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It seems that in the paper, $S$ is a Polish space, which is homeomorphic to a dense subset of a compact metric space denoted by $\overline S$. Without loss of generality, we shall work with this subsp …

The fact that $b_n\to \infty$ is quite easy to check: if not, there is a $M$ and a subsequence $(b_{n'})$ which remains below $M$, hence $\frac 1{b_{n'}}\max_{1\leqslant j\leqslant n'}|X_j|\geqslant \ …

For $\omega\in\Omega$, define $S(\omega):=\(Y_n(\omega),n\in\mathbb N\)$. We have to show that $P(\omega\mid S(\omega)\mbox{ is relatively compact})=1$. Considering $\varepsilon=1/j$, we can see tha …

A good reference for mixing condition is Bradley's book or paper . The notion of $\lambda$-mixing is introduced, and defined by $$\lambda(\mathcal A,\mathcal B):=\sup_{A\in\mathcal A,B\in\mathcal B} …

As Christian Remgling's example $\mu_n:=\delta_{e_n}$ shows, the convergence of the characteristic function of $\mu_n$ to some characteristic function does not even guarantee tightness. It's worth poi …

As a immediate corollary of the real-valued case, a necessary condition is that for all $f\in H$, $\langle X,f\rangle$ should be centered and have a finite moment of order two. For $n\geqslant 3$, den …

Convergence holds in any $L^p$ for $p\geqslant 1$. By Theorem III.4.3 page 106 in Garsia, Adriano M. Martingale inequalities: Seminar notes on recent progress. Math. Lecture Note Ser. W. A. Benjamin, …

If you are interested in non-asymptotic bounds, the following references can be useful (of course, the list is far from being complete). The martingale case is addressed in Nagaev, S. V. On probabi …

Let us state Corollary 2.1 of these notes. Let $p>1$, $X\in\mathbb L^p$ and let $\left(\mathcal F_n\right)_{n\geqslant 1}$ be a filtration. Denote by $\mathcal F$ the $\sigma$-algebra generated by …

In this context, we have $L^2$ convergence if and only if $X_n\to X$ in probability. Indeed, $L^2$ convergence always implies convergence in probability. Conversely, if $X_n\to X$ in probability and …

Since $1/X_j$ has a finite moment generating function, the random variable $\frac 1{X_1+\dots+X_{n-1}}$ has moments of any order. Using independence, we thus have that $Y_n\in\mathbb L^p$ if and only …

Notice that for a non negative random variable $Y$, we have $$ \mathbb E(Y)=\int_0^{\infty}\mu(Y\geqslant s)\mathrm ds.$$ Fix $t\geqslant 0$. We have for $s\leqslant t$ that $\{X\mathbf 1_{X\geqslant …

The functional $F$ is continuous at $\nu:=0$, the null measure. The set $F^{-1}(-1,1)$ is open. Therefore, the exists a positive $r$, an integer $J$ and $g_1,\dots,g_J\in\mathcal C_b$ such that $$V:= …

Let $(\Omega,\mathcal B,\mu)$ be a probability space and $\mathcal A$ a sub-sigma-algebra of $\mathcal B$. The following statements are equivalent: $\mathcal A$ admits a countable set of generators …

There is a general result, which is Theorem 4.2 of Billingsley's book Convergence of probability measures, 1968: Theorem. For each integers $m$ and $n$, let $X_n$, $X_n^{(m)}$ and $X^{(m)}$ be real-v …