grandes-ecoles

Q1 Indefinite & Definite Integrals Integral Inequalities and Limit of Integral Sequences View

Show that $$I _ { n } \geqslant \frac { 1 } { 2 ^ { n } }.$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.

Q2 Indefinite & Definite Integrals Convergence and Evaluation of Improper Integrals View

Justify the existence of $K _ { n } = \int _ { 0 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$ and give the exact value of $K _ { 1 }$.

Q3 Indefinite & Definite Integrals Estimation or Bounding of a Sum View

Show that $$\int _ { 1 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t = O \left( \frac { 1 } { n 2 ^ { n } } \right) .$$ One may lower bound $1 + t ^ { 2 }$ by a polynomial of degree 1.

Q4 Indefinite & Definite Integrals Determine Asymptotic Behavior or Limits of Sequences Defined by Integrals View

Deduce that, as $n$ tends to $+ \infty$, $$I _ { n } \sim K _ { n } .$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$ and $K _ { n } = \int _ { 0 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.

Q5 Indefinite & Definite Integrals Derive a Reduction/Recurrence Formula via Integration by Parts View

Establish the recurrence relation $K _ { n } = K _ { n + 1 } + \frac { 1 } { 2 n } K _ { n }$, where $K _ { n } = \int _ { 0 } ^ { + \infty } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.

Q6 Indefinite & Definite Integrals Evaluate a Closed-Form Expression Using the Reduction Formula View

Using the recurrence relation $K _ { n } = K _ { n + 1 } + \frac { 1 } { 2 n } K _ { n }$ and the fact that $I_n \sim K_n$, deduce a simple equivalent of $I _ { n }$ as $n$ tends to $+ \infty$, where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.

Q7 Indefinite & Definite Integrals Substitution to Transform Integral Form (Show Transformed Expression) View

Justify that $$\sqrt { n } I _ { n } = \int _ { 0 } ^ { \sqrt { n } } \frac { 1 } { \left( 1 + u ^ { 2 } / n \right) ^ { n } } \mathrm {~d} u$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.

Q8 Indefinite & Definite Integrals Substitution to Evaluate Limit of an Integral Expression View

Show that $$\lim _ { n \rightarrow \infty } \sqrt { n } I _ { n } = \int _ { 0 } ^ { + \infty } \mathrm { e } ^ { - u ^ { 2 } } \mathrm {~d} u$$ where $I _ { n } = \int _ { 0 } ^ { 1 } \frac { 1 } { \left( 1 + t ^ { 2 } \right) ^ { n } } \mathrm {~d} t$.

Q9 Indefinite & Definite Integrals Substitution to Evaluate a Definite Integral (Numerical Answer) View

Deduce the values of $$\int _ { 0 } ^ { + \infty } \mathrm { e } ^ { - u ^ { 2 } } \mathrm {~d} u \quad \text { then of } \quad \int _ { - \infty } ^ { + \infty } \mathrm { e } ^ { - u ^ { 2 } / 2 } \mathrm {~d} u .$$

Q10 Indefinite & Definite Integrals Probability Inequality and Tail Bound Proof View

Let $x > 0$. By writing that $\varphi ( t ) \leqslant \frac { t } { x } \varphi ( t )$ for all $t \geqslant x$, show that $$\int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t \leqslant \frac { \varphi ( x ) } { x }$$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.

Q11 Differentiating Transcendental Functions Probability Inequality and Tail Bound Proof View

Let $x > 0$. Using the study of a well-chosen function, show that $$\frac { x } { x ^ { 2 } + 1 } \varphi ( x ) \leqslant \int _ { x } ^ { + \infty } \varphi ( t ) \mathrm { d } t$$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.

Q12 Normal Distribution Tail Bound or Concentration Inequality Proof View

Deduce a simple equivalent of $1 - \Phi ( x )$ as $x$ tends to $+ \infty$, where $\Phi ( x ) = \int _ { - \infty } ^ { x } \varphi ( t ) \mathrm { d } t$ and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.

Q13 Discrete Probability Distributions Event Expression and Partition View

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$. In the case where $n \geqslant 2$, define the events $$A _ { 1 } = \left\{ \left| R _ { 1 } \right| \geqslant 3 x \right\} \quad \text { and } \quad A _ { p } = \left\{ \max _ { 1 \leqslant i \leqslant p - 1 } \left| R _ { i } \right| < 3 x \right\} \cap \left\{ \left| R _ { p } \right| \geqslant 3 x \right\}$$ for $p \in \llbracket 2 , n \rrbracket$.
Express the event $A$ using the events $A _ { 1 } , A _ { 2 } , \ldots , A _ { n }$.

