Question

Let \(X\) be a random variable satisfying (a) \(E[X] \leq m<0\), and (b) \(\operatorname{Pr}\\{-1 \leq X \leq+1\\}=1\). Suppose \(X_{1}, X_{2}, \ldots\) are jointly distributed random variables for which the conditional distribution of \(X_{n+1}\) given \(X_{1}, \ldots, X_{n}\) always satisfies (a) and (b). Let \(S_{n}=X_{1}+\cdots+X_{n}\left(S_{0}=0\right)\) and for \(a

Short answer

The short answer to the problem is: In this specific setup, given the conditions (a) and (b) on the sequence \(X_i\), we can calculate the bound on the expected value of \(T_a\) as follows: $$ E[T_a] \leq (1 + x - a) / |m|, \quad a < x $$

Step-by-step solution

Define Notations and Variables

Define random variables in terms of given data: 1. \(X_i\): the \(i\)-th random variable in the sequence 2. \(S_n\): the sum of the first \(n\) random variables, given by \(S_n = X_1 + \cdots + X_n\) 3. \(T_a\): the smallest \(n\) such that \(x + S_n \leq a\) 4. \(m\): a value related to the expected value of each \(X_i\), satisfying \(E[X] \leq m < 0\)

Understand the Meaning of Given Conditions (a) and (b)

Condition (a) states that the expected value of each \(X_i\) is less than \(m < 0\). This means each \(X_i\) has a negative expected value, which implies that they will mostly have negative values. Condition (b) states that the probability of each \(X_i\) being within the interval \([-1,1]\) is \(1\). This means each \(X_i\) will always take a value within the interval \([-1,1]\).

Analyze Sequence of Sums: \(S_n\)

As mentioned above, since each \(X_i\) has a negative expected value, the sums of individual variables in the sequence \(S_n\) will show a mostly negative trend (more negative values are contributed with each subsequent \(X_i\)). Now, condition (b) ensures that \(X_i\) will always take a value within the interval \([-1,1]\). Thus, the sequence \(S_n = X_1 + \cdots + X_n\) will always take values within the sum of these intervals.

Estimate a Bound on \(E[T_a]\)

Now, we need to estimate a bound on the expected value of \(T_a\). Observe that \(T_a\) is a random variable that measures the minimum number of steps to move from \(x\) to a point less than or equal to \(a\). Since we are moving to the left (due to negative expected values of \(X_i\)), the distance we move in each step is bounded between \((1-m)\) and \(1\). At each step, we move to the left a distance of at least \(|m|\). Therefore, \(|m|\) can be considered as the average distance moved to the left per step. To move from \(x\) to a point less than or equal to \(a\), on average, there should be at most \((x-a) / |m|\) steps (some steps might not be as large as \(|m|\)). Considering that at least one step is needed to move from \(x\) to \(a\), we get the following inequality for the number of steps: $$ T_{a} \leq (1 + x - a) / |m| $$

Apply Expectation on Both Sides of the Inequality

Now, we will apply the expectation to both sides of the inequality to find the bound on \(E[T_a]\): $$ E[T_a] \leq E[(1 + x - a) / |m|] $$ Since \((1 + x - a) / |m|\) is a constant, \(E[(1 + x - a) / |m|] = (1 + x - a) / |m|\). Thus, $$ E[T_a] \leq (1 + x - a) / |m|, \quad a < x $$ This is the desired inequality.