grandes-ecoles 2019 Q3

grandes-ecoles · France · centrale-maths2__mp Probability Generating Functions Characteristic function product or trigonometric identity

Let $n$ be a non-zero natural number and $t$ be a real number. We have $\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right)$ and $\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}$.
Determine the pointwise limit of the sequence of functions $\left(\Phi_{X_n}\right)_{n \geqslant 1}$.

Let $n$ be a non-zero natural number and $t$ be a real number. We have $\Phi_{X_n}(t) = \prod_{k=1}^{n} \cos\left(\frac{t}{2^k}\right)$ and $\sin\left(\frac{t}{2^n}\right) \Phi_{X_n}(t) = \frac{\sin(t)}{2^n}$.

Determine the pointwise limit of the sequence of functions $\left(\Phi_{X_n}\right)_{n \geqslant 1}$.

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