Abstract
This chapter provides the mathematical framework and example illustrations of stochastic processes as families of random variables with values in some measurable (state) space S, such as the integers or the real line or higher dimensional Euclidean space, and indexed by some set Λ. Examples include i.i.d. sequences, random walks, Brownian motion, Poisson processes, branching processes, queue processes, Markov processes, and various martingale processes. Special emphasis is given to existence and constructions of these important classes of stochastic processes.
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Notes
- 1.
For a proof using Caratheodory’s extension theorem see BCPT p. 168. Another elegant proof due to Edward Nelson using the Riesz Representation theorem may be found in Nelson (1959), or BCPT, p. 169.
- 2.
For a general statement and proof of Tulcea’s theorem, see Neveu (1965), pp. 162–167.
- 3.
See Gikhman and Skorokhod (1969), pp. 159–169.
- 4.
See Caratheodory’s extension theorem in BCPT p. 226.
- 5.
The theory of martingales is primarily due to Doob (1953).
- 6.
Under some extra conditions on the offspring distribution, it is shown in Agresti (1975) that the condition \(\sum _{n=0}^\infty ({\mathbb E}Y_n)^{-1} = \infty \) is both necessary and sufficient for extinction in the non-homogeneous case.
References
Agresti A (1975) On the extinction times of varying and random environment branching processes. J Appld Probab 12(1):39–46.
Bhattacharya R, Waymire E (2016) A basic course in probability theory. Springer universitext series, 2nd edn. Springer, Berlin. (ERRATA: http://sites.science.oregonstate.edu/~waymire/)
Doob JL (1953) Stochastic processes. Wiley, New York.
Gikhman II, Skorokhod AV (1969) An introduction to the theory of random processes. W. B. Saunders, Philadelphia.
Neveu J (1965) Mathematical foundations of the calculus of probability. San Francisco, Holden-Day.
Nelson E (1959) Regular probability measures on function space. Ann Math 630–643.
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Bhattacharya, R., Waymire, E.C. (2021). What Is a Stochastic Process?. In: Random Walk, Brownian Motion, and Martingales. Graduate Texts in Mathematics, vol 292. Springer, Cham. https://doi.org/10.1007/978-3-030-78939-8_1
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