Convergence in the Mean and Convergence of Means

Prove that convergence in the mean of order $\alpha > 0$ implies convergence in the mean of order $\beta$ for every $\beta \in (0, \alpha )$.

Prove that convergence in the mean of some order implies convergence in probability.

Give an example confirming that convergence in the mean of order $\alpha$ does not imply, in general, almost sure convergence.

Give an example confirming that convergence in probability does not imply, in general, convergence in the mean of order $\alpha$.

Give an example confirming that almost sure convergence does not imply, in general, convergence in the mean of order $\alpha$.

Prove that a sequence of random variables $\xi_n$ converges in the mean of order $\alpha$ if and only if the sequence $\left|\xi_n\right|^{\alpha }$ is uniformly integrable and $\xi_n \xrightarrow {p} \xi$.

Let $\sum_{n=1}^{\infty } \mathbb {E}\left[\left|\xi_n - \xi \right|^{\alpha }\right] < \infty$. Prove that $\xi_n \rightarrow \xi$ a.s.

Let $\left|\xi_n\right| \leqslant c < \infty$ a.s. for all $n$, and let $\xi_n \rightarrow \xi$ in probability as $n \rightarrow \infty$. Prove that $\mathbb {E}\left[\left|\xi_n - \xi \right|^{\alpha }\right] \rightarrow 0$ for every $\alpha > 0$.

Let $\xi , \xi_1, \xi_2, \ldots$ be nonnegative random variables such that $\xi_n \rightarrow \xi$ in probability and $\mathbb {E}\left[\xi_n\right] \rightarrow \mathbb {E}\left[\xi \right] < \infty$ as $n \rightarrow \infty$. Prove that $\mathbb {E}\left[\left|\xi_n - \xi \right|\right] \rightarrow 0$.

Let $\alpha > 0$ and $\mathbb {E}\left[\left|\xi_n\right|^{\alpha }\right] < \infty$ for all $n$. Prove that the following statements are equivalent:

1. $\xi_n \rightarrow \xi$ in probability and $\mathbb {E}\left[\left|\xi_n\right|^{\alpha }\right] \rightarrow \mathbb {E}\left[\left|\xi \right|^{\alpha }\right] < \infty$ as $n \rightarrow \infty$;

2. $\mathbb {E}\left[\left|\xi_n - \xi \right|^{\alpha }\right] \rightarrow 0$ as $n \rightarrow \infty$.

Let the family $\{ \xi_n\}$ of random variables be uniformly integrable. Prove that $\sup_n \mathbb {E}\left[\left|\xi_n\right|\right] < \infty$.

Let the sequence $\{ \xi_n\}$ of random variables be such that $\sup_n \mathbb {E}\left[\left|\xi_n\right|^{1+\delta }\right] < \infty$ for some $\delta > 0$. Prove that this sequence is uniformly integrable.

Let $\{ \xi_n\}$ be a sequence of random variables. Prove that the following conditions are equivalent:

1. the family $\{ \xi_n\}$ is uniformly integrable;

2. there exists a nonnegative function $g(x)$ such that $g(x) \rightarrow \infty$ as $x \rightarrow \infty$ and $\sup_n \mathbb {E}\left[\left|\xi_n\right| g(\left|\xi_n\right|)\right] < \infty$.

Let the sequence $\{ \xi_n\}$ of random variables be such that $\mathbb {P}\left(\left|\xi_n\right| \geqslant x\right) \leqslant \mathbb {P}\left(\left|\eta \right| \geqslant x\right)$ for every $x > 0$, with $\mathbb {E}\left[\left|\eta \right|\right] < \infty$. Prove that $\{ \xi_n\}$ is uniformly integrable.

Let $\{ \xi_n\}$ be a sequence of nonnegative random variables with finite means such that the distribution of $\xi_n$ converges weakly to the distribution of a random variable $\xi$ and $\mathbb {E}\left[\xi_n\right] \rightarrow \mathbb {E}\left[\xi \right] < \infty$ as $n \rightarrow \infty$. Prove that this sequence is uniformly integrable.

