Talk:Stochastic process/Archive 2

random variables are not natural numbers ?
The article says:"If both t and X_t belong to N, the set of natural numbers, ..." Is this correct ? How can the random variable X_t be a natural number ? — Preceding unsigned comment added by 213.54.36.209 (talk) 10:22, 14 June 2013 (UTC)


 * It is non-rigorous but quite usual, to say such things as "almost surely, X is a number between a and b", "almost surely, X is an integer" etc., when X is a random variable. Even more non-rigorous and still usual (especially among non-mathematicians) is, to omit the "almost surely". Boris Tsirelson (talk) 10:51, 14 June 2013 (UTC)

Heterogeneous process explanation
The Heterogeneous process page routes to this one, but then is not included in the explanation. This would be good to include for those rewriting this. Thanks. Alrich44 (talk) 15:10, 30 July 2014 (UTC)

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The Introduction is flawed in a number of ways.
1. Wiki recommend starting with a simple explanation, and gradually adding more complexity; the present Introduction is too complex. Casual readers will be content to read the Introduction, and skim through the rest of the article, but interested students, and we are all students, will move on and read and re-read the article, in conjunction with others.

2. There are factual errors in the Introduction, which need to be rectified. This is important, because the concept under discussion is fundamental, not just to study of probability and statistics, but to engineering as well.

I propose to rewrite the Introduction, but I do not want to do so, without getting agreement from those who feel that they have a stake in maintaining this page; can anyone with such a stake make themselves known? - we can then work jointly on revising the Introduction, to achieve consensus. Systems Analyst 2 (talk) 03:07, 21 September 2016 (UTC)


 * As for me, it does start with a simple explanation. Do you think otherwise? And what are the factual errors? Boris Tsirelson (talk) 05:15, 21 September 2016 (UTC)


 * Firstly, I propose that we address four levels of reader: 1. casual reader 2. new student 3. intermediate student 4. advanced student. In my view, the Introduction is confusing to a casual reader. I further propose that we address this by using the following model: 1. Provide an example. 2. Formalise the approach. 3. Give examples of use. This can be done for each level of reader. Secondly, there are factual errors: 1. As the term Stochastic Process is used in a generic sense, it is not true to say that it is a time sequence; think of casting dice, for example. The four classifications need to be covered in the Introduction in an orderly way. 2. The Introduction does not clarify the distinction between a Stochastic system and a Deterministic system; they have a degree of overlap; for example, a specific sample from a stochastic process could be indistinguishable from the output of Finite State Machine, or the equivalent Turing Machine, both being purely deterministic; in fact, there is a non-denumerable set of such samples, and their equivalent FSMs. This is of practical significance in cryptography, where keys or one-time-pads need to be 'strong'. I can provide more detail, if this approach meets with agreement.Systems Analyst 2 (talk) 06:03, 22 September 2016 (UTC)


 * On the first: your "1. Provide an example. 2. Formalise the approach. 3. Give examples of use." is good, but should all that be done in the lead? Isn't it too long and detailed for the lead (especially, "3")? Well, try to do so, and we'll see.
 * On the second: I am puzzling, why "casting dice" is not a time sequence? Isn't it a sequence of i.i.d random variables? Also, the problem of pseudo-randomness versus true randomness is good, but again, why already in the lead? Doesn't it overload the lead? Boris Tsirelson (talk) 10:06, 22 September 2016 (UTC)
 * Now I note that you speak about "introduction", not "lead". Many articles contain a lead, then introduction section, then other sections. Do you mean this architecture? Boris Tsirelson (talk) 10:09, 22 September 2016 (UTC)


 * I will try a first draft Introduction, over the next week, and see how long it is; as you say, there could be a Lead, which could be a short summary of the Introduction.
 * As for the dice; you could cast six fair dice simultaneously, and that would be one sample from a Stochastic Process; there does not have to be an independent variable (time). This is an example of a discrete/discrete case.
 * Downloading a share-price history is also a sample from an S.P. for that share; it is not a Random Variable, though it is a time history of course; both are simple examples that are intelligible to a general reader, I would think. I tend to find that the formalism is off-putting at first; often it seems too abstract, and even obtuse, and an example breaks the ground.
 * I checked for other Maths articles of level-B quality and Top Importance, and found the following:
 * Probability Theory refers to Stochastic Process, hence we need to fit in with that. (I noticed a 'howler' in there.)
 * Probability Space has an intersection with Probability Theory, but does not refer to S.P.
 * I am not sure why those two articles are not combined, or why there is not a common 'Head' Article for the two.
 * Below S.P. in the development of ideas lie: Bayes Theorem, Probability Density Function, Probability Distribution and Random Variable, all level-B & Top Importance.
 * Markov Chain sits to one side of S.P. Again, I am not sure why it is not combined with S.P.
 * We need to fit in with this context, I think, as per Wiki's recommendation. — Preceding unsigned comment added by Systems Analyst 2 (talk • contribs) 03:37, 24 September 2016 (UTC)

