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Results tagged with pr.probability
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user 317937
Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory.
0
votes
Submartingales bounded in $L^p$, $p>1$
A stronger condition that does guarantee $L^p$ convergence for submartingales is as follows:
$\sup_k \{E|M_k|^p + E|A_k|^p\} < \infty.$
where $X_k = M_k + A_k$ is Doob's decomposition.
For a proof, …
- 625
1
vote
$L^p$-convergence of submartingale
Here is one alternative set of conditions (sufficient but I am not sure if they are necessary). Essentially, I move the $L^p$-uniform integrability requirement of $X_k$ to the predictable component $A …
- 625
1
vote
Accepted
Covering number of the conditional distribution function
You need a different approach. Each function in your function space can be written as
$$F_{Y|W}(y|W) = \int 1(s \leq y) P(Y = ds|W)$$
for some $y$. Thus,
$$\|F_{Y|W}(y_2|W) - F_{Y|W}(y_1|W)\|_{L^1} = …
- 625