Q14 Discrete Probability Distributions Proof of a Probability Identity or Inequality View

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$, and $A_1, \ldots, A_n$ as defined in Q13.
Show that we have $$\mathbb { P } ( A ) \leqslant \mathbb { P } \left( \left\{ \left| R _ { n } \right| \geqslant x \right\} \right) + \sum _ { p = 1 } ^ { n } \mathbb { P } \left( A _ { p } \cap \left\{ \left| R _ { n } \right| < x \right\} \right) .$$

Q15 Discrete Probability Distributions Proof of Set Membership, Containment, or Structural Property View

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A_1, \ldots, A_n$ be as defined in Q13.
Justify that for all $p \in \llbracket 1 , n \rrbracket$, we have the inclusion $$A _ { p } \cap \left\{ \left| R _ { n } \right| < x \right\} \subset A _ { p } \cap \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} .$$

Q16 Discrete Probability Distributions Probability Inequality and Tail Bound Proof View

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$. Let $A$ denote the event $\left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\}$.
Deduce that $$\mathbb { P } ( A ) \leqslant \mathbb { P } \left( \left\{ \left| R _ { n } \right| \geqslant x \right\} \right) + \max _ { 1 \leqslant p \leqslant n } \mathbb { P } \left( \left\{ \left| R _ { n } - R _ { p } \right| > 2 x \right\} \right) .$$

Q17 Discrete Probability Distributions Probability Inequality and Tail Bound Proof View

In this subsection, $n$ is a non-zero natural number and $Z _ { 1 } , \ldots , Z _ { n }$ are discrete random variables independent on a probability space $(\Omega , \mathcal { A } , \mathbb { P })$. For all $p \in \llbracket 1 , n \rrbracket$, we denote $R _ { p } = \sum _ { i = 1 } ^ { p } Z _ { i }$.
Conclude that $$\forall x > 0 , \quad \mathbb { P } \left( \left\{ \max _ { 1 \leqslant p \leqslant n } \left| R _ { p } \right| \geqslant 3 x \right\} \right) \leqslant 3 \max _ { 1 \leqslant p \leqslant n } \mathbb { P } \left( \left\{ \left| R _ { p } \right| \geqslant x \right\} \right) .$$

Q18 Complex Numbers Arithmetic Direct Proof of a Stated Identity or Equality View

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$.
Compare the real numbers $- x _ { n , k }$ and $x _ { n , n - k }$.

Q19 Curve Sketching Range and Image Set Determination View

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0$$ and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Justify the existence of the real number $\Delta _ { n }$ for all $n \in \mathbb { N } ^ { * }$.

Q20 Curve Sketching Convergence in Distribution / Central Limit Theorem Application View

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n } : \mathbb { R } \rightarrow \mathbb { R }$ is defined by $$\forall x \in ] - \infty , - \sqrt { n } - \frac { 1 } { \sqrt { n } } [ , \quad B _ { n } ( x ) = 0$$ $$\forall k \in \llbracket 0 , n \rrbracket , \forall x \in \left[ x _ { n , k } - \frac { 1 } { \sqrt { n } } , x _ { n , k } + \frac { 1 } { \sqrt { n } } [ , \right. \quad B _ { n } ( x ) = \frac { \sqrt { n } } { 2 } \binom { n } { k } \frac { 1 } { 2 ^ { n } }$$ $$\forall x \in \left[ \sqrt { n } + \frac { 1 } { \sqrt { n } } , + \infty [ , \right. \quad B _ { n } ( x ) = 0$$ and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$ where $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
Show that, for all $n \in \mathbb { N } ^ { * }$, we have the equality $$\Delta _ { n } = \sup _ { x \geqslant 0 } \left| B _ { n } ( x ) - \varphi ( x ) \right| .$$

Q22 Sequences and series, recurrence and convergence Asymptotic Equivalents and Growth Estimates for Sequences/Series View

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. We introduce the set $I _ { n } = \left\{ k \in \llbracket 0 , n \rrbracket \mid x _ { n , k } \in [ 0 , \ell + 1 ] \right\}$ and assume that $n$ and $k$ vary such that $k \in I _ { n }$.
Show that we have $$k ! ( n - k ) ! = 2 \pi \mathrm { e } ^ { - n } k ^ { k + 1 / 2 } ( n - k ) ^ { n - k + 1 / 2 } \left( 1 + O \left( \frac { 1 } { n } \right) \right)$$ as $n$ tends to infinity. One may use Stirling's formula: $n ! = \left( \frac { n } { \mathrm { e } } \right) ^ { n } \sqrt { 2 \pi n } \left( 1 + O \left( \frac { 1 } { n } \right) \right)$.