The concept of the Stochastic Process is based on axiomatic Probability Theory [Maybeck, chapter 3]. There are four classifications of Stochastic Process, summarised here [Jazwynski]:
 * 1. Discrete time and discrete state-space, known as Markov Chains.

Examples include repeatedly selecting a card from a full, well shuffled pack, tossing a coin, or repeatedly tossing a die [Maybeck, chapter 3]; the result is a sequence of heads or tails for a fair coin, or a sequence of integers, each from one to six, for a fair die, for example. Any such experiment, giving a specific sequence of fixed length, generates a sample ω from a sample space Ω, of probability P(ω), where P is the Probability Function; it is the ensemble of such samples that constitutes the Stochastic Process [Maybeck, chapter 3,4]. It is important to realise that the intuitive concept of what is 'random' can be misleading here; for example, the coin-tossing experiment could yield a sample which is all 'heads'; this does not seem random, in itself. Any other recurring sequence of head or tails could occur with equal probability, or the sequence could be non-recurring; this is analogous to the occurence of rational numbers in the real domain. It is just that there are many more non-recurring sequences in a long but finite sequence, and hence the non-recurring sequences outnumber the recurring sequences, but all individual sequences are equally likely, given a fair coin or a fair die. There is no requirement for the points of the sample space to be numbers, or sets of numbers; each could be made up of letters or arbitrary symbols, or sets of symbols, depending on how the card, die or coin is marked. Applications include a Roulette Wheel, a Pack of Cards or a Slot Machine; other areas of application are Simulation and Cryptography. It is worth noting here that there is an overlap between the behaviour of Deterministic Systems, and Stochastic Systems; for example, Pseudo-random Number Generators [Knuth, 1981] are deterministic Finite State Machines [Knuth], or Turing machines [Turing], that can produce finite runs of numbers, of more or less 'random' appearance, but in parts of the sequence the numbers do not look 'random'; such sequences are used in running numerical simulations of systems [Marse and Roberts], where noise needs to be simulated. The 'randomness of such numbers is considered in [Hull and Dobell], in an earlier survey of the literature. In Cryptography [Welsh], the intention is to convert a meaningful message into an apparently random sequence, a sample from a stochastic process, using an algorithm; there is also an algorithm to reverse the encryption. One method is to use a one-time pad, but this relies on the 'randomness' of the pad, which has to be generated in some way. In other words, deterministic systems in the form of random-number generators, are used to mimic samples from a Stochastic Process; thus stochastic systems and deterministic systems are not entirely separate from each other, in terms of the output samples generated. 2. Continuous time and continuous state-space. Here, for example, time is specified by a real number, and the values of the state variable are also real numbers. An alternative example could involve distance, instead of time. In this case, there is no discrete sequence of samples being taken from the sample space; instead, there is a continuum of values, taken continuously in time, say. Here, time is increasing monotonically, and the state variable is varying, in general. Denoting the ensemble of samples taken from the sample space by X(\[FilledSmallCircle],\[FilledSmallCircle]), where the first argument denotes time, and the second denotes the sample taken, then X(t,\[FilledSmallCircle]) is the ensemble of samples taken at the specific time t, and X(\[FilledSmallCircle],\[Omega]) is the specific sample \[Omega], taken over time. The function X(t,\[FilledSmallCircle]) is referred to as a Random Variable, at the time t, with a mean value and variance at time t. X(\[FilledSmallCircle],\[Omega]) will look like a specific time plot, the properties of which can be specified; for example, it has a mean-value function, which can itself vary with time, and it has a variance, also varying with time, in