Abstract
In this chapter the mathematical foundations of statistics and stochastic processes are presented. Probability spaces, probability measures and densities as well as random variables are defined and examples for these mathematical structures in physics are given. The maximum entropy principle is presented as the basis to obtain a priori probabilities for a given problem. The central limit theorem is discussed and the theory of extreme value distributions is presented. For the theory of stochastic processes we focus on Markov processes and Martingales as these classes of processes underlie all applications discussed in later chapters. The master equation, the Fokker-Planck equation and Itô stochastic differential equations are discussed as the equations of motion for these processes.
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Notes
- 1.
Actually when information (or lack thereof) is measured in bits, the logarithm to basis 2 instead of the natural logarithm needs to be used, but this just leads to a constant shift in (2.20).
- 2.
Generally, ‘iff’ is taken to mean ‘if and only if’, a necessary and sufficient condition.
- 3.
A non-degenerate distribution is the opposite of a degenerate distribution. A distribution of a random variable is called degenerate, if it is a delta function, i.e., it is 1 for one value of the random variable and 0 otherwise. The corresponding cumulative distribution is the Heaviside step function.
References
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970)
L. Arnold, Stochastic Differential Equations: Theory and Applications (Wiley, New York, 1974)
N.H. Bingham, R. Kiesel, Risk-Neutral Valuation (Springer, London, 1998)
L. de Haan, A. Ferreira, Extreme Value Theory: An Introduction (Springer, Berlin, 2006)
E.B. Dynkin, Markov Processes, vol. 1 (Springer, Berlin, 1965)
E.B. Dynkin, Markov Processes, vol. 2 (Springer, Berlin, 1965)
P. Embrechts, C. Klüppelberg, T. Mikosch, Modelling Extremal Events for Insurance and Finance (Springer, Berlin, 2008)
W. Feller, An Introduction to Probability Theory and its Applications, vol. 1 (Wiley, New York, 1966)
W. Feller, An Introduction to Probability Theory and its Applications, vol. 2 (Wiley, New York, 1966)
R. Frigg, C. Werndl, Entropy—a guide for the perplexed, in Probabilities in Physics, ed. by C. Beisbart, S. Hartmann (Oxford University Press, Oxford, 2011)
C.W. Gardiner, Stochastic Methods: A Handbook for the Natural and Social Sciences (Springer, Berlin, 2009)
B.W. Gnedenko, A.N. Kolmogorov, Limit Distributions for Sums of Independent Random Variables (Addison-Wesley, Reading, 1954)
P. Gregory, Bayesian logical data analysis for the physical sciences: a comparative approach with mathematica support (Cambridge University Press, Cambridge, 2005)
P. Hänggi, H. Thomas, Stochastic processes: time evolution, symmetries and linear response. Phys. Rep. 88, 207 (1982)
J. Honerkamp, Statistical Physics (Springer, Berlin, 1998)
N. Ikeda, S. Watanabe, M. Fukushima, H. Kunita, Itô’s Stochastic Calculus and Probability Theory (Springer, Tokyo, 1996)
K. Itô, H.P. McKean Jr., Diffusion Processes and Their Sample Paths (Springer, Berlin, 1996)
E.T. Jaynes, Information theory and statistical mechanics. Phys. Rev. 106, 620 (1957)
E.T. Jaynes, Information theory and statistical mechanics. II. Phys. Rev. 108, 171 (1957)
E.T. Jaynes, Probability Theory: The Logic of Science (Cambridge University Press, Cambridge, 2003)
N.G. van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1985)
I. Karatzas, S.E. Shreve, Brownian motion and stochastic calculus (Springer, Berlin, 1999)
A. Khintchine, Korrelationstheorie der stationären stochastischen Prozesse. Math. Ann. 109, 604 (1934)
P.E. Kloeden, E. Platen, Numerical Solution of Stochastic Differential Equations (Springer, Berlin, 1995)
A. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung. Ergebnisse der Math., vol. 2 (Springer, Berlin, 1933)
H.C. Öttinger, Stochastic Processes in Polymeric Fluids (Springer, Berlin, 1996)
I. Oppenheim, K. Shuler, G.H. Weiss, The Master Equation (MIT Press, Cambridge, 1977)
L.E. Reichl, A Modern Course in Statistical Physics (VCH, Weinheim, 2009)
H. Risken, The Fokker-Planck Equation: Methods of Solution and Applications (Springer, Berlin, 1996)
C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379 (1948). See also Bell Syst. Tech. J. 27, 623 (1948)
A. Wehrl, General properties of entropy. Rev. Mod. Phys. 50, 221 (1978)
D. Williams, Probability with Martingales (Cambridge University Press, Cambridge, 1997)
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Paul, W., Baschnagel, J. (2013). A Brief Survey of the Mathematics of Probability Theory. In: Stochastic Processes. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00327-6_2
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