Distribution Functions and Their Basic Properties, Helly-Bray Type Results

Let $F: \mathbb {R} \rightarrow [0,1]$ be nondecreasing, right continuous, $F(-\infty ) = 0$ and $F(\infty ) = 1$ (i.e., a d.f. of a r.v.), and let $F^{-1}$ be defined by

Next, consider the probability space $(\Omega , \mathcal{A}, P) = \left([0,1], \mathcal{B}_{[0,1]}, P\right)$, where $P = \lambda$ is the Lebesgue measure, and on $\Omega$, define the function $X$ by $X(\omega ) = F^{-1}(\omega )$. Then show that $X$ is a r.v. and that its d.f. is $F$.

Let $F_{n}, n \geq 1$, be d.f.s of r.v.s and suppose that $F_{n} \underset {n \rightarrow \infty }{\Rightarrow } F$ where $F$ is a d.f. with $\operatorname {Var}\left[F\right] = 1$. Then show that $F$ is the d.f. of a r.v.

Show that the Weak Compactness Theorem (Theorem 5) still holds, if condition 1 in Definition 1 is replaced by: $(0 \leq ) F(x) \leq B, x \in \mathbb {R}, n \geq 1$, for some $B < \infty$.

Show that Theorems $6-8$ hold true if, for all $x \in \mathbb {R}, 0 \leq F(x) \leq B$ and $0 \leq F_{n}(x) \leq B, n = 1,2, \ldots$, for some $B > 0$, as indicated in Remark 6.

For $n = 1,2, \ldots$, consider the r.v.s $X_{n}, X$ and $Y$ such that $\left|X_{n}\right| \leq Y, \mathbb {E}\left[Y^{r}\right] < \infty$ (for some $r > 0$), and $X_{n} \xrightarrow [n \rightarrow \infty ]{d} X$. Then show that $\mathbb {E}\left[\left|X_{n}\right|^{r}\right] \underset {n \rightarrow \infty }{\longrightarrow } \mathbb {E}\left[\left|X\right|^{r}\right]$.

Let $X$ be a r.v. with d.f. $F$; i.e., $F(x) = \mathbb {P}\left(X \leq x\right), x \in \mathbb {R}$. Then show that

In the following expression, determine the constants $\alpha$ and $\beta$, so that the function $F$ defined is the d.f. of a r.v.

For $n = 1,2, \ldots$, let $F_{n}$ and $F$ be d.f.s such that $F_{n} \xrightarrow [n \rightarrow \infty ]{c} F$, and let $F$ be continuous. Then show that $F_{n}(x) \underset {n \rightarrow \infty }{\longrightarrow } F(x)$ uniformly in $x \in \mathbb {R}$.

Let $X$ be a r.v. with continuous d.f. $F_{X}$, and define the r.v. $Y$ as follows: $Y = X$ if $X \leq C$, and $Y = C$ if $X > C$, where $C$ is a constant. Then

By a simple example demonstrate that it is possible that $F_{n}, n \geq 1$, are d.f.s of r.v.s with $F_{n} \underset {n \rightarrow \infty }{\Rightarrow } F$, a d.f., but not that of a r.v.

For $n \geq 1$, let $X_{n}$ and $X$ be r.v.s defined on the probability space $(\Omega , \mathcal{A}, P)$, and suppose that, as $n \rightarrow \infty , \mathbb {E}\left[\left|X_{n}\right|\right] \rightarrow \mathbb {E}\left[\left|X\right|\right] < \infty$ and $X_{n} \xrightarrow {d} X$. Then show that $\left|X_{n}\right|, n \geq 1$, are uniformly integrable.

For $n \geq 1$, let $F_{n}$ and $F$ be d.f.s of r.v.s, let $f_{n}$ be real-valued measurable functions defined on $\mathbb {R}$, let $f: \mathbb {R} \rightarrow \mathbb {R}$ be continuous, and let $g: \mathbb {R} \rightarrow [0, \infty )$ be continuous. Assume that: