, algebra, etc. That is, statistics is based solidly on theorems and proofs, sometimes relatively deep ones.

   Statistics as theorems and proofs is called 'mathematical statistics'. Long standard has been (with TeX markup):

   Alexander M.\ Mood, Franklin A.\ Graybill, and Duane C.\ Boas, {\it Introduction to the Theory of Statistics, Third Edition,\/} McGraw-Hill, New York.\ \

   The main prerequisite for this book is just a not very good course in calculus, and the book actually makes not much use of calculus. Mostly all a student will need from calculus is that it can find the area under a curve. Since the book has long been standard, can't really ignore it, but it's ugly. And often, with just calculus, the book doesn't really give solid proofs of the results. E.g., their treatment of sufficient statistics has some nice intuition, but their proof is junk. The subject cries out for a good book, but I'm not trying to write one or waiting for someone else to.

   Can get some of the flavor of mathematical statistics done with high quality, as mathematics, in, say (with TeX markup):

   Jean-Ren\'e Barra, {\it Mathematical Basis of Statistics,\/} ISBN 0-12-079240-0, Academic Press, New York.\ \

   Robert J.\ Serfling, {\it Approximation Theorems of Mathematical Statistics,\/} ISBN 0-471-02403-1, John Wiley and Sons, New York.\ \

   P.\ Billingsley, {\it Convergence of Probability Measures, 2\raise0.5ex\hbox{ed},\/} ISBN: 0-471-19745-9, John Wiley, New York.\ \

   R.\ S.\ Lipster and A.\ N.\ Shiryayev, {\it Statistics of Random Processes I, II,} ISBN 0-387-90226-0, Springer-Verlag, New York.\ \

   However, pursued mathematically, statistics has some relatively advanced prerequisites some of which curiously are not popular in US university mathematics departments.

   For the prerequisites,

   High School. Should have had high school first and second year algebra (reasonable facility with algebraic manipulations, the binomial theorem, complex numbers, both of which will see again in important ways), plane geometry (where nearly all the work was proofs -- first place to learn about proofs), trigonometry (usually assumed in calculus and important in, say, analysis of organ tone harmonics and, thus, the most important example of an infinite dimensional Hilbert space), analytic geometry (especially the conic sections, especially ellipses which definitely will see again), and, if can, solid geometry (for more intuition in three dimensions).

   College. Need a standard calculus course, not necessarily a very comprehensive or difficult one because will do the subject all over again, maybe two or three times, and more later, WITH the proofs!

   Then need linear algebra, that is, how to work with data of several dimensions, which is just crucial. The big result is the polar decomposition, and there get to think about ellipses and get to use complex numbers. Also the course is an introduction to functional analysis and Hilbert space. Use any popular book to get started but in the end cover the classic, Halmos, "Finite Dimensional Vector Spaces'. Halmos wrote this when he was an assistant to von Neumann and intended it to be a finite dimensional introduction to Hilbert space (which once von Neumann had to explain to Hilbert) which it is. It also has some multi-linear algebra, of interest to exterior algebra now popular in relativity, but likely for nearly any business applications of statistics for the next several decades can skip that chapter.

   Then need some advanced calculus. That is a poorly organized, huge, catch-all subject beyond any one course. The usual start is 'baby' Rudin, 'Principles of Mathematical Analysis'. So, that's calculus with the proofs. Warning: The book is severe, succinct, with zero pictures. Have to draw your own pictures in your head. The book is packed solidly with powerful material just awash in important applications from statistics, economics, and engineering to physics, but there is hardly a hint of the applications in the book. I enjoyed the book, but few people will enjoy it or even get through it. Hint: Get a really good teacher! Then for more, popular is Spivak, 'Calculus on Manifolds', mostly because it is short. Actually, it's too short. I prefer Fleming, 'Functions of Several Variables' until get to the exterior algebra chapter at which time, if care, can now get the thin

   Henri Cartan, {\it Differential Forms,\/} ISBN 0-486-45010-4, Dover, Mineola, NY, 2006.\ \

   in English.

   Between Halmos, baby Rudin, and Spivak, you will have covered Harvard's Math 55 with a colorful description at

   http://www.american.com/archive/2008/march-april-magazine-co...

   Harvard tries to cover these three for freshman, but in most math departments the material will take you through all or nearly all of a focused undergraduate pure math major.

   If somewhere take a course in abstract algebra, e.g., with a little group theory, then that might help!

   Graduate School. Might learn a little more about topology, say, from Simmons, 'Introduction to Topology and Modern Analysis'. So, get good with metric spaces and get started on duality.

   The next big step is a course in measure theory and functional analysis. The Simmons work will help. Baby Rudin will be crucial; Halmos is recommended. So, with measure theory, do calculus over again and in a very different and much more powerful way and a way just crucial, even central, for mathematical approaches to statistics. The functional analysis will concentrate on representation theorems, the Radon-Nikodym theorem, and Hilbert and Banach spaces. Long popular, from Stanford and sometimes aimed at statistics students, is Royden, 'Real Analysis'. It's gorgeous. Should also read the real half of Rudin, 'Real and Complex Analysis'; it's a few steps up in difficulty from baby Rudin. Again, hint: Get a good course from a good teacher who can get you over the material without getting stuck. Then go back and study the material 2-3 more times, apply it, do some original research in it, and finally begin to understand it.

   We're talking high, top, center crown jewels of civilization here; the stuff is of just awesome power; my view is that it is, for the rest of this century, one of the main pillars of increases in economic productivity via the exploitation of Moore's law; on 'what to program', computer science is stuck and this material is the most promising way forward; as a famous restaurant owner once said about some Morey St. Denis, "you won't find better".

