Please forgive me if it is a simple answer or if I didn't get something completely, because I don't understand various types of convergence clearly. In a theorem, I can prove convergence in probability that say $\hat{\theta}_n \rightarrow \theta$ as $n\rightarrow \infty$. But what I need is the following \begin{eqnarray} \vert \hat{\theta}_{n}-\theta \vert \lt \vert \hat{\theta}_{n_0}-\theta \vert,\qquad \forall n \geq n_0, \text{possibly with probability} \quad 1-\epsilon \end{eqnarray} What kind of convergence, such as convergence in probability, convergence with probability one or other types of convergence implies this inequality and how to reach from that type of convergence to the above inequality?
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Kind of looks like a Fejer monotone sequence. – Jürgen Sukumaran May 20 '18 at 01:05
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Depending on how I choose to interpret it, it seems either to be an equivalent way of writing regular convergence, or perhaps Cauchy convergence, but it also looks like the ratio test for convergence. – CogitoErgoCogitoSum May 20 '18 at 01:10
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@CogitoErgoCogitoSum, actually What I need is a type of probability convergence. I don't quite understand the relation of these type of convergence with convergence in probability. I think it is regular convergence in probability. – user85361 May 20 '18 at 01:15