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Let $\{X_i\}_{i=0}^{\infty}$ be an i.i.d. Bernoulli sequence with $P[X_i=1]=P[X_i=0]=1/2$. Consider: $$\{X_1, X_0, X_2, X_1, X_3, X_2, X_4, X_3, X_5, X_4, X_6, X_5, ...\}$$

This looks like a "Wald equality" question. Define $Y=\sum_{i=1}^{T_n} X_i$. Then: \begin{align} 1 + x_{max} \geq Y =\sum_{i=1}^{\infty} X_i1\{T_n\geq i\}\\ \end{align} where $1\{T_n\geq i\}$ is a …

Let $\{X_1, X_2, \dotsc, X_n\}$ be independent and identically distributed (i.i.d.) random variables sampled from a common distribution with density $f_{\theta}(x)$, where $\theta$ is an unknown param …

For each $i \in \{1, 2, 3, ...\}$ define $Y_i = X_i^{-\alpha/2}$. Define $Z_k$ by: $$Z_k = \left[ 1 + \left( \frac{1}{1+\sum_{i=1}^kY_i} \right)^{\alpha/2} \right]^{k+1} $$ Note that $1 \leq Y_i < …