Q23 Sequences and series, recurrence and convergence Asymptotic Equivalents and Growth Estimates for Sequences/Series View

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n }$ is defined as in Q19. We assume $k \in I _ { n } = \left\{ k \in \llbracket 0 , n \rrbracket \mid x _ { n , k } \in [ 0 , \ell + 1 ] \right\}$.
Deduce that, as $n$ tends to $+ \infty$, we have $$B _ { n } \left( x _ { n , k } \right) = \frac { 1 } { \sqrt { 2 \pi } } \frac { 1 + O \left( \frac { 1 } { n } \right) } { \left( \frac { 2 k } { n } \right) ^ { k + 1 / 2 } \left( 2 - \frac { 2 k } { n } \right) ^ { n - k + 1 / 2 } }$$

Q24 Sequences and series, recurrence and convergence Limit evaluation using series expansion or exponential asymptotics View

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n }$ is defined as in Q19. We assume $k \in I _ { n } = \left\{ k \in \llbracket 0 , n \rrbracket \mid x _ { n , k } \in [ 0 , \ell + 1 ] \right\}$.
Deduce that $$B _ { n } \left( x _ { n , k } \right) = \frac { 1 } { \sqrt { 2 \pi } } \frac { 1 + O \left( \frac { 1 } { n } \right) } { \left( 1 - \frac { x _ { n , k } ^ { 2 } } { n } \right) ^ { \frac { n + 1 } { 2 } } \left( 1 + \frac { x _ { n , k } } { \sqrt { n } } \right) ^ { \frac { x _ { n , k } } { 2 } \sqrt { n } } \left( 1 - \frac { x _ { n , k } } { \sqrt { n } } \right) ^ { - \frac { x _ { n , k } } { 2 } \sqrt { n } } }$$ then that $$B _ { n } \left( x _ { n , k } \right) = \frac { 1 } { \sqrt { 2 \pi } } \exp \left( - \frac { x _ { n , k } ^ { 2 } } { 2 } \right) \left( 1 + O \left( \frac { 1 } { \sqrt { n } } \right) \right)$$

Q25 Sequences and series, recurrence and convergence Convergence in Distribution / Central Limit Theorem Application View

For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, we set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B _ { n }$ is defined as in Q19, and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$. We fix $\varepsilon > 0$ and $\ell \in \mathbb{R}^+$ such that $\varphi(\ell) \leqslant \frac{\varepsilon}{2}$.
Show that there exists a natural number $n _ { 1 }$ such that, for all integers $n \geqslant n _ { 1 }$, $$\sup _ { x \in [ 0 , \ell ] } \left| B _ { n } ( x ) - \varphi ( x ) \right| \leqslant \frac { \varepsilon } { 2 }$$

Q26 Sequences and series, recurrence and convergence Tail Bound or Concentration Inequality Proof View

The function $B _ { n }$ is defined as in Q19, and $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$.
For all $\ell > 0$, show that there exists a natural number $n _ { 2 }$, such that, for all $n \geqslant n _ { 2 }$, $$B _ { n } ( \ell ) \leqslant 2 \varphi ( \ell )$$

Q27 Sequences and series, recurrence and convergence Sequence of functions convergence View

The function $B _ { n }$ is defined as in Q19, $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$, and $\Delta _ { n } = \sup _ { x \in \mathbb { R } } \left| B _ { n } ( x ) - \varphi ( x ) \right|$.
Conclude that the sequence $\left( \Delta _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ converges to 0.

