Norm-like Function

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Norm-like Function

Definition (Norm-like function)

A function $V$ is called norm-like (or Lyapunov) if $V (x) → ∞$ as $x → ∞$ : this means that the sublevel sets $\{x : V (x) ≤ r\}$ are Precompact for each $r > 0$ .

an equivalent definition is as follows:

Definition (E.7)

Let $X$ be a Metric Space. A nonnegative Measurable Function $v$ on $X$ is said to be a moment if there exists a nondecreasing sequence of Compact sets $K_{n}\uparrow X$ such that $\lim_{ n \to \infty } \inf_{x\not\in K_{n}}v(x)=\infty$

Proposition (E.8)

Let $\mathcal{P}$ be a family of Probability Measures on a Metric Space $X$ . If there exists a $v$ on $X$ such that $\sup_{\mu \in \mathcal{P}}\int\limits v \, d\mu <\infty$ then $\mathcal{P}$ is Tight.

or an even stronger Lemma:

Lemma (D.5.3)

A sequence of probabilities $\{\nu_{k}\}_{k\ge1}$ is Tight if and only if there exists a Norm-like Function $V$ such that $\limsup_{ k \to \infty } \nu_{k}(V)<\infty$
If for each $x ∈ X$ there exists a Norm-like Function $V_{x}( · )$ on $X$ such that $\limsup_{ k \to \infty } \mathbb{E}_{x}[V_{x}(\Phi_{k})]<\infty$ then the chain is bounded in probability.

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Norm-like Function