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# 数学代写|概率论代考Probability Theory代写|GENERAL ORIENTATION. THE REFLECTION PRINCIPLE

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## 数学代写|概率论代考Probability Theory代写|GENERAL ORIENTATION. THE REFLECTION PRINCIPLE

From a formal point of view we shall be concerned with arrangements of finitely many plus ones and minus ones. Consider $n=p+q$ symbols $\epsilon_1, \ldots, \epsilon_n$, each standing either for +1 or for -1 ; suppose that there are $p$ plus ones and $q$ minus ones. The partial sum $s_k=\epsilon_1+\cdots+\epsilon_k$ represents the difference between the number of pluses and minuses occurring at the first $k$ places. Then
$$s_k-s_{k-1}=\epsilon_c= \pm 1, \quad s_0=0, \quad s_n=p-q,$$
where $k=1,2, \ldots, n$.
We shall use a geometric terminology and refer to rectangular coordinates $t, x$; for definiteness we imagine the $t$-axis is horizontal, the $x$-axis vertical. The arrangement $\left(\epsilon_1, \ldots, \epsilon_n\right)$ will be represented by a polygonal line whose $k$ th side has slope $\epsilon_k$ and whose $k$ th vertex has ordinate $s_k$. Such lines will be called paths.

Definition. Let $n>0$ and $x$ be integers. A path $\left(s_1, s_2, \ldots, s_n\right)$ from the origin to the point $(n, x)$ is a polygonal line whose vertices have abscissas $0,1, \ldots, n$ and ordinates $s_0, s_1, \ldots, s_n$ satisfying (1.1) with $s_n=x$.

We shall refer to $n$ as the length of the path. There are $2^n$ paths of length $n$. If $p$ among the $\epsilon_k$ are positive and $q$ are negative, then
$$n=p+q, \quad x=p-q .$$

## 数学代写|概率论代考Probability Theory代写|RANDOM WALKS: BASIC NOTIONS AND NOTATIONS

The ideal coin-tossing game will now be described in the terminology of random walks which has greater intuitive appeal and is better suited for generalizations. As explained in the preceding example, when a path $\left(s_1, \ldots, s_\rho\right)$ is taken as record of $\rho$ successive coin tossings the partial sums $s_1, \ldots, s_\rho$ represent the successive cumulative gains. For the geometric description it is convenient to pretend that the tossings are performed at a uniform rate so that the $n$th trial occurs at epoch ${ }^6 n$. The successive partial sums $s_1, \ldots, s_n$ will be marked as points on the vertical $x$-axis; they will be called the positions of a “particle” performing a random walk. Note that the particle moves in unit steps, up or down, on a line. A path represents the record of such a movement. For example, the path from $O$ to $N$ in figure 1 stands for a random walk of six steps terminating by a return to the origin.

Each path of length $\rho$ can be interpreted as the outcome of a random walk experiment; there are $2^\rho$ such paths, and we attribute probability $2^{-\rho}$ to each. (Different assignments will be introduced in chapter XIV. To distinguish it from others the present random walk is called symmetric.)
We have now completed the definition of the sample space and of the probabilities in it, but the dependence on the unspecified number $\rho$ is disturbing. To see its role consider the event that the path passes through the point $(2,2)$. The first two steps must be positive, and there are $2^{\rho-2}$ paths with this property. As could be expected, the probability of our event therefore equals $\frac{1}{4}$ regardless of the value of $\rho$. More generally, for any $k \leq \rho$ it is possible to prescribe arbitrarily the first $k$ steps, and exactly $2^{\rho-k}$ paths will satisfy these $k$ conditions. It follows that an event determined by the first $k \leq \rho$ steps has a probability independent of $\rho$. In practice, therefore, the number $\rho$ plays no role provided it is sufficiently large. In other words, any path of length $n$ can be taken as the initial section of a very long path, and there is no need to specify the latter length. Conceptually and formally it is most satisfactory to consider unending sequences of trials, but this would require the use of nondenumerable sample spaces. In the sequel it is therefore understood that the length $\rho$ of the paths constituting the sample space is larger than the number of steps occurring in our formulas. Except for this we shall be permitted, and glad, to forget about $\rho$.

# 概率论代写

## 数学代写|概率论代考Probability Theory代写|GENERAL ORIENTATION. THE REFLECTION PRINCIPLE

$$s_k-s_{k-1}=\epsilon_c= \pm 1, \quad s_0=0, \quad s_n=p-q,$$

$$n=p+q, \quad x=p-q .$$

## 数学代写|概率论代考Probability Theory代写|RANDOM WALKS: BASIC NOTIONS AND NOTATIONS

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