Q28 Sequences and series, recurrence and convergence Integral Inequalities and Limit of Integral Sequences View

Let $I$ be an interval of $\mathbb { R }$ and $\left( f _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ a sequence of functions piecewise continuous on $I$ that converges uniformly on $I$ to a function $f$ also piecewise continuous on $I$.
If $\left( u _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$ (respectively $\left( v _ { n } \right) _ { n \in \mathbb { N } ^ { * } }$) is a sequence of real numbers belonging to $I$ that converges to $u \in I$ (respectively $v \in I$), show that $$\lim _ { n \rightarrow + \infty } \left( \int _ { u _ { n } } ^ { v _ { n } } f _ { n } ( x ) \mathrm { d } x \right) = \int _ { u } ^ { v } f ( x ) \mathrm { d } x$$

Q29 Discrete Probability Distributions Binomial Distribution Identification and Application View

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$. The function $B_n$ is defined as in Q19.
Show that, for all $j \in \llbracket 0 , n \rrbracket$, $$\mathbb { P } \left( \left\{ T _ { n } = j \right\} \right) = \int _ { x _ { n , j } - 1 / \sqrt { n } } ^ { x _ { n , j } + 1 / \sqrt { n } } B _ { n } ( x ) \mathrm { d } x$$

Q30 Discrete Probability Distributions Limit and Convergence of Probabilistic Quantities View

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Set $Y _ { i } = \frac { X _ { i } + 1 } { 2 }$ and $T _ { n } = \sum _ { i = 1 } ^ { n } Y _ { i }$. For all $n \in \mathbb { N } ^ { * }$ and all $k \in \llbracket 0 , n \rrbracket$, set $x _ { n , k } = - \sqrt { n } + \frac { 2 k } { \sqrt { n } }$.
Consider a pair $( u , v )$ of real numbers such that $u < v$, and denote $$J _ { n } = \left\{ j \in \llbracket 0 , n \rrbracket \left\lvert \, \frac { n + u \sqrt { n } } { 2 } \leqslant j \leqslant \frac { n + v \sqrt { n } } { 2 } \right. \right\}$$
Justify that $$\mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \sum _ { j \in J _ { n } } \mathbb { P } \left( \left\{ T _ { n } = j \right\} \right)$$

Q31 Discrete Random Variables Properties of Probability Measures and Convergence of Measures View

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. The functions $\varphi$ and $\Phi$ are defined by $\varphi ( x ) = \frac { 1 } { \sqrt { 2 \pi } } \mathrm { e } ^ { - x ^ { 2 } / 2 }$ and $\Phi ( x ) = \int _ { - \infty } ^ { x } \varphi ( t ) \mathrm { d } t$.
Deduce that we have $$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \leqslant v \right\} \right) = \int _ { u } ^ { v } \varphi ( x ) \mathrm { d } x$$ then that $$\lim _ { n \rightarrow + \infty } \mathbb { P } \left( \left\{ u \leqslant \frac { S _ { n } } { \sqrt { n } } \right\} \right) = 1 - \Phi ( u )$$

Q32 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Fix $\varepsilon \in ] 0 , 1 [$.
Show that there exists $x _ { 0 } \geqslant 1$ such that we have $$\forall x \geqslant x _ { 0 } , \quad \exists n _ { x } \in \mathbb { N } , \quad \forall n \geqslant n _ { x } , \quad x ^ { 2 } \mathbb { P } \left( \left\{ \left| S _ { n } \right| \geqslant x \sqrt { n } \right\} \right) \leqslant \varepsilon .$$

Q33 Discrete Random Variables Probability Bounds and Inequalities for Discrete Variables View

Let $( \Omega , \mathcal { A } , \mathbb { P } )$ be a probability space and $X$ a discrete random variable such that $\mathbb { P } ( X = - 1 ) = 1 / 2$ and $\mathbb { P } ( X = 1 ) = 1 / 2$. Consider a sequence $\left( X _ { i } \right) _ { i \in \mathbb { N } ^ { * } }$ of mutually independent discrete random variables with the same distribution as $X$. Set $S _ { 0 } = 0$ and $S _ { n } = \sum _ { i = 1 } ^ { n } X _ { i }$. Fix $\varepsilon \in ] 0 , 1 [$.
For $x _ { 0 }$ and $x$ as in the previous question, we fix $N \geqslant \frac { n _ { x } } { \varepsilon }$ and we choose $n \geqslant N$. Show that then $$x ^ { 2 } \mathbb { P } \left( \left\{ \max _ { 1 \leqslant p \leqslant n } \left| S _ { p } \right| \geqslant 3 x \sqrt { n } \right\} \right) \leqslant 3 \varepsilon$$

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