Lecture

Continuous-Time Stochastic Processes: Ergodicism Examples

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Description

This lecture introduces continuous-time stochastic processes and provides examples illustrating ergodicism. The examples involve different functions of time and random variables, showcasing the concept of ergodicity in stochastic processes.

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Ontological neighbourhood

Statistics

Statistical inference: Mathematical statistics

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Random variable

A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. The term 'random variable' can be misleading as it is not actually random nor a variable, but rather it is a function from possible outcomes (e.g., the possible upper sides of a flipped coin such as heads and tails ) in a sample space (e.g., the set ) to a measurable space (e.g., in which 1 corresponding to and −1 corresponding to ), often to the real numbers.

Exchangeable random variables

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Complex random variable

In probability theory and statistics, complex random variables are a generalization of real-valued random variables to complex numbers, i.e. the possible values a complex random variable may take are complex numbers. Complex random variables can always be considered as pairs of real random variables: their real and imaginary parts. Therefore, the distribution of one complex random variable may be interpreted as the joint distribution of two real random variables.

Independent and identically distributed random variables

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Convergence of random variables

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