INDEPENDENT ACADEMIC REFERENCE · stochastic-series-lab.org
The mathematics of probability and stochastic series
Stochastic Series Lab is an independent academic reference devoted to mathematical statistics, the theory of stochastic time series, bivariate density matrices and the Gaussian normal law. The pages that follow are written for students and researchers who value careful notation, honest derivations and a calm, theoretical voice.
This portal is a theoretical academic reference on mathematical statistics and probability theory. It is not affiliated with any standards body or institution. No files, datasets or products are distributed or hosted. For educational and scientific study only; no advisory or consulting services are provided
Why a laboratory of series?
A series in this reference is an ordered family of random variables, written X1, X2, …, and studied as a single mathematical object. Such a family may describe the sequence of daily temperatures at a station, the successive deviations of a pendulum from rest, or the fluctuations of a measured voltage. The early chapters build the probability vocabulary that makes these objects tractable: sample spaces, events, random variables, expectation, variance and the density function. Only after that vocabulary is steady do we turn to the dependence between observations, which is where the theory of series truly begins.
Our treatment is deliberately theoretical. We do not present datasets, we do not provide numerical recipes for any particular field, and we do not recommend procedures for any particular setting. Instead we give the reader the shapes of the arguments: the theorem, the lemma, the asymptotic statement, the inequality. A reader who understands these shapes can later approach any specialised text with confidence.
The four movements of the reference
The reference is arranged in five states that a reader may visit in any order. State 01 gathers the foundations of probability: the measure that assigns to each event a number between zero and one, the discipline of conditioning, and the algebra of independent events. State 02 introduces the Gaussian law and the matrix language of joint density, covariance and correlation. State 03 studies processes and series, including white noise, random walks, the moving-average and autoregressive families, and the spectral view of dependence. State 04 examines dynamics proper: the Markov chain with its transition matrix, and the continuous-time Wiener process. State 05 presents reference material, estimation theory, inference and tests, a glossary, and a note on the editorial desk.
Foundations, in brief
Every argument in this reference rests on one primitive idea: a probability measure assigns a number to each event, and that assignment obeys a short list of axioms. From the axioms we derive the rule of the complement, the addition rule for disjoint events, and the definition of conditional probability. Independence is then a statement about the product of measures, and Bayes’ rule is nothing more than a change of conditioning. These four ideas, taken together, carry the whole edifice. The foundations page develops them with the patience they deserve.
After the foundations, the random variable enters as a measurable function from the sample space to the real line. Its distribution is the push-forward of the measure, and its density, where it exists, is the local weight of that distribution. Expectation is the integral of the variable against its distribution, variance is the expectation of the squared deviation from the mean, and the higher moments complete the portrait of the law. Generating functions package all the moments into a single object and thereby turn many manipulation problems into algebra.
An excerpt from the reader’s questions
Does a density always exist?
No. Densities belong to the absolutely continuous family. A discrete law places its mass on a countable set and is summarised by a probability mass function instead. The cumulative distribution function, by contrast, exists for every law, and it is often the safest starting point.
Why does the Gaussian law appear so often?
Partly because of the central limit theorem, which describes the behaviour of sums of many small, weakly dependent contributions, and partly because the Gaussian family is closed under linear operations, which makes it unusually convenient.
Is stationarity the same as independence?
Not at all. Independence forbids any dependence at every lag. Stationarity merely asks that the probabilistic character of the family not shift as time advances. A stationary series may still carry strong dependence from one observation to the next.
We invite the reader to move next to probability foundations, or to browse the glossary whenever a term needs a quick restatement.