general. The stochastic process is the ensemble of X(\[FilledSmallCircle],\[Omega]). The mean-value function is the mean of the ensemble, and the variance is the variance of the ensemble. It should be noted that a specific sample X(\[FilledSmallCircle],\[Omega]) could be a 'random constant', a sinusoid of any frequency, a square-wave or any regular cyclic function, but it is far more likely to be what conforms to the intuitive notion of a random quantity, varying in time. However, there is no requirement for X(\[FilledSmallCircle],\[Omega]) to be continuous, as time varies; it is only continuous if X is correlated in time. Brownian motion is the classic example of the stochastic process X(\[FilledSmallCircle],\[FilledSmallCircle]). The scalar X can be generalised to the vector X. The scalar mean is then replaced by a mean-vector, and the scalar variance is replaced by the covariance matrix. If the vector stochastic process is correlated in time, then the scalar correlation function is replaced by the auto-cross correlation matrix, and the covariance matrix generalises to the auto-cross covariance matrix. Applications could involve modelling the noise from an analogue electronic sensor, or circuit. If the noise from a sensor is recorded and plotted, this would be a sample from the stochastic process involved. Re-running the experiment repeatedly would give an ensemble of samples from the stochastic process, and its statistics could be investigated. Another application could involve modelling the tilts of the local gravity vector, relative to the earth's reference ellipsoid; in this example, the tilts will be correlated in distance, as a vehicle moves over the ellipsoid, which can affect the accuracy of an inertial navigation system. Itô stochastic calculus is based on this notion of a stochastic process, and enables the modelling of stochastic systems of diverse forms, using stochastic differential equations.The differential equations can combine systematic 'signal', due to the deterministic nature of the differential equations, with added 'noise'. The histories of state variables of such a system are samples from the stochastic processes; thus a stochastic process can include a deterministic contribution, along with random noise. In addition, if the system model includes 'random' constants, these can integrate up to a ramp in a particular sample time history X(\[FilledSmallCircle],\[Omega]), but the gradient of the ramp will vary from sample to sample. Feed-back loops can also generate sinusoids, and these too can vary from sample to sample. These examples indicate that a stochastic process can model a wide range of dynamic variables, hence the power of the concept. 3. Discrete time and continuous state-space. A sensor may measure a quantity at a fixed sampling rate; this can involve measurement noise, due to the sensor itself, but also process noise, due to the system being monitored. The sampling process yields discrete-time data. Many engineering systems are implemented in this way. They can be modelled as stochastic difference quations. This has led to the development of the Kalman Filter, which is used in diverse disciplines. If a satellite's position is measured in orbit, then the measurements involve noise, but in place of time, the noise arises at different positions of the satellite measured. An equity or share price is available in discrete time; a history of a share price is a sample from a discrete-time stochastic process; it is not the stochastic process irself. Investors may use price-data to estimate the mean of a share-price, and look for deviations to trade on; there is an assumption that there will be no rapid change in the mean, but this does occur occasionally, and is usually unpredictable. To reduce risk, portfolios of shares are employed, usually in different sectors of the market; the linear combination of share holdings at a particular time averages noise across the portfolio. There is often dynamics involved in a share price; for example it may ramp up, oscillate, or even exponentiate for a time. The price is still a sample from a stochastic process; a Kalman filter can be used to model linear dynamics in the presence of noise. Systems Analyst 2 (talk) 21:35, 1 October 2016 (UTC)


