Law of large numbers about non identical random sequence.

Asked Jul 29 '20 at 05:56

Active Jul 29 '20 at 05:56

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I want to show the following statement.

Statement :

$Z \to 0 $, when $N \to \infty $.

where $Z = \frac{{\left| {{e^{j{\theta _1}}}\left| {{x_1}} \right| + {e^{j{\theta _2}}}\left| {{x_2}} \right| + ,..., + {e^{j{\theta _N}}}\left| {{x_N}} \right|} \right|}}{N}$ and ${\theta _n}\sim {\text{uniform}}\left[ {0,2\pi } \right]$.

$x_n$ is not random variable but just sequence (it means that it does not have any specific distribution.)

$x_n$ is bounded such that $\left| {{x_n}} \right| < B < \infty $.

I think the law of large numbers can be used but I cannot prove that clearly.

Thanks for your help!

asked Jul 29 '20 at 05:56

rasidz

1

What type of convergence are you taking about? Are $\theta_i$'s independent? – Kavi Rama Murthy Jul 29 '20 at 06:01
yes, $\theta_n$ are i.i.d – rasidz Jul 29 '20 at 06:13
Are you asking for weak law or strong law? Convergence in probability or convergence almost surely? – Kavi Rama Murthy Jul 29 '20 at 06:17
I mean weak law, almost surely, wanna show $Z$ goes to zero – rasidz Jul 29 '20 at 06:19
1

You have uniform bound on $x_i$, so just approach the same way as you would with bounded fourth moment. – user10354138 Jul 29 '20 at 06:20
1

Weak law does not talk about almost sure convergence. – Kavi Rama Murthy Jul 29 '20 at 06:23

Law of large numbers about non identical random sequence.

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