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Problem 978

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Random Walk Skewness

In this problem we consider a random walk on the integers $\mathbb{Z}$, in which our position at time $t$ is denoted as $X_t$.

At time $0$ we start at position $0$. That is, $X_0=0$.

At time $1$ we jump to position $1$. That is, $X_1=1$.

Thereafter, at time $t=2,3,\dots$ we make a jump of size $|X_{t-2}|$ in either the positive or negative direction, with probability $1/2$ each way. If $X_{t-2}=0$ we stay put at time $t$.

At $t=5$ we find our position $X_5$ has the following distribution:
$$
X_5=\begin{cases}
-1\quad&\text{with probability }3/8\\
1\quad&\text{with probability }3/8\\
3\quad&\text{with probability }1/8\\
5\quad&\text{with probability }1/8\\
\end{cases}
$$
The standard deviation $\sigma$ of a random variable $X$ with mean $\mu$ is defined as
$$\sigma=\sqrt{\mathbb{E}[X^2]-\mu^2}$$
Furthermore the skewness of $X$ is defined as
$$\text{Skew}(X)=\mathbb{E}\biggl[\Bigl(\frac{X-\mu}{\sigma}\Bigr)^3\biggr]$$
For $X_5$, which has mean $1$ and standard deviation $2$, we find $\text{Skew}(X_5)=0.75$. You are also given $\text{Skew}(X_{10})\approx2.50997097$.

Find $\text{Skew}(X_{50})$. Give your answer rounded to eight digits after the decimal point.

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随机游走的偏度

在本题中，我们考虑整数集$\mathbb{Z}$上的一个随机游走，并将我们在时刻$t$的位置记为$X_t$。

在时刻$0$，我们从位置$0$出发，即$X_0=0$。

在时刻$1$，我们跳到位置$1$，即$X_1=1$。

此后，在时刻$t=2,3,\dots$，我们以$|X_{t-2}|$为步长，各$1/2$的概率向正方向或负方向跳跃；如果$X_{t-2}=0$，则在时刻$t$保持不动。

在$t=5$时，所在位置$X_5$的分布如下：
$$
X_5=\begin{cases}
-1\quad&\text{概率为}3/8\\
1\quad&\text{概率为}3/8\\
3\quad&\text{概率为}1/8\\
5\quad&\text{概率为}1/8\\
\end{cases}
$$
记随机变量$X$的均值为$\mu$，则其标准差$\sigma$的定义是
$$\sigma=\sqrt{\mathbb{E}[X^2]-\mu^2}$$
进而$X$的偏度的定义是
$$\text{Skew}(X)=\mathbb{E}\biggl[\Bigl(\frac{X-\mu}{\sigma}\Bigr)^3\biggr]$$
对于$X_5$，其均值为$1$，标准差为$2$，可得$\text{Skew}(X_5)=0.75$。已知$\text{Skew}(X_{10})\approx2.50997097$。

求$\text{Skew}(X_{50})$，并四舍五入保留八位小数作为你的答案。

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