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Let $X_1,X_2,\dots$ be independent and identically distributed random variables with mean zero and variance one, and let $S_k={\sum }_{j=1}^k{X}_j$. Define the linearly interpolated processThe Donsker invariance principle, also called the functional central limit theorem, states that $W_n$ converges weakly in the space $C[0,1]$ with the uniform norm to standard Brownian motion.

For independent and identically distributed random variables with continuous distance distribution function $p_x$, the probability integral transform makes the probability content of the order-$k$ nearest ball a uniform order statistic, with the Beta distribution $\mathrm{Beta}(k,n+1-k)$.

For independent and identically distributed random variables $Z_1,\dots ,Z_n$ and independent Rademacher signs ${\epsilon }_1,\dots ,{\epsilon }_n$, the Rademacher complexity of a function class $\mathcal{F}$ isIt measures how strongly the class can correlate with independent random signs on a random sample.

For an i.i.d. sample and an integrable function class $\mathcal{F}$,Introduce an independent ghost sample, replace each expectation by its ghost-sample average using Jensen inequality, and then multiply each paired difference by an independent Rademacher sign. Splitting the resulting supremum into its two sample contributions gives the factor two.

For independent and identically distributed random variables with distribution function $F$, the sample maximum $M_n={\max }_{1\le i\le n}{X}_i$ has distribution function $F(x{)}^n$.

A collection of observed random variables used to infer a statistical model or its parameters. A common assumption is that the observations are independent and identically distributed random variables; dependence must otherwise be included in the model. Its randomness induces the sampling distribution of a statistic.

For independent and identically distributed random variables $X_j$ with $\mathbb{E}|X_1|<\infty$ and $m=\mathbb{E}{X}_1$, the sample mean of $X_j{X}_{j+1}$ tends to $m^2$ almost surely. Split into odd and even indexed products; each subsequence consists of independent and identically distributed random variables that are integrable random variables. Apply the strong law of large numbers separately.

If $(X_n)$ are independent and identically distributed random variables with $\mathbb{E}|X_1|<\infty$, thenalmost surely.

A sequence of independent and identically distributed random variables with zero mean and finite variance is strong white noise. It is weak white noise, but need not have a normal distribution.

If independent and identically distributed random variables have characteristic function $\phi (t)=1+iat+o(|t|)$ at zero, then their sample mean has convergence in distribution to $a$: its characteristic function is $\phi (t/n{)}^n\to e^{iat}$. This is also convergence in probability by convergence in distribution to a constant implies convergence in probability.

For independent and identically distributed random variables with finite expected value, the sample mean converges in probability to that expected value.