 * Wow... Interesting. And problematic.


 * Interesting, since it could be a helpful introductory/explanatory section in a textbook.


 * Problematic, since Wikipedia is not a textbook.


 * Likewise, you could rewrite the article "function" like that: "One classification (or do you mean, class?) of functions, x=f(t), are functions from time to space; these describe a motion of a point (in particular, the barycenter of a material body). Another class..."


 * However, given a function, say $$\sin:\mathbb R \to \mathbb R,$$ you cannot say, whether its argument is time, or money, or something else. Thus, your-style classification is not a classification of functions (as mathematical objects), but rather a classification of their real-world applications.


 * Yes, I am a mathematician and you are not, and accordingly, I treat mathematical objects and you treat their real-world applications. This itself is OK. However, your thinking (cogitation) may appear in Wikipedia only if it can be sourced. Have you a "reliable source" (preferably textbook) that conforms to your text? If you have, please provide. Otherwise, try to publish your cogitation elsewhere. Maybe, on Wikiversity? Boris Tsirelson (talk) 05:37, 2 October 2016 (UTC)


 * A good reference is "Stochastic Models, Estimation and Control, Volume 1" by Peter S. Maybeck, Academic Press 1979. What I wrote is a bit long for an Intro, but it could be used in the section "Classification" of the main article; a precis might serve as an Introduction. I agree that the mathematical viewpoint and the systems analysis viewpoint are different, as in s.a. we use less rigour, but stochastic processes are very important in engineering and financial engineering, hence we need to embrace both viewpoints in the article, I feel.  I could use the above to revise the Classification section, and provide a draft here in Talk. Systems Analyst 2 (talk) 20:26, 10 October 2016 (UTC)


 * OK, I'll look at this book. Boris Tsirelson (talk) 21:18, 10 October 2016 (UTC)
 * Looking at Sect. 4 of the book by Maybeck, I do not see any essential deviation of his terminology from the terminology used by mathematicians. In particular, "stochastic process" is still a mathematical object (rather than a real-world application of the mathematical object). I do not see coin tossing there, but still, a finite family of independent random variables, each with two equiprobable values (0 and 1), is a mathematical object (often called Bernoulli process); its real-world applications include both repeated toss of a single coin, and a one-time toss of a finite collection of coins.
 * Many other thoughts contained in your text above also are not contained in that book, as far as I see for now. Boris Tsirelson (talk) 06:00, 11 October 2016 (UTC)

Applications Section
I think the entirety of the article is well-cited and gives a good picture of what stochastic processes are. My one concern after reading the article is that I don't know how they are used. I'd like to propose an applications section, with a couple of examples from different fields. For example, we could have a few sentences about financial applications, a few about weather prediction models, and a few more sentences about some scientific (biology or chemistry?) application. Anyone who has worked with an application should feel free to contribute to that. — Preceding unsigned comment added by Chaley17 (talk • contribs) 17:41, 11 October 2016 (UTC)

Martingale
"A martingales is a discrete-time or continuous-time stochastic processes..." — wow! singular or plural? Boris Tsirelson (talk) 19:56, 8 January 2017 (UTC)

"...there are two known martingales based on the martingale the Wiener process, forming in total three continuous-time martingales..." — wow again! I did not take this hint. Boris Tsirelson (talk)

Modification
Here one must be very careful. The phrase "two stochastic processes that are modifications of each other have the same law" is true or false, depending on the used space of functions and the sigma-algebra on it. Boris Tsirelson (talk) 21:48, 8 January 2017 (UTC)

Stationarity
What happens to stationarity in the first paragraph of Sect. 8.3? In the first phrase, only one-dimensional distributions matter. In the second phrase, n random variables are mentioned, but still, only one-dimensional distributions matter. In the second paragraph, the definition is OK (through finite-dim distributions), with the reservation that the index set is interpreted as time. Hmmm... does it mean that for a stationary random field only one-dimensional distributions matter? Boris Tsirelson (talk) 14:15, 8 January 2017 (UTC)

Where/how does it say (or imply) only "one-dimensional distributions matter"? I tried to keep it for general T. Improbable keeler (talk) 13:22, 10 January 2017 (UTC)

Filtration
Sect. 8.4: "A filtration is an increasing sequence of sigma-algebras..." — sometimes sequence, but generally family. If this is intended, then the second phrase probably is not only "more formally" but also "more generally" than the first. Boris Tsirelson (talk) 14:20, 8 January 2017 (UTC)

OK. We can change it. I just need a reference. I don't like writing anything without a citation or two, so people can look it up.Improbable keeler (talk) 13:25, 10 January 2017 (UTC)

Random walk
"...random walks, defined on different mathematical objects, such as lattices and groups..." — I guess, S is a group, not T, and so the walk is in S (rather than on S); though, being not a native English speaker, I am not sure. Boris Tsirelson (talk) 19:31, 8 January 2017 (UTC)

I'm a native speaker and I actually have difficulties with prepositions in mathematics. People talk about a random walk on a group (so, yes, then the state space S would be the group), but, then I don't know how to describe the corresponding stochastic process. It's a random walk/stochastic process defined on group? Probably not. I would say it's a random walk on a group (that's how I've read it in articles), but a stochastic process with a group as its state space. Improbable keeler (talk) 13:28, 10 January 2017 (UTC)

