Show that $Z_n$ converges in probability to $\frac{1}{\lambda}$.

Question

Let $(X_n)_n$ a sequence of iid random variables $(\sim \exp(\lambda), \lambda >0)$ and $F$ their common distribution function. Let $$M_n = \sup(X_1,\cdots, X_n) \text{ and } Z_n=\frac{M_n}{\ln(n)}$$

1. Show that $F_{Z_n}(x)=(1−e^{−λx \ln(n)})^n \chi_{\mathbb R^+}(x)$

2. Show that $Z_n$ converges in probability to $\frac{1}{\lambda}$

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For the first question : \begin{align*} F_{Z_n}(x)&=P(Z_n\leq x)\\&=P(\sup(X_1,\cdots,X_n) \leq x\ln(n)) \\&=P(X_i \leq x\ln(n) : \forall i=1:n) \\&=\prod_{i=1}^n P(X_i \leq x\ln(n)) \\&=\prod_{i=1}^n (1-e^{-\lambda x\ln(n)})\chi_{\mathbb R^+}(x) \\&=(1-e^{-\lambda x \ln(x)})^n \chi_{\mathbb R^+}(x)\end{align*} For the second question, I don't know how to show it! Any help is appreciated !

Answer 1

For convergence in probability, you must show that $$ \Pr\left\{|Z_n-\frac{1}{\lambda}|>\epsilon\right\}<\epsilon, $$ so, find the probability $\Pr\left\{\frac{1}{\lambda}-\epsilon<Z_n<\frac{1}{\lambda}+\epsilon\right\}$ and proceed.