Random discretization of stationary continuous time processes

Abstract

This paper investigates second order properties of a stationary continuous time process after random sampling. While a short memory process always gives rise to a short memory one, we prove that long-memory can disappear when the sampling law has very heavy tails. Despite the fact that the normality of the process is not maintained by random sampling, the normalized partial sum process converges to the fractional Brownian motion, at least when the long memory parameter is preserved.

Appendix

To prove Lemma 4.1, we need the following intermediate result:

Lemma 5.1

If ${\mathbb {E}}[T_1]<\infty$ and $\mathbf {X}$ has a regularly varying covariance function

$$\begin{aligned} \sigma _X(t) = L(t) t^{-1+2d} \end{aligned}$$

with $0<d<1/2$ and L slowly varying at infinity and ultimately non-increasing. Then,

$$\begin{aligned} {\mathrm {Var}}( \sigma _X(T_h))=\circ (L(h)^2h^{-2+4d}), \qquad \text {as } h\rightarrow \infty . \end{aligned}$$

(5.1)

Proof

By Theorem 3.1, we have ${\mathbb {E}}[\sigma _X(T_h)] \underset{h\rightarrow \infty }{\sim }L(h)(h{\mathbb {E}}[T_1])^{-1+2d}$. To get the result, it is enough to prove that

$$\begin{aligned} {\mathbb {E}}[\sigma _X(T_h)^2] \underset{h\rightarrow \infty }{\sim } L(h)^2(h{\mathbb {E}}[T_1])^{-2+4d}. \end{aligned}$$

To prove the asymptotic behavior of ${\mathbb {E}}[\sigma _X(T_h)^2]$, we will follow a similar proof as theorem 3.1:

• Let $0<c<{\mathbb {E}}[T_1]$, and $h\in {\mathbb {N}}$ such that $ch\ge 1$,

  $$\begin{aligned} {\mathbb {E}}[\sigma _X(T_h)^2]\ge & {} {\mathbb {E}}\left[ \sigma _X(T_h)^2\,\mathbb {I}_{T_h>ch}\right] \ge {\mathbb {E}}\left[ L(T_h)^2T_h^{-2+4d}\,\mathbb {I}_{T_h> ch}\right] \\\ge & {} \inf _{t>ch}\{L(t)^2t^{4d}\}{\mathbb {E}}\left[ \frac{\,\mathbb {I}_{T_h> ch}}{T_h^2}\right] . \end{aligned}$$

  Thanks to Jensen and Hölder inequalities,

  $$\begin{aligned} {\mathbb {E}}\left[ \frac{\,\mathbb {I}_{T_h> ch}}{T_h^2}\right] \ge {\mathbb {E}}\left[ \frac{\,\mathbb {I}_{T_h> ch}}{T_h}\right] ^2 \text { and } P(T_h> ch)^2 \le {\mathbb {E}}[T_h] {\mathbb {E}}\left[ \frac{\,\mathbb {I}_{T_h> ch}}{T_h}\right] , \end{aligned}$$

  that is

  $$\begin{aligned} {\mathbb {E}}\left[ \frac{\,\mathbb {I}_{T_h> ch}}{T_h^2}\right] \ge \frac{P(T_h> ch)^4}{{\mathbb {E}}[T_h]^2}. \end{aligned}$$

  Summarizing,

  $$\begin{aligned} \frac{{\mathbb {E}}[\sigma _X(T_h)^2] }{ L(h)^2(h{\mathbb {E}}[T_1])^{-2+4d}}\ge \frac{\inf _{t>ch}\{L(t)^2t^{4d}\}}{L(h)^2h^{4d}{\mathbb {E}}[T_1]^{4d}}P(T_h> ch)^4. \end{aligned}$$

  (5.2)

  Then, for $c<{\mathbb {E}}[T_1]$, we have $P(T_{h}>ch)\rightarrow 1$ and $\inf _{t>ch}\{L(t)^2t^{4d}\}\sim L(ch)^2(ch)^{4d}$. Finally, for all $c<{\mathbb {E}}[T_1]$,

  $$\begin{aligned} \liminf _{h\rightarrow \infty }\frac{{\mathbb {E}}[\sigma _X(T_h)^2] }{ L(h)^2(h{\mathbb {E}}[T_1])^{-2+4d}}\ge \left( \frac{c}{{\mathbb {E}}[T_1]}\right) ^{4d}. \end{aligned}$$

  Taking the limit as $c\rightarrow {\mathbb {E}}[T_1]$, we get

  $$\begin{aligned} \liminf _{h\rightarrow \infty }\frac{{\mathbb {E}}[\sigma _X(T_h)^2] }{ L(h)^2(h{\mathbb {E}}[T_1])^{-2+4d}}\ge 1. \end{aligned}$$