The lead
"...disciplines including physical sciences such as biology..." — being not acquainted with the term "physical sciences", I have looked at "Outline of physical science", and there I see "Natural science can be broken into two main branches: life science, for example biology and physical science. Each of these branches, and all of their sub-branches, are referred to as natural sciences." Then probably "...disciplines including natural sciences such as biology..."? Indded, in Sect. 1.1 I see "many problems from the natural sciences". Boris Tsirelson (talk) 18:38, 8 January 2017 (UTC)

I don't know the difference between "natural sciences" and "physical sciences". To my ears, the former sounds more old fashioned or classic eg Newton was a professor of natural sciences. I didn't think "natural sciences" would cover "neuroscience". In Section 1.1, I did use "natural sciences" -- I believe that was the exact phrase that the Russian writer (Borokov) used in his book written in English.Improbable keeler (talk) 13:38, 10 January 2017 (UTC)

Random field
"If the specific definition of a stochastic process requires the index set to be a subset of the real line, then the random field is considered as a generalization of stochastic process." — but I fail to understand, why "generalization"? Boris Tsirelson (talk) 20:12, 8 January 2017 (UTC)

I thought Applebaum and also the random field wiki say it's a generalization. What would you call it? They only consider things on R, then decide, lets extend its definition to R^2 or R^3. An extension? Improbable keeler (talk) 13:44, 10 January 2017 (UTC)

Silent majority?
I am a bit discouraged by the silent community. Major changes are made. The topic should be important. Different positions are voiced. So what? Is anyone interested? Otherwise it is pointless anyway. Boris Tsirelson (talk) 16:20, 10 January 2017 (UTC)

It's very discouraging. It's a major problem across all of Wikipedia. Apparently its number of editors peaked in 2005 or so, according to this [|piece]. When I first completely re-wrote the Poisson (point) process article, which had remained the same for years, I thought there would be major reactions -- good or bad. Almost nothing. So I am not surprised that there's silence here too. It's hard to think of a more important topic in probability theory. And I was hoping by me doing all the tedious work of finding various citations and implementing the code for them, then people could use those citations in other more specific articles on stochastic processes. I am seriously contemplating about writing a serious article somewhere trying to encourage more mathematicians to contribute to wiki. Improbable keeler (talk) 18:17, 10 January 2017 (UTC)


 * Yes. But, frankly, this is very natural, for an evident reason: millions of articles were created in the first several years, and then, the extensive scope of work greatly narrowed, workers become redundant... Boris Tsirelson (talk) 19:10, 10 January 2017 (UTC)


 * On the other hand:
 * Number of page watchers = 331
 * Number of page watchers who visited recent edits = 23
 * (data obtained just now). Where are they all? Boris Tsirelson (talk) 19:16, 10 January 2017 (UTC)

That's one explanation. Many of the original editors retired, while Wikipedia has had great difficulty attracting new editors. Yes, many articles were created. I actually think the number of articles is a big problem -- there are too many. For example, do we really need an article on a subordinator? Surely that's just a subsection of the article on Lévy processes. I am considering merging such articles. Generally, you should do it the "democratic" way by stating your intention etc. But since there are so few people actively editing (and we all have day jobs, I suppose), I think one should just do it, and wait for the reaction. A merging can always be undone. Improbable keeler (talk) 19:26, 10 January 2017 (UTC)

The number page watchers doesn't mean much if everybody is logged out of the Wikipedia account -- you don't see notifications. I didn't log in for a few months for various reasons. I suspect a lot of students become wiki editors, and then finish university and stop editing. Improbable keeler (talk) 19:28, 10 January 2017 (UTC)


 * But note the second formulation: "page watchers who visited recent edits". Boris Tsirelson (talk) 19:56, 10 January 2017 (UTC)

That's a point. But some (perhaps most) are non-mathematical (or non-probability) people. Just general wikipedians. Perhaps they just don't untick the "Watch this page" when they click "Save changes". Improbable keeler (talk) 19:32, 11 January 2017 (UTC)


 * Ah, yes, I see. Boris Tsirelson (talk) 20:17, 11 January 2017 (UTC)