Let $\xi_n \Rightarrow \xi$ as $n \rightarrow \infty$. Prove that the following statements are equivalent:

1. the sequence $\{ \xi_n\}$ is uniformly integrable;

2. $\mathbb {E}\left[\left|\xi_n\right|\right] < \infty$, $\mathbb {E}\left[\left|\xi \right|\right] < \infty$, and as $n \rightarrow \infty$

   $$\mathbb {E}\left[\max (0, \xi _n)\right] \rightarrow \mathbb {E}\left[\max (0, \xi )\right], \qquad \mathbb {E}\left[\min (0, \xi _n)\right] \rightarrow \mathbb {E}\left[\min (0, \xi )\right].$$

Let the sequences $\xi_n$ and $\eta_n$ be uniformly integrable. Prove that the following sequences are uniformly integrable:

Let the independent random variables $\xi_1, \xi_2, \ldots$ be identically distributed with finite mean; set $S_n = \xi_1 + \ldots + \xi_n$. Prove that the sequence of random variables $\{ S_n / n, n \geqslant 1\}$ is uniformly integrable.

Let $x \in \mathbb {R}$ and let $\{ \xi_n\}$ be a sequence of random variables with finite means such that $\mathbb {E}\left[\left|\xi_n\right|\right] \rightarrow \mathbb {E}\left[\left|\xi \right|\right]$ as $n \rightarrow \infty$. Determine whether this implies the convergence $\mathbb {E}\left[\left|\xi_n + x\right|\right] \rightarrow \mathbb {E}\left[\left|\xi + x\right|\right]$.

Let $\xi , \xi_1, \xi_2, \ldots$ be nonnegative random variables such that $\mathbb {E}\left[\xi_n\right] \rightarrow \mathbb {E}\left[\xi \right]$ and $\mathbb {P}\left(\xi > \xi_n + \varepsilon \right) \rightarrow 0$ as $n \rightarrow \infty$ for every $\varepsilon > 0$. Prove that $\mathbb {E}\left[\left|\xi_n - \xi \right|\right] \rightarrow 0$.

Let $\left|\xi_n - \xi \right| \leqslant \eta_n$ for every $n$, $\mathbb {E}\left[\left|\xi \right|\right] < \infty$, and $\mathbb {E}\left[\eta_n\right] \rightarrow 0$ as $n \rightarrow \infty$. Prove that $\xi_n \rightarrow \xi$ in probability and $\mathbb {E}\left[\xi_n\right] \rightarrow \mathbb {E}\left[\xi \right]$.

Let the random variables $\xi , \xi_1, \xi_2, \ldots$ be such that $\xi_n \Rightarrow \xi$ and $\operatorname {Var}\left[\xi_n\right] \rightarrow \operatorname {Var}\left[\xi \right]$. Prove that $\mathbb {E}\left[\xi_n\right] \rightarrow \mathbb {E}\left[\xi \right]$.

Let $\{ \xi_n\}$ be a uniformly integrable sequence of random variables such that the distribution of $\xi_n$ converges weakly to the distribution of a random variable $\xi$. Prove that $\mathbb {E}\left[\xi \right]$ exists and $\mathbb {E}\left[\xi_n\right] \rightarrow \mathbb {E}\left[\xi \right]$ as $n \rightarrow \infty$.

Let $\xi_n \Rightarrow \xi$. Prove that $\liminf_{n \rightarrow \infty } \operatorname {Var}\left[\xi_n\right] \geqslant \operatorname {Var}\left[\xi \right]$.

Let $\xi_n \xrightarrow {p} \xi$, with $\mathbb {E}\left[\left|\xi_n\right|\right] < \infty$ for all $n$, and suppose

Prove that the sequence of random variables $\{ \xi_n\}$ is uniformly integrable and $\mathbb {E}\left[\left|\xi_n - \xi \right|\right] \rightarrow 0$ as $n \rightarrow \infty$.

Let $\xi_n \xrightarrow {p} \xi$, with $\mathbb {E}\left[\left|\xi_n\right|\right] < \infty$ for all $n$, and suppose

Prove that the sequence of random variables $\{ \xi_n\}$ is uniformly integrable and $\mathbb {E}\left[(\xi_n - \xi )^2\right] \rightarrow 0$ as $n \rightarrow \infty$.

Prove that the series $\sum_{n=1}^{\infty } \xi_n$ of uncorrelated random variables converges in the mean square if and only if the series $\sum_{n=1}^{\infty } \mathbb {E}\left[\xi_n\right]$ and $\sum_{n=1}^{\infty } \operatorname {Var}\left[\xi_n\right]$ converge.