Some Modes of Convergence of Sequences of Random Variables and their Relationships

If $X_{n} \xrightarrow [n \rightarrow \infty ]{\mu } X$, then show directly (that is, without reference to other results) that: $\left(X_{n}-X\right)^{+} \xrightarrow [n \rightarrow \infty ]{\mu } 0,\left(X_{n}-X\right)^{-} \xrightarrow [n \rightarrow \infty ]{\mu } 0, X_{n}^{+} \xrightarrow [n \rightarrow \infty ]{\mu } X^{+}$ and $X_{n}^{-} \xrightarrow [n \rightarrow \infty ]{\mu } X^{-}$.

By means of examples, show that:

For any sequence of events $\left\{ A_{n}\right\} , n \geq 1$, show that $\sum_{n = 1}^{\infty } \mathbb {P}\left(A_{n}\right) < \infty$ implies $\mathbb {P}\left(\limsup_{n \rightarrow \infty } A_{n}\right) = 0$.

The sequence $\left\{ X_{n}\right\} , n \geq 1$, of r.v.s is said to converge completely to 0 , if, for every $\varepsilon > 0, \sum_{n = 1}^{\infty } \mathbb {P}\left(\left|X_{n}\right| \geq \varepsilon \right) < \infty$.

Show that $X_{n} \xrightarrow [n \rightarrow \infty ]{P} X$ if and only if, for every $\left\{ n^{\prime }\right\} \subseteq \left\{ n\right\}$, there exists a further subsequence $\left\{ n^{\prime \prime }\right\} \subseteq \left\{ n^{\prime }\right\}$ such that $X_{n^{\prime \prime }} \xrightarrow [n^{\prime \prime } \rightarrow \infty ]{\text{a.s.}} X$ and any two limiting r.v.s are a.s. equal.

By means of an example, show that $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.s. }} c \neq 0$ need not imply that $\mathbb {P}\left(X_{n} \neq 0\right) = 1$ for all $n$.

If for some $\varepsilon_{n} > 0$ with $\sum_{n = 1}^{\infty } \varepsilon_{n} < \infty$, it holds that $\sum_{n = 1}^{\infty } \mathbb {P}\left(\left|X_{n+1}-X_{n}\right| \geq \varepsilon_{n}\right) < \infty$, then show that $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.s. }}$ to a r.v.

For $n = 1,2, \ldots$, let $X_{n}$ and $X$ be (real-valued) r.v.s and let $g: \mathbb {R} \rightarrow \mathbb {R}$ be continuous. Then show that $X_{n} \xrightarrow [n \rightarrow \infty ]{P} X$ implies $g\left(X_{n}\right) \xrightarrow [n \rightarrow \infty ]{P} g(X)$ (so that continuity preserves convergence in probability).

For $n = 1,2, \ldots$, let $X_{n}, Y_{n}, X$, and $Y$ be (real-valued) r.v.s and let $g: \mathbb {R}^{2} \rightarrow \mathbb {R}$ be continuous. Then show that $X_{n} \xrightarrow [n \rightarrow \infty ]{P} X$ and $Y_{n} \xrightarrow [n \rightarrow \infty ]{P} Y$ implies $g\left(X_{n}, Y_{n}\right) \xrightarrow [n \rightarrow \infty ]{P} g(X, Y)$ (so that continuity preserves convergence in probability).

Show that $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.s. }} X$ if and only if there is a sequence $0 < \varepsilon_{n} \xrightarrow [n \rightarrow \infty ]{\rightarrow } 0$ such that

(Egorov's Theorem). Show that, if $\mu$ is finite, then $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.e. }} X$ implies that $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.u. }} X$.

Show that the complement $A^{c}$ of the set $A$ of (pointwise) convergence of a sequence of r.v.s $\left\{ X_{n}\right\} , n \geq 1$, is expressed as follows: $A^{c} = \cup_{r, s}\left\{ \omega \in \Omega \right.$; $\lim_{n \rightarrow \infty } \inf^{n} \left.X_{n}(\omega ) \leq r < s \leq \limsup_{n \rightarrow \infty } X_{n}(\omega )\right\}$, where the union is taken over all rationals $r$ and $s$ with $r < s$.