• Let $\frac{1}{2}<s<\tau <1$, $t_0$ such that L(.) is non-increasing and positive on $[t_0,\infty )$ and h such that $\mu _{h,s}-\mu _{h,s}^\tau \ge t_0$, with the same notation as Theorem 3.1,

  $$\begin{aligned} {\mathbb {E}}[\sigma _X(T_h)^2]= & {} {\mathbb {E}}\left[ L(T_h)^2T_h^{-2+4d} \,\mathbb {I}_{T_{h,s}\ge \mu _{h,s}-\mu _{h,s}^\tau }\right] +{\mathbb {E}}\left[ \sigma (T_h)^2\,\mathbb {I}_{T_{h,s}<\mu _{h,s}-\mu _{h,s}^\tau }\right] \\\le & {} L(\mu _{h,s}-\mu _{h,s}^\tau )^2\left( \mu _{h,s}-\mu _{h,s} ^\tau \right) ^{-2+4d}+\sigma _X(0)^2 P \left( T_{h,s}<\mu _{h,s}-\mu _{h,s}^\tau \right) . \end{aligned}$$

  We get

  $$\begin{aligned} \frac{{\mathbb {E}}[\sigma _X(T_h)^2] }{ L(h)^2(h{\mathbb {E}}[T_1])^{-2+4d}}&\le \left( \frac{L(\mu _{h,s}-\mu _{h,s}^\tau )}{L(h)}\right) ^{2} \left( \frac{\mu _{h,s}-\mu _{h,s}^\tau }{h{\mathbb {E}}[T_1]}\right) ^{-2+4d}\\ {}&\quad +\sigma _X(0)^2 \frac{P \left( T_{h,s}<\mu _{h,s}-\mu _{h,s}^\tau \right) }{L(h)^2(h{\mathbb {E}}[T_1])^{-2+4d}}, \end{aligned}$$

  and finally

  $$\begin{aligned} \limsup _{h\rightarrow \infty }\frac{{\mathbb {E}}[\sigma _X(T_h)^2] }{ L(h)^2(h{\mathbb {E}}[T_1])^{-2+4d}}\le 1. \end{aligned}$$

$\square$

Proof of Lemma 4.1:

Denote

$$\begin{aligned} W_n&=L(n)^{-1} n^{-1-2d}\sum _{i=1}^n\sum _{j=1}^n\sigma _X(T_j-T_i) =L(n)^{-1}n^{-1-2d}{\mathrm {Var}}(X_{T_1}\\&\quad +\dots +X_{T_n}\vert T_1~,\dots ,~T_n). \end{aligned}$$

We want to prove that $W_n$ converges in probability to $\gamma _d$. To do this, we will show that ${\mathbb {E}}[W_n]\xrightarrow [n\rightarrow \infty ]{} \gamma _d$ and ${\mathrm {Var}}(W_n)\xrightarrow [n\rightarrow \infty ]{} 0$.

• As $\mathbf {X}$ is a centered process $E[W_n]=L(n)^{-1}n^{-1-2d}{\mathrm {Var}}(Y_1+\dots +Y_n)$. By Theorem 3.1, we have

  $$\begin{aligned} \sigma _Y(h) \sim L(h)(h{\mathbb {E}}[T_1])^{-1+2d} \qquad h\rightarrow \infty , \end{aligned}$$

  then

  $$\begin{aligned} L(n)^{-1}n^{-1-2d}{\mathrm {Var}}(Y_1+\dots +Y_n)\xrightarrow [n\rightarrow \infty ]{} \gamma _d, \end{aligned}$$

  (5.3)

  [see Giraitis et al. (2012) Proposition 3.3.1, page 43]. Therefore we obtain

  $$\begin{aligned} E[W_n]\xrightarrow [n\rightarrow \infty ]{} \gamma _d . \end{aligned}$$

• Furthermore,

  $$\begin{aligned} {\mathrm {Var}}(W_n)&=L(n)^{-2}n^{-2-4d}{\mathrm {Var}}\left( \sum _{i=1}^n\sum _{j=1}^n \sigma _X(T_j-T_i)\right) \\&\le L(n)^{-2}n^{-2-4d}\left( \sum _{i=1}^n\sum _{j=1}^n \sqrt{{\mathrm {Var}}(\sigma _X(T_j-T_i))}\right) ^2\\&=\left( 2n^{-1-2d}L(n)^{-1} \sum _{h=1}^n(n-h)\sqrt{{\mathrm {Var}}(\sigma _X(T_h))}\right) ^2. \end{aligned}$$

  Then, by Lemma 5.1, $\sqrt{{\mathrm {Var}}(\sigma _X(T_h))}=\circ (L(h)h^{-1+2d})$ and $2 \sum _{h=1}^n (n-h)L(h)h^{-1+2d}\sim \frac{L(n)n^{1+2d}}{d(1+2d)}$. We get

  $$\begin{aligned} 2 \sum _{h=1}^n(n-h)\sqrt{{\mathrm {Var}}(\sigma _X(T_h))}=\circ (L(n)n^{1+2d}). \end{aligned}$$

  Finally, ${\mathrm {Var}}(W_n)=\circ (1)$ which means that ${\mathrm {Var}}(W_n)\xrightarrow [n\rightarrow \infty ]{} 0$. We obtain

  $$\begin{aligned} W_{n}\xrightarrow [n\rightarrow \infty ]{L^2,~ p} \gamma _d. \end{aligned}$$