Seriesvaries

Seriesvaries is a term used in some mathematical and data-analytic discussions to describe a class of numerical series in which one or more defining characteristics vary with the index. It is not a standard, formally defined term in most textbooks, but it serves as a convenient label for phenomena where the behavior of a series changes over time or across terms.

Definition and scope

In a broad sense, a seriesvaries refers to a numeric series {a_n} in which the generating rule or the statistical properties of the terms do not remain constant as n grows. This can involve deterministic changes in coefficients, piecewise or time-dependent definitions, or stochastic variability such as a changing variance. Because the concept is informal, seriesvaries can encompass several concrete forms, from nonhomogeneous recurrence relations to time series with non-constant volatility.

Examples

A deterministic example is a_n = c_n b_n with c_n varying with n, such as a_n = (1 + 0.1n)^{-1} sin(n). A stochastic example is a_n = sigma_n z_n, where z_n are standardized random variables and sigma_n changes with n, producing time-varying variance.

Relation to established concepts

Seriesvaries is closely related to non-stationary behavior and heteroscedasticity in time series analysis. It highlights cases where convergence properties, variance, or dependency structures are not fixed, which contrasts with stationary or homoskedastic models. In formal settings, related frameworks include time-varying parameter models and volatility models like GARCH, which address changing variance explicitly.

See also

Series, sequence, non-stationary process, heteroscedasticity, time-varying parameter models, volatility modeling.