Articles of probability limit theorems

Kolmogorov's sufficient and necessary condition for SLLN – What about pairwise uncorrelated RV?

Kolmogorov proved, that, as one considers independent (not necessary equally distributed) Random Variables: $\{X_n\}_{n\ge0}\subseteq \mathcal L^2$ With $\mathrm{Var} (X_n)=\sigma^2_n$ and without loss of generality $E[X_n]=0$. If $\sum_{n=0}^\infty \frac{\sigma^2_n}{n^2} \lt \infty$ then SLLN holds, that is: $$\frac1n\sum_{k=0}^n X_k \rightarrow 0 \,\,\,\,\,\,\,\,\,\,\,\,\,\text{a.s.}$$ Something different: Also it is known, that if $\sup_{n\ge0}\sigma^2_n =: v \lt \infty$ pairwise uncorrelated […]

Evolution of a discrete distribution of probability

I am designing a virtual card game and I defined an evolution of probabilities, but I don’t have the knowledge on this matter to find out how they will evolve. I hope you help me here, with bibliography and teaching me some things. This is the system: I have a discrete distribution of probability of […]

Integral with quadratic square root inside trigonometric functions

Is there anyway to solve $\displaystyle \int t \frac{\sin \left(\frac{t}{2} \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}\right) }{ \sqrt{ a \left(t+ \frac{b}{2a}\right)^2-\frac{b^2-4ac}{4a}}} \operatorname{d}t \tag8$ either by analytical method or from geometrical method with out using numerical methods? means looking for a closed form with out infinite series expansion in the result NB: Main issue is the lack of […]

Weak/strong law of large numbers for dependent variables with bounded covariance

Let $(X_i)_{i\in\mathbb{N}}$ be a sequence of $L^2$ random variables with expected value $m$ for all $n$. Let $S_n=\sum_{i=1}^n X_i$ and $|\mathrm{Cov}(X_i,X_j)|\leq\epsilon_{|i-j|}$ for finite, non-negative constants $\epsilon_k$. Show that: (1) If $\lim_{n\to\infty} \epsilon_n=0$ then $S_n/n\to m$ in $L^2$ and probability (2) If $\sum_{k=1}^\infty \epsilon_k<\infty$, then $\mathrm{Var}(S_n/n)$ is of order $O(1/n)$ and $S_n/n\to m$ almost surely (1) […]

Rate of convergence in the central limit theorem (Lindeberg–Lévy)

There are similar posts to this one on stackexchange but none of those seem to actually answer my questions. So consider the CLT in the most common form. Let $(X_n)_{n \in \mathbb{N}}$ be a sequence of i.i.d. random variables with $X_1 \in L^2(P)$ and $\mathbb{E}[X_i]= \mu$ and $\mathbb{V}ar[X_i] = \sigma^2>0$. Denote with $\widehat{X}:= \frac{(X_1+\dots+X_n)}{n}$. Then […]

How often was the most frequent coupon chosen?

In the coupon collector‘s problem, let $T_n$ denote the time of completion for a collection of $n$ coupons. At time $T_n$, each coupon $k$ has been collected $C_k^{n}\geqslant 1$ times. Consider how often the most frequently chosen coupon, was chosen, that is, the random variable $$C^*_n=\max_{1\leqslant k\leqslant n}C_k^{n}. $$ Can one compute $E(C^*_n)$? What is […]

Does really convergence in distribution or in law implies convergence in PMF or PDF?

Ref :Introduction to Mathematical Statistics-Prentice Hall (1994) by Robert V. Hogg, Allen Craig. Now , in the above problem it has been shown that a sequence converges to a random variable X in distribution but the sequence of PMF doesn’t converge to the PMF of X. but we know that “a sequence {Xn} with PDF/PMF […]

Rate of convergence of mean in a central limit theorem setting

I recently asked a question here that was the following: If $Z_1,Z_2,Z_3,\ldots$ are i.i.d. with $P(Z_i=-1) = P(Z_i=+1) = \frac 12,$ then we have by the Central Limit Theorem that $\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\stackrel{d}{\to} \mathcal{N}(0,1),$ so that for any continuous bounded function $f,$ we have $\mathbb{E}f\left(\frac{\sum_{i=1}^n Z_i}{\sqrt{n}}\right)\to\mathbb{E}f(W)$ where $W\sim\mathcal{N}(0,1).$ Now, $|\cdot|$ is not a bounded function, so […]

If $(X_n)$ is i.i.d. and $ \frac1n\sum\limits_{k=1}^{n} {X_k}\to Y$ almost surely then $X_1$ is integrable (converse of SLLN)

Let $(\Omega,\mathcal F,P)$ be a finite measure space. Let $X_n:\Omega \rightarrow \mathbb R$ be a sequence of iid r.v’s I need to prove that if: $ n^{-1}\sum _{k=1}^{n} {X_k} $ converges almost surely to $Y$ then all $X_k$ have expectation. If I understand correctly then $X_k$ has expectations means $X_k$ is in $\mathcal L^1(\Omega)$. And […]

Assume $A,B,C,D$ are pairwise independent events. Decide if $A\cap B$ and $B\cap D$ are independent

Assume A,B,C,D, are pairwise independent events. Decide if $(A \cap B)$ and $(B \cap D)$ are independent events? Then repeat this assuming the four events are mutually independent. Well, what I’m thinking is $P(A \cap B) = P(A) \cdot P(B) \\P(A \cap C) = P(A) \cdot P(C) \\P(A \cap D) = P(A) \cdot P(D) \\P(B […]