Implications of an expected value on almost sure convergence

Mathematics Asked by qp212223 on September 8, 2020

The question I’m trying to solve is shown below – from a practice qualifying exam in probability which has no posted solution (self study). I really don’t know where to start (even for part (a)) and am looking for some major help.

Am I supposed to apply one of Kolmogorov’s two/three series theorem? Thanks for any ideas you have!

b) For $$phi in (0, 1/e)$$ you can modify @Michael's hint and note that if

$$sum_{n=1}^infty P(log^+(Z) > rn) < infty$$ for any $$r > 0$$, then $$mathbb{E}[log^+(Z)] < infty$$, which means you can just modify your calculation in the comment for $$phi in (0, 1/e)$$ by

$$sum_{n=1}^infty P(log^+(Z) > rn) = sum_{n=1}^infty P(Z > e^{rn} )$$ and now pick $$r$$ s.t. $$e^r = 1/phi$$.

c) Provided the indexing on $$Z_t$$s is from $$-infty$$ to $$infty$$, yes, this sequence is strictly stationary. To see that, note that shifting the indices of $$Z_i$$s does not change anything about the joint distribution of finitely many $$X_{t_1}, X_{t_2}, ..., X_{t_n}$$ and the joint distribution does not depend on the absolute values of $$t_1, t_2, ..., t_n$$, but only their differences.

I think$$^*$$ sequence, since $$phi$$ is a random variable, is not ergodic. To see that, formally, we have

$$lim_{n rightarrow infty} frac{1}{2n} sum_{t=-n}^{n} X_n = lim_{n rightarrow infty} frac{1}{2n} sum_{t=-n}^{n} sum_{j=0}^infty phi^j Z_{t-j} = lim_{nrightarrow infty} frac{1}{2n} sum_{k=-infty}^{infty} sum_{j= max(|k| - n, 0) }^{|k| + n} phi^j Z_k rightarrow frac{1}{1 - phi} mathbb{E} Z$$

where we can swap sums because everything is non-negative. Since this limit is not trivial provided $$phi$$ is not, by the ergodic theorem, this sequence is not ergodic.

$$*$$: I am not fully sure; in particular, I am very unsure about the limit. I am quite sure it isn't ergodic though, since the limit of this running average is going to depend on the value of $$phi$$.

Correct answer by E-A on September 8, 2020

Related Questions

Can Laplace’s transformation be equal to a Gaussian for any integer?

2  Asked on November 2, 2021

Is there analytical solution to this heat equation?

1  Asked on November 2, 2021 by titanium

Proving Threshold Properties of a Dynamic Programming Problem

0  Asked on November 2, 2021

Assumptions in converting between nominal/effective interest/discount

1  Asked on November 2, 2021 by minyoung-kim

How can I prove that 3 planes are arranged in a triangle-like shape without calculating their intersection lines?

7  Asked on November 2, 2021

Detailed analysis of the secretary problem

1  Asked on November 2, 2021 by saulspatz

Are all finite-dimensional algebras of a fixed dimension over a field isomorphic to one another?

6  Asked on November 2, 2021 by perturbative

Why does the plot of $f(x)=|cos x|-|sin x|$ look almost piecewise linear?

2  Asked on November 2, 2021 by meowdog

Excluded middle, double negation, contraposition and Peirce’s law in minimal logic

2  Asked on November 2, 2021 by lereau

Does iterating the complex function $zmapstofrac{2sqrt z}{1+z}$ always converge?

3  Asked on November 2, 2021 by mr_e_man

If an infinite set $S$ of positive integers is equidistributed, is $S+S$ also equidistributed?

1  Asked on November 2, 2021 by vincent-granville

How to evaluate $int frac{dx}{sin(ln(x))}$?

6  Asked on November 2, 2021

$lfloorfrac12+frac1{2^2}+frac1{2^3}+cdotsrfloor;$ vs $;lim_{ntoinfty}lfloorfrac12+frac1{2^2}+cdots+frac1{2^n}rfloor$

2  Asked on November 2, 2021 by drift-speed

Finding the Center of Mass of a disk when a part of it is cut out.

6  Asked on November 2, 2021

Do functions with the same gradient differ by a constant?

4  Asked on November 2, 2021

What loops are possible when doing this function to the rationals?

2  Asked on November 2, 2021 by user808945

Is there an explicit construction of this bijection?

2  Asked on November 1, 2021 by gregory-j-puleo

How can I determine the radius of 4 identical circles inside an equilateral triangle $ABC$?

5  Asked on November 1, 2021 by user766881

Prove that $tan^{-1}frac{sqrt{1+x^2}+sqrt{1-x^2}}{sqrt{1+x^2}-sqrt{1-x^2}}=frac{pi}{4}+frac 12 cos^{-1}x^2$

4  Asked on November 1, 2021

Why is my value for the length of daylight wrong?

2  Asked on November 1, 2021 by user525966