   Now are ready for probability. I recommend:

   Leo Breiman, {\it Probability,\/} ISBN 0-89871-296-3, SIAM, Philadelphia.\ \

   M.\ Lo\`eve, {\it Probability Theory, I and II, 4th Edition,\/} Springer-Verlag, New York.\ \

   Kai Lai Chung, {\it A Course in Probability Theory, Second Edition,\/} ISBN 0-12-174650-X, Academic Press, New York.\ \

   Jacques Neveu, {\it Mathematical Foundations of the Calculus of Probability,\/} Holden-Day, San Francisco.\ \

   Neveu is succinct, gorgeous, but not easy. This material is NOT popular in US departments of mathematics. At Princeton, see Cinlar.

   Then should make some progress with stochastic processes: The big book is Gihman and and Skorohod, right, in three volumes, but mostly people settle for shorter treatments. Whatever, should learn about Poisson processes, Markov processes (discrete time, finite state space is enough to get started), Brownian motion, and martingales. Might also learn about second order stationary processes. A good course in stochastic processes is NOT easy to find, especially in mathematics departments.

   Now are ready to attack statistics mathematically! I don't know of a good, single 'mathematical statistics' book at this level. Instead, there are many books -- I gave some above -- and then the journals. Thankfully, the field is relatively close to applications; so can take a practical problem and concentrate on what is relevant to it. One of my papers was some new work, at this level, in mathematical statistics for a problem in practical computing and computer science. The computer science community struggled terribly with the mathematics. So, it was some progress in computer science that community will have to struggle to understand.

   One approach to work in computing is just to try things, that is, just to throw things against the wall and see if they appear to stick. Or, maybe the truth of the situation really is a simple statistical model. Likely that model will fit the data well. So, try many simple statistical models. We use these models mostly ignoring the mathematical assumptions; mostly we are proceeding 'heuristically', that is, with guessing. If any of the models fit well, then they can be considered candidates for the truth. So, are throwing things against the wall to see if they fit. This approach also called 'data mining'.

   Problems:

   (1) Will be quite limited in what statistical models can use. That is, will be drawing from a cookbook instead of being a real chef who can create good, new dishes appropriate for the available ingredients and customers!

   (2) Don't have much, e.g., have not proceeded mathematically where from the deductive logic of assumptions and proofs actually know in advance some good things about the results. Something like breaking into a pharmacy, mixing up a lot of pills, taking them, and seeing if feel better! Uh, I'll pass and let you do that without me!

   (3) May have gone through a lot of computer time in an 'exponential, combinatorial explosion' of efforts throwing against the wall.

   (4) Have ignored a LOT in statistics that can add to what know about the results.

   (5) Will be tempted to conclude have have found 'causality' but will likely not have.

   (6) Will be tempted to conclude that have a model that predicts, but that is on shaky ground and risky and needs more work.

   Applied to important problems, this approach can be dangerous.

   There are not many healthy statistics departments. Much of the career interest is in biostatistics, especially related to FDA rules.

   It appears that among the top statistics departments are Berkeley and UNC. Since Breiman and Brillinger are at Berkeley and since Stanford, long good in statistics, is not far away, if I were looking for a Ph.D. in statistics then I'd pick Berkeley.

   There is a general problem getting a 'job' in a technical field and likely also with statistics. The assumption in US business is still as in factories 150 years ago: The supervisor knows more than the subordinate; the subordinate is supposed just to add common labor to what the supervisor says. In particular job descriptions are written by the supervisors, not the subordinates!

   Well, there are nearly no supervisors in US business who have even as much as a weak little hollow hint of a tiny clue about the material described here. So, won't need that material to qualify for the job descriptions. Moreover, if actually know such material and let that fact leak out, then will likely not make it past the first level HR English major phone screen person who will tremble and conclude that you are not like the employees they have! If you do get hired and someone in your management chain discovers that you have used some mathematics they don't understand, you might be on the way out the door, especially is your work was valuable for the company!

   Of course, the solution is to find a valuable application and start your own business. While maybe biomedical venture capital can understand crucial, core technical content, in information technology venture capital, likely you will be trying to explain this stuff to, say, history majors who worked in fund raising, marketing, general management, or financial analysis or have a background in just relatively elementary parts of computing. Just will NOT find more than six people, maybe not more than zero people, in US venture capital who can work the exercises in Royden or explain the strong law of large numbers. Sorry 'bout that! So, if you explain that the value of your venture is the powerful material in your 'secret sauce', then you will be regarded as a kook, far outside the mainstream of venture funded entrepreneurs, discarded, maybe even laughed at. As it is, some of the venture people are making money now, and the rest just want to be more like the ones who are making money. Looking for anything really new, powerful, and valuable is just NOT in the picture.

   So, once you have some results in users, customers, revenue, etc., then maybe you can get some venture funding; just why at that point, owning 100% of the business, you would take venture funding, a Board that can fire you, etc. is less clear! Or venture funding is not for everyone! Or venture firms prefer to give money to people who don't need it!

   For the real power of the 'secret sauce', you have just to keep that a secret!

   Once mathematicians have yachts, at the venture firms math will be to info tech like biochem is to biotech. In the meanwhile note that a valuable application of statistics can put you on the Forbes 400 where there are not many people! Generally if you are making a valuable application of advanced or new statistics, then you will not know many people who understand what you are doing. Or, if lots of people understood it, then it wouldn't be valuable!

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