Mediaspace scheduled maintenance: Aug 25, 2026 07:00 - 12:00 AM. During this time, videos will be temporarily unavailable. Check status updates.

Lecture

Continuous-Time Stochastic Processes: Stationarity

Description

This lecture covers the concept of weak-sense stationarity for continuous-time stochastic processes, discussing properties such as autocovariance functions, cross-correlation, and joint weak-sense stationarity. It also explores the relationship between different processes and their stationarity.

This video is available exclusively on Mediaspace for a restricted audience. Please log in to MediaSpace to access it if you have the necessary permissions.

Watch on Mediaspace

Official source

Related concepts (27)

Stationary process

In mathematics and statistics, a stationary process (or a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose unconditional joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance also do not change over time. If you draw a line through the middle of a stationary process then it should be flat; it may have 'seasonal' cycles around the trend line, but overall it does not trend up nor down.

Cross-correlation

In signal processing, cross-correlation is a measure of similarity of two series as a function of the displacement of one relative to the other. This is also known as a sliding dot product or sliding inner-product. It is commonly used for searching a long signal for a shorter, known feature. It has applications in pattern recognition, single particle analysis, electron tomography, averaging, cryptanalysis, and neurophysiology. The cross-correlation is similar in nature to the convolution of two functions.

Autocovariance

In probability theory and statistics, given a stochastic process, the autocovariance is a function that gives the covariance of the process with itself at pairs of time points. Autocovariance is closely related to the autocorrelation of the process in question. With the usual notation for the expectation operator, if the stochastic process has the mean function , then the autocovariance is given by where and are two instances in time.

Correlation

In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistics it usually refers to the degree to which a pair of variables are linearly related. Familiar examples of dependent phenomena include the correlation between the height of parents and their offspring, and the correlation between the price of a good and the quantity the consumers are willing to purchase, as it is depicted in the so-called demand curve.

Stochastic process

In probability theory and related fields, a stochastic (stəˈkæstɪk) or random process is a mathematical object usually defined as a sequence of random variables, where the index of the sequence has the interpretation of time. Stochastic processes are widely used as mathematical models of systems and phenomena that appear to vary in a random manner. Examples include the growth of a bacterial population, an electrical current fluctuating due to thermal noise, or the movement of a gas molecule.

Related lectures (33)

Time Series: Fundamentals and Models

Explores the fundamentals of time series analysis, including stationarity, linear processes, forecasting, and practical aspects.

Stochastic Processes: Basics & Stationarity

Covers signal generation, statistical relations, stationarity, white noise, and orthogonality in stochastic processes.

Describing Data: Statistics and Hypothesis Testing

Covers descriptive statistics, hypothesis testing, and correlation analysis with various probability distributions and robust statistics.

Dependence Concepts and Copulas

Explores dependence concepts, copulas, correlation fallacies, and rank correlations in statistics.

Time Series: Linear Filtering and Spectral Estimation