For $n = 1,2, \ldots$, let $X_{n}, X$ be r.v.s defined on the measurable space $(\Omega , \mathcal{A})$, and suppose that $X_{n} \underset {n \rightarrow \infty }{\longrightarrow } X$ (pointwise). Then show that

For a sequence of r.v.s $\left\{ X_{n}\right\} , n \geq 1$, show that the set $A = \left\{ \omega \in \Omega ; \sum_{n = 1}^{\infty } X_{n}(\omega ) \text{ converges}\right\}$ is in the $\sigma \left(X_{m}, X_{m+1}, \ldots \right)$ for every $m \geq 1$.

Let $X_{n}, n \geq 1$, be r.v.s defined on the measure space ($\Omega , \mathcal{A}, P$), and suppose that $\sum_{n = 1}^{\infty } \mathbb {P}\left(\left|X_{n}\right| > n\right) < \infty$. Then show that $\varlimsup_{n \rightarrow \infty } \frac{\left|X_{n}\right|}{n} \leq 1$ a.s.

For $n = 1,2, \ldots$, let $\mathbf{Z}_{n} = \left(Z_{n 1}, \ldots , Z_{n k}\right)$ and $\mathbf{Z} = \left(Z_{1}, \ldots , Z_{k}\right)$ be $k$ dimensional r.v.s. Then we say that $\mathbf{Z}_{n} \xrightarrow [n \rightarrow \infty ]{P} \mathbf{Z}$ if $\left\| \mathbf{Z}_{n}-\mathbf{Z}\right\| \underset {n \rightarrow \infty }{P} 0$, or $\left[\sum_{j = 1}^{k}\left(Z_{n j}-Z_{j}\right)^{2}\right] \underset {n \rightarrow \infty }{\xrightarrow {P}} 0$. Then show that $\mathbf{Z}_{n} \underset {n \rightarrow \infty }{P} \mathbf{Z}$ if and only if $Z_{n j} \xrightarrow [n \rightarrow \infty ]{P} Z_{j}, j = 1, \ldots , k$.

If $X_{n} \xrightarrow [n \rightarrow \infty ]{P} X$ and $g: \mathbb {R} \rightarrow \mathbb {R}$ is continuous, then $g\left(X_{n}\right) \xrightarrow [n \rightarrow \infty ]{P} g(X)$ (see Exercise 8). Show that this need not be true, if $g$ is not continuous.

If $X_{n} \xrightarrow [n \rightarrow \infty ]{\mu } X$ and $\left|X_{n}\right| \leq Y$ a.e. $n \geq 1$, then show that $\left|X\right| \leq Y$ a.e. with respect to the measure $\mu$.

Refer to Exercise 13 and continue as follows: For $n = 1,2, \ldots$, let $X_{n}, X$ be r.v.s defined on the measure space $(\Omega , \mathcal{A}, \mu )$, and suppose that $X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.e. }} X$. Then, by means of concrete examples, show that:

Show that the convergence $X_{m}-X_{n} \xrightarrow {\text{ a.e. }} 0$ as $m, n \rightarrow \infty$ is equivalent to the convergence $X_{n+v}-X_{n} \xrightarrow [n \rightarrow \infty ]{\text{ a.e. }} 0$ uniformly in $v \geq 1$.

Show that, for every $\varepsilon > 0: \mu \left(\left|X_{m}-X_{n}\right| \geq \varepsilon \right) \rightarrow 0$ as $m, n \rightarrow \infty$ if and only if $\mu \left(\left|X_{n+v}-X_{n}\right| \geq \varepsilon \right) \underset {n \rightarrow 0}{\longrightarrow } 0$ uniformly in $v \geq 1$.