Increasing numbers
Well, in English, I think a set of increasing numbers could be any set of numbers that increase in value. Of course, all the subsets of the real line are ordered, but I thought one should say the numbers are ordered or increasing in the lead. Improbable keeler (talk) 17:23, 13 January 2017 (UTC)
 * "My understanding of Wikipedia is that you need to support every claim" (a quote from you). As for me, definitions are even more important than theorems ("claims"); a theorem can be deduced from definitions, but a definition cannot be deduced from anything else. "Set of increasing numbers" is undefined.
 * Also, it may confuse. Someone may think that you mean an increasing sequence (and so, an interval does not fit). Someone may think that the set of all negative integers does not fit (while in fact this case is actively used in the theory of filtrations).
 * "a set of increasing numbers could be any set of numbers that increase in value" — as for me, the phrase "set of numbers that increase in value" is absolutely incomprehensible. Have you an example of a set of numbers that do not increase in value? Boris Tsirelson (talk) 18:26, 13 January 2017 (UTC)

This is not a mathematical issue, surely. Isn't this just a language thing? When I say a "set of increasing numbers", the set is not, say, (3,1,11,4,2,-12,...), which are not increasing or ordered. I think that in everyday English, people would understand the expression a "set of increasing numbers". Support every claim, yes, but not everyday language. Perhaps it's not an issue. I'm sure when people say a set of numbers, they don't think of something like (3,1,11,4,2,-12,...).

Perhaps I should just replace "increasing" with "ordered" or remove it completely. .Improbable keeler (talk) 19:54, 13 January 2017 (UTC)


 * "the set is not, say, (3,1,11,4,2,-12,...)" — sure; this is not a set but a sequence. "Increasing sequence" is defined; "increasing function" is defined; "increasing set" is not. A set is rather {3,1,11,4,2,-12}, and it is equal to {-12,1,2,3,4,11}, as well as {11,4,3,2,1,-12}.
 * You are extremely careful with textbooks. Did you ever see a textbook (in math) that confuses "set" and "sequence"? Did you ever see there that a random process is a family of random variables indexed by a sequence of real numbers? Why are you so uneven (sometimes too much accurate, sometimes not enough accurate)?
 * "I had no idea such a simple phrase would cause any confusion" — but I gave you two examples; what do you think of these? Boris Tsirelson (talk) 20:08, 13 January 2017 (UTC) Strange: I did not enter "edit conflict" with you, while I should...

Perhaps you're right people may think an interval does not fit. OK. Removed. Improbable keeler (talk) 19:59, 13 January 2017 (UTC)


 * Everyday English just cannot be used on this level. If you really want to use it, say something like "A stochastic process is a mathematical formalization of the idea of something that is random and extended in time (or space)". Well, surely you can say it better than me. But if you use "random variables" and "set", you are already beyond the everyday English anyway. Boris Tsirelson (talk) 20:26, 13 January 2017 (UTC)

Construction issues
"the distribution of the stochastic process does not uniquely specify the properties of the sample functions" — this is true or false, depending on the space of functions and the sigma-field on it. No problem when we are lucky to have a standard Borel space. (See also Sect. 2c of my course.) Boris Tsirelson (talk) 21:41, 8 January 2017 (UTC)

See also Talk:Law (stochastic processes). It is quite usual to say that the law (in other words, the distribution) of the standard Brownian motion is the classical Wiener measure (on the space of continuous functions). On the other hand, it is quite usual to say that, in general, the law of a stochastic process is a probability measure on the space of all functions from T to S endowed with the product sigma-algebra (generated by evaluations). But in the latter case the probability of being continuous (for a Brownian sample function) is undefined, since the set of all continuous functions is not measurable. Thus, we have two different terminological approaches. Boris Tsirelson (talk) 07:05, 9 January 2017 (UTC)

True, a separable modification exists, always. So what? It is not unique (up to indistinguishability, I mean). A simple example: the left-continuous and the right-continuous modifications of the Poisson process are both separable. A harder example: a stationary Gaussian process with a continuous correlation function may seem to be a simple matter, but it is not. For some correlation function the paths are unbounded on every interval; in this case separability means nothing. (See the end of Sect.21 of my course.) Bad news...

Good news: practically, we do not need to bother, whether a Poisson path is left-continuous or right-continuous; we need not bother at all about its values at points of discontinuity. This is instructive: beyond continuity, it is a good idea to treat sample functions up to some equivalence relation. Just like Lebesgue integration theory; there it is quite usual to say "function" but to mean "equivalence class of functions" (up to equality almost everywhere). In this framework the "bad" stationary Gaussian process (mentioned above) becomes tractable: its path is well-defined almost everywhere, and is Lebesgue integrable. But, again, the set of all integrable functions is not measurable w.r.t. the product sigma-algebra.

