📂Probability Theory

Convergence of Probabilities Defined by Measure Theory

Probability Convergence Defined Rigorously

Given a probability space $( \Omega , \mathcal{F} , P)$.

A sequence of random variables $\left\{ X_{n} \right\}_{n \in \mathbb{N}}$ is said to converge in probability to a random variable $X$ if it converges in measure to $X$, denoted as $X_{n} \overset{P}{\to} X$.

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• If you’re not yet familiar with measure theory, the term probability space can be disregarded.

Explanation

The convergence of $\left\{ X_{n} \right\}_{n \in \mathbb{N}}$ to $X$ means, for all $\varepsilon > 0$, $$\lim_{n \to \infty} P \left( \left\{ \omega \in \Omega : | X_{n}(\omega) - X(\omega) | \ge \varepsilon \right\} \right) = 0$$ and in a more familiar form, it can be shown as follows: $$\lim_{n \to \infty} P \left( | X_{n}(\omega) - X(\omega) | < \varepsilon \right) = 1$$ Since a sequence of random variables is a stochastic process, it is inferred to be useful in the theory of stochastic processes.

Properties of probability convergence from measure convergence:

• [3] If $X_{n}$ converges to $X$ almost surely, it implies probability convergence.
• [4] If $X_{n}$ converges to $X$ in $\mathcal{L}_{p}$ convergence, it implies probability convergence.

Since probability $P$ is a measure, it inherits the properties of measure convergence.

See Also

• Probability Convergence Defined in Mathematical Statistics
• Almost Sure Convergence $\implies$ Probability Convergence $\implies$ Distribution Convergence
• $\mathcal{L}_{p}$ Convergence $\implies$ Probability Convergence