Show that limn →∞ P(|Xn|>ϵ)=0 for every ϵif and only if given any ϵ, there exists n0(ϵ) such tha

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Show that limn →∞ P(|Xn|>ϵ)=0 for every ϵif and only if...

Show that $\lim _{n \rightarrow \infty} P\left(\left|X_{n}\right|>\epsilon\right)=0$ for every $\epsilon$ if and only if given any $\epsilon$, there exists $n_{0}(\epsilon)$ such that $$ P\left(\left|X_{n}\right|>\epsilon\right)<\epsilon \text { for } n>n_{0}(\epsilon) $$ This is also equivalent to: given any $\delta$ and $\epsilon$, there exists $n_{0}(\delta, \epsilon)$ such that $$ P\left(\left|X_{n}\right|>\epsilon\right)<\delta \text { for } n>n_{0}(\delta, \epsilon) . $$ [Hint: consider $\epsilon^{\prime}=\delta \wedge \epsilon$ and apply the first form.

Show that $\lim _{n \rightarrow \infty} P\left(\left|X_{n}\right|>\epsilon\right)=0$ for every $\epsilon$ if and only if given any $\epsilon$, there exists $n_{0}(\epsilon)$ such that
$$
P\left(\left|X_{n}\right|>\epsilon\right)<\epsilon \text { for } n>n_{0}(\epsilon)
$$
This is also equivalent to: given any $\delta$ and $\epsilon$, there exists $n_{0}(\delta, \epsilon)$ such that
$$
P\left(\left|X_{n}\right|>\epsilon\right)<\delta \text { for } n>n_{0}(\delta, \epsilon) .
$$
[Hint: consider $\epsilon^{\prime}=\delta \wedge \epsilon$ and apply the first form.

Key Concepts

Equivalence of Asymptotic Conditions

In the context of convergence in probability, different forms of the definition—such as ensuring that P(|X?| > ?) becomes arbitrarily small or directly less than any given threshold after some index—are shown to be equivalent. This equivalence shows that multiple natural formulations of the concept capture the same limiting behavior, reinforcing the robustness of the notion of convergence in probability in theoretical and applied settings.

Convergence in Probability

This concept refers to the idea that a sequence of random variables {X?} converges to a limit (often a constant) if for every ? > 0, the probability that X? deviates from the limit by more than ? tends to 0 as n tends to infinity. It is a fundamental mode of convergence in probability theory and underlies many results in statistics and stochastic processes.

Epsilon-Delta Formulation in Probability

This formulation involves translating the idea of convergence into a language involving ? (and sometimes ?) to quantify how small the tail probability can be made as the sequence progresses. Instead of just saying that the limit probability is zero, we express that for any predetermined threshold (? or ?), there is a stage in the sequence beyond which the deviation probability is kept below that threshold. This type of formulation is crucial for rigorous definitions and proofs in analysis and probability.

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