A conclusion: the approach presented in our article is widely used but not universal. Different cases need different approaches to the idea of a stochastic process. Most generally it is a random element of a measurable space whose elements are either functions, or equivalent classes of functions, or generalized functions of some kind (notable examples: white noise, Gaussian free field), etc. (see list of types of functions). Maybe some day we'll have "list of types of stochastic processes" too. :-) Boris Tsirelson (talk) 07:37, 9 January 2017 (UTC)

Two approaches
I tried to make it clear that there are different approaches. I just wrote that the "collection of random variables" is a popular approach, but it is also mentioned twice, at least, that a stochastic process is also a random element of a measurable space with functions as elements.Improbable keeler (talk) 20:48, 13 January 2017 (UTC)

I see what you mean now in your lectures in section 2c. I didn't introduce equivalent classes of functions. But we could, I suppose. But I need citations, preferably books. I can't just cites some lectures notes on the web, though they can be very helpful in clarifying matters.Improbable keeler (talk) 20:48, 13 January 2017 (UTC)

Citation needed?
"both the left-continuous modification and the right-continuous modification of a Poisson process have the same finite-dimensional distributions" — well, maybe this is my original research (hope not) but the proof is more than evident; we observe the two processes at chosen (fixed, nonrandom) points $$t_1,\dots,t_n$$ and see different values only when at least one of these points is a point of discontinuity of the Poisson process, which is a negligible event (of probability zero and therefore unable to influence any probability). We speak math, not politics here... Boris Tsirelson (talk) 20:18, 9 January 2017 (UTC)

Even simpler: these two processes are equivalent; and "Two stochastic processes that are modifications of each other have the same law". (Though I do not like this phrase since it presupposes the product sigma-algebra; I'd say "Two stochastic processes that are modifications of each other have the same finite-dimensional distributions" which is unambiguous. Indeed, the Wiener process has one continuous modification, whose law is the Wiener measure, and a lot of discontinuous modifications... whose laws cannot be the Wiener measure, if the set of all continuous functions is included into the sigma-algebra, in particular, when Borel sets are used rather than Baire sets). Boris Tsirelson (talk) 20:42, 9 January 2017 (UTC)

I am not disagreeing. My understanding of Wikipedia is that you need to support every claim, even if the proof of something is 'more than evident'. Of course you can't cite every step of reasoning, but if the reader is unfamiliar with the material eg modifications etc, I don't think we can say this and that is simple or evident. Improbable keeler (talk) 13:50, 10 January 2017 (UTC)

I am just uncomfortable with writing any sentence without one or two citations. I did want to put an example in this section. There's a typical example of two different stochastic processes being equal in distribution I've seen a couple of times (I think it uses sup of the process), but I can't give a citation right now. Improbable keeler (talk) 13:50, 10 January 2017 (UTC)

We can change it to "Two stochastic processes that are modifications of each other have the same finite-dimensional distributions".Improbable keeler (talk) 13:50, 10 January 2017 (UTC) Improbable keeler (talk) 13:50, 10 January 2017 (UTC)

Further definitions: Modification
Oops, I did not note this yesterday... "some authors use the term version when the above condition is met, but the two stochastic processes X and Y are defined on different probability spaces" — no, this could not happen, since "the above condition" involves the equality $$X_t=Y_t$$ between two functions... defined on different sets?? Boris Tsirelson (talk) 20:50, 9 January 2017 (UTC)


 * Mmm I think Yor and... somebody else wrote that. Probably Revuz. Perhaps I misquoted/misunderstood them.Improbable keeler (talk) 13:40, 10 January 2017 (UTC)


 * OK, Revuz and Yor use the term version when two stochastic processes defined on different probability spaces have the same finite-dimensional distributions, and not the above expression. So I'll try to correct that now. Improbable keeler (talk) 21:10, 13 January 2017 (UTC)

Finite-dimensional probability distributions
"...each set $$t_i$$ is a non-empty finite subset of the index set $$T$$, so each $$t_i\subset T$$, which means that $$t_1,\dots,t_n$$ is any finite collection of subsets of the index set $$T$$..." — Does someone understand this phrase?? Each $$t_i$$ is a point, not a set. Finite collection of subsets? Where? I see a finite collection of points, that is, a finite subset. Or, if we mean all such objects together, then an infinite collection of (finite) subsets. And of course, $$t_i\in T.$$ Boris Tsirelson (talk) 20:45, 8 January 2017 (UTC)

