grandes-ecoles 2024 Q6

grandes-ecoles · France · mines-ponts-maths1__psi Continuous Probability Distributions and Random Variables Integrability, Boundedness, and Regularity of Density/Distribution-Related Functions

Let $t \in \mathbf{R}_{+}$. Show that if $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$, then $P_{t}(f) \in C^{0}(\mathbf{R})$. Also show that $P_{t}(f)$ is bounded in absolute value by a polynomial function in $|x|$ independent of $t$. Deduce that $P_{t}(f) \in L^{1}(\varphi)$.

Let $t \in \mathbf{R}_{+}$. Show that if $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$, then $P_{t}(f) \in C^{0}(\mathbf{R})$. Also show that $P_{t}(f)$ is bounded in absolute value by a polynomial function in $|x|$ independent of $t$. Deduce that $P_{t}(f) \in L^{1}(\varphi)$.

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