What is the Uniform Convergence Problem in Stochastic Functions?

In summary, the speaker is seeking help with a problem involving the convergence of a stochastic function, with the goal of showing that the expected value of this function approaches a known value. They mention using the triangle inequality and adding/subtracting terms, but are unsure of how to approach the expectation aspect. They ask for any possible approaches to solving this problem and express gratitude for any assistance.
  • #1
St41n
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First of all, hello all this is my first post i think. Congratulations on this great community
Please move my post if I'm not posting on the right forum and I'm sorry for any inconvenience.

I have this problem that I need to solve and I don't have a clue. I hope you could give me some ideas.

I need to show this: [tex]\mathop {\sup }\limits_\theta \left\| {E_\theta \left( {\hat \beta _T } \right) - b\left( \theta \right)} \right\| \to 0[/tex]

knowing that:
[tex]\hat \beta _T \stackrel{T\rightarrow\infty}{\rightarrow} b\left( {\theta _0 } \right)[/tex]

where [tex]\hat \beta _T[/tex] is a stochastic function of [tex]y_T[/tex] that comes from a distribution with true parameter [tex]\theta_0[/tex]

θ and β belongs in a compact subset of R^p and R^q respectively.

The convergence apparently is non-stochastic as we've taken expectation.
A hint is to add and subtract something into the norm and use the triangle inequality to show the above claim. But, I have no idea how to treat the expectation.

I haven't supplied all info there is, but please tell me if you can think of any possible approaches for this. Any help is much appreciated
 
Last edited:
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  • #2
I get the feeling you should add and subtract [tex] \hat \beta_T[/tex] and use its convergence and the strong law of large numbers maybe.
 

What is the definition of uniform convergence?

Uniform convergence is a type of convergence in which a sequence of functions converges to a limiting function in such a way that the rate of convergence is independent of the point in the domain. In other words, the convergence is uniform across the entire domain of the functions.

Why is uniform convergence important?

Uniform convergence is important because it allows us to interchange the order of limits and integrals, which is crucial in many mathematical and scientific applications. It also ensures that the limiting function is continuous, which is a desirable property in many fields.

How is uniform convergence different from pointwise convergence?

Pointwise convergence is a weaker form of convergence in which the convergence only needs to hold at each individual point in the domain, rather than uniformly across the entire domain. Uniform convergence implies pointwise convergence, but the converse is not necessarily true.

What are some common examples of uniform convergence?

Some common examples of uniform convergence include the sequence of functions f_n(x) = x^n on the interval [0,1], which converges uniformly to the function f(x) = 0. Another example is the sequence of functions f_n(x) = n^2x(1-x^2)^n on the interval [0,1], which converges uniformly to the function f(x) = 0.

How can uniform convergence be tested?

Uniform convergence can be tested using a variety of methods, such as the Weierstrass M-test, the Cauchy criterion, or the Dini's theorem. These tests involve checking the behavior of the sequence of functions at various points in the domain, and if certain conditions are met, then the convergence is deemed to be uniform.

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