Let X1, X2, …be independent, equidistributed random...

Let $X_{1}, X_{2}, \ldots$ be independent, equidistributed random variables, and set $S_{n}=X_{1}+\cdots+X_{n}, n \geq 1$. The sequence $\left\{S_{n}, n \geq 0\right\}$ (where $S_{0}=0$ ) is called a random walk. Consider the following "perturbed" random walk. Let $\left\{\varepsilon_{n}, n \geq 1\right\}$ be a sequence of random variables such that, for some fixed $A>0$, we have $P\left(\left|\varepsilon_{n}\right| \leq A\right)=1$ for all $n$, and set $$ T_{n} \equiv S_{n}+\varepsilon_{n}, \quad n=1,2, \ldots $$ Suppose that $E X_{1}=\mu$ exists. Show that the law of large numbers holds for the perturbed random walk, $\left\{T_{n}, n \geq 1\right\}$.

Let $X_{1}, X_{2}, \ldots$ be independent, equidistributed random variables, and set $S_{n}=X_{1}+\cdots+X_{n}, n \geq 1$. The sequence $\left\{S_{n}, n \geq 0\right\}$ (where $S_{0}=0$ ) is called a random walk. Consider the following "perturbed" random walk. Let $\left\{\varepsilon_{n}, n \geq 1\right\}$ be a sequence of random variables such that, for some fixed $A>0$, we have $P\left(\left|\varepsilon_{n}\right| \leq A\right)=1$ for all $n$, and set
$$
T_{n} \equiv S_{n}+\varepsilon_{n}, \quad n=1,2, \ldots
$$
Suppose that $E X_{1}=\mu$ exists. Show that the law of large numbers holds for the perturbed random walk, $\left\{T_{n}, n \geq 1\right\}$.

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Key Concepts

Perturbation of Random Walk

Perturbation in a random walk involves modifying the process by adding an additional sequence of random variables, often subject to certain bounds. If these perturbations are uniformly bounded, their effect vanishes in the limit when normalized by the number of steps, thereby preserving the convergence results established by the law of large numbers.

Law of Large Numbers

The law of large numbers is a key theorem in probability theory which states that, as the number of trials increases, the sample average converges to the expected value of the underlying random variable. This result ensures the stability and predictability of long-term averages in random processes.

Random Walk

A random walk is a sequence of partial sums of independent and identically distributed random variables. It serves as a fundamental model in probability theory for describing a path that consists of a succession of random steps, and is widely used in fields such as physics, economics, and statistics.

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