z****e
发帖数: 702 | 1 有一个kernel funcition，值域在[0,1]之间。
K(x*,x_k), 其中x*为定点，x_k是从概率分布f(x)中独立同分布draw出来的。
{x_k}从1到无穷。
这样将无穷个x_k点对应的K(x*,x_k)记做矢量K.
问：若使矢量K在L1和L2空间内，那么kernel function K和概率f应分别满足什么条件？
呼唤大牛，有包子酬谢。 | a***s
发帖数: 616 | 2 This seems impossible except some degenerated cases.
Let $Y_k := K(x^*, X_k)$. Since $\{X_k\}_{k \geq 1}$ are i.i.d., $\{Y_k\}_{k
\geq 1}$ are also i.i.d. Since the value of the kernal is between 0 and 1,
$Y_k$'s are nonnegative. Then Borel-Cantelli lemma says $\sum_{k=1}^{\infty}
Y_k = \infty$ with probability one unless $Y_k = 0$ with probability one.
Therefore for the infinite sequence $K$ to be in $l^1$ with positive probabi
lity (one, actually), we must have $Y_k = 0$ with probability one. This mean
s the range of $X_k$ is not in the support of $K$.
件？
【在 z****e 的大作中提到】
: 有一个kernel funcition，值域在[0,1]之间。
: K(x*,x_k), 其中x*为定点，x_k是从概率分布f(x)中独立同分布draw出来的。
: {x_k}从1到无穷。
: 这样将无穷个x_k点对应的K(x*,x_k)记做矢量K.
: 问：若使矢量K在L1和L2空间内，那么kernel function K和概率f应分别满足什么条件？
: 呼唤大牛，有包子酬谢。
| z****e
发帖数: 702 | 3 用borel-cantelli引理，似乎证明的是事件概率的和吧，
此处是随机变量的和。
{k
,
infty}
probabi
mean
【在 a***s 的大作中提到】
: This seems impossible except some degenerated cases.
: Let $Y_k := K(x^*, X_k)$. Since $\{X_k\}_{k \geq 1}$ are i.i.d., $\{Y_k\}_{k
: \geq 1}$ are also i.i.d. Since the value of the kernal is between 0 and 1,
: $Y_k$'s are nonnegative. Then Borel-Cantelli lemma says $\sum_{k=1}^{\infty}
: Y_k = \infty$ with probability one unless $Y_k = 0$ with probability one.
: Therefore for the infinite sequence $K$ to be in $l^1$ with positive probabi
: lity (one, actually), we must have $Y_k = 0$ with probability one. This mean
: s the range of $X_k$ is not in the support of $K$.
:
: 件？
| a***s
发帖数: 616 | 4 Let $\{ X_k \}_{k \geq 1}$ be i.i.d. random variables such that they are non
negative and $\Pr( X_1 > 0 ) > 0$.
In particular, we can assume $\Pr( X_1 > c ) = \delta > 0$ with some nonrand
om positive constants $c$ and $\delta$.
Now B-C lemma tells us that with probability one, events $\{ X_k > c \}$ hap
pens infinitely many times. Thus with probability one, $\sum_{k=1}^{\infty}
X_k = \infty$.
【在 z****e 的大作中提到】
: 用borel-cantelli引理，似乎证明的是事件概率的和吧，
: 此处是随机变量的和。
:
: {k
: ,
: infty}
: probabi
: mean
| z****e
发帖数: 702 | 5 en, 你这个是对的。
non
nonrand
hap
}
【在 a***s 的大作中提到】
: Let $\{ X_k \}_{k \geq 1}$ be i.i.d. random variables such that they are non
: negative and $\Pr( X_1 > 0 ) > 0$.
: In particular, we can assume $\Pr( X_1 > c ) = \delta > 0$ with some nonrand
: om positive constants $c$ and $\delta$.
: Now B-C lemma tells us that with probability one, events $\{ X_k > c \}$ hap
: pens infinitely many times. Thus with probability one, $\sum_{k=1}^{\infty}
: X_k = \infty$.
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