"the Cartesian power $$S^n=S_1\times\dots \times S_n$$" — either Cartesian power $$S^n=S\times\dots \times S,$$ or Cartesian product $$S^n=S_1\times\dots \times S_n$$ (if we have more than one S). Boris Tsirelson (talk) 20:48, 8 January 2017 (UTC)


 * Corrected. I chose Cartesian power $$S^n=S\times\dots \times S,$$.Improbable keeler (talk) 21:16, 13 January 2017 (UTC)

Classifications
"...if the index set T of a stochastic process has a finite or countable number of elements ... then the stochastic process is said to be in discrete time, and the process is sometimes called a random sequence..." — I doubt that is called so if T is (for example) the (countable!) set of all rational numbers. Boris Tsirelson (talk)


 * Good point! Revised. Improbable keeler (talk) 21:28, 13 January 2017 (UTC)

Definitions
"...indexed by or depends on some mathematical set..." — well, "indexed by a set" sounds good, but "depends on a set" does not; rather, depends on an element of the set? Say, the measure of a set depends indeed on the set; but here we do not mean a set function. Boris Tsirelson (talk) 18:54, 8 January 2017 (UTC)


 * "I agree that "indexed by a set". But I read the other phrases somewhere -- whatever the citation I give -- so I wanted to offer two possible ways for people to interpret it. Improbable keeler (talk) 13:35, 10 January 2017 (UTC)

"...defined on a common probability space $$(P, {\cal F}, \Omega)$$..." — I am used to $$(\Omega, {\cal F}, P)$$; how usual is this inverse order? Boris Tsirelson (talk) 19:00, 8 January 2017 (UTC)


 * Actually, I think you're right with the order. You always define the space first.Improbable keeler (talk) 13:35, 10 January 2017 (UTC)

"...space $$S$$, which must be measurable with respect to some $$\sigma$$-algebra..." — rather, S is endowed with the sigma-algebra (and some subsets of S are measurable w.r.t the sigma-algebra). Boris Tsirelson (talk) 19:04, 8 January 2017 (UTC)

"Collection" may mean "set" or "family"; here family is meant, not set. Boris Tsirelson (talk) 19:07, 8 January 2017 (UTC)


 * I thought all those words are used interchangeably, though sometimes people prefer one over the other. (Recalling Halmos' Naive Set Theory)Improbable keeler (talk) 13:35, 10 January 2017 (UTC)

"Each random variable ... and, consequently, the stochastic process ... are actually functions of the two variables" — no, each random variable is a function of one variable. Boris Tsirelson (talk) 19:15, 8 January 2017 (UTC)


 * True. Revised.Improbable keeler (talk) 21:37, 13 January 2017 (UTC)

Lévy process
"...the corresponding increments are all identically distributed and independent of each other..." — oops, no; they are always independent, but not identically distributed unless the time intervals are of equal length. Boris Tsirelson (talk) 20:07, 8 January 2017 (UTC)


 * Good point. Thinking how to phrase that without using mathematics. Improbable keeler (talk) 13:33, 14 January 2017 (UTC)

Autoregressive and moving average processes
The autoregressive and moving average processes are types of stochastic processes that are used to model discrete-time empirical time series data, especially in economics. The autoregressive process or model treats a stochastic variable as depending on its own prior values and on a current independently and identically distributed stochastic term. The moving average model treats a stochastic variable as depending on the current and past values of an iid stochastic variable.


 * This section was added to the main article. I have put it here until/if some issues can be resolved. a) Complete lack of citations b) inconsistency with the rest of the article (eg stochastic variable, whereas the rest of the article uses random variable) c) Are they important/fundamental enough in the theory of stochastic processes to be listed here? They are important tools in statistics (not just economics, though the GARCH got the attention of the Swedish Bank, giving Engel a Nobel Prize) d) Stochastic process or statistical model? I am not quite sure what to call the AR and MA models. They are statistical models based on the assumptions of an underlying random process -- but a random process in the everyday sense. Improbable keeler (talk) 09:11, 8 